Alexey Vasilyevich Pogorelov, the mathematician of an incredible power

Alexey V asily evic h P ogorelo v, the mat hematician of an incredible p o w er A.A.Borisenk o Khark ov Karazin Nat ional Univ ersity , e-mail: b orisenk@univ er.khark ov.ua Ma y 31, 2018 Abstract Life and the mathematical legacy of the great mathematician A.V . P ogorelov. 1 Mathematical Sub ject Classification 2000: 01A70, 53C45, 35J60 Keywor ds: conv ex su rface, elliptic differential equation In tro duction. By the b eginning of the 2 0th century , the lo ca l differe n tial geometr y of s urfaces was very well- developed, mostly by the means of the lo cal ana lysis. In contrast, there was an appar e n t la ck of metho ds and results in the g lobal geometry , “ g eometry in the large” , in which b oth geo metr y a nd analysis were equally helpless. A typical example of such a question is the class ic a l problem of the rigidity of a clo sed convex surface (ov alo id). The co nt ribution to this pr o blem was ma de b y br il- liant mathematicians , s uch as Liebmann, Minko w s ki, Hilb ert, W eyl, Bla schk e. A ma jor progres s was achiev ed by Cohn-V o ssen at the beg inning of the 192 0-th. He prov ed that isomeric C 3 ov aloids of po sitive Ga uss curv ature are congruent. Meanwhile, b y a classic a l result of Cauch y , isomeric closed conv ex p olyhedr ons are congruent. It seemed that these two res ults are the particular ca s es of a general theore m of co ngruency of any tw o (genera lly sp eaking, irreg ular) isomeric ov aloids. No ap- proaches to the pr oblem in such a ge nerality were visible. Similarly , it was not clear how to pr ov e an isometric defor mability of an ov aloid with a part remov ed, and to estimate the deformability of an incomplete ov aloid. Under an ass umption of a sufficient regula rity , these pro ble ms can b e formulated in the langua ge of nonlinear PDE’s, but t he theory of such equations was f ar from b eing well developed at that time. In many cases, ana lytic to ols were sp ecifically developed to study a particular ge ometric problem. This is illustra ted b y the pro blem of the existence of a closed con vex sur face with a prescrib ed ana- lytic metric of positive Gaus s curv a ture defined on a top olo gical sphere. The genera l a pproach given by W eyl in 1916 was succ e ssfully co mpleted in the 19 30-th by G.Levi who developed a very delicate techn iques fro m the analy tic theo ry of t he Monge- Amp ` ere equations. How e ver, in the G.Levi’s pap ers, analysis was developed sepa rately from geometr y and w as applied to geometr y as a re a dy-made to o l. Many o ther pap ers just gav e ad ho c metho ds for separate problems. In fact, at that time, the fun- damental problems of deformation of surfaces a nd many other pr oblems of global geo metry remained unapproachable. Another area of active resear ch at the b eginning o f the 20th ce n tury was the theor y of co nv ex b o dies in the E uclidean space, including v arious geometric pro p e r ties, the mixed volumes and inequalities. 1 This pa per is the Eng lish translation of [25] 1 The cornerstone bo ok “Theory o f c o nv ex bo dies” by T.Bonnezen and T.F enchel’s published in 1934 in German contained a complete up-to-date ac count o f the rese a rch in that area and an extensive bibliography . This bo o k has not lost its v alue till now adays. In 2001, it w as transla ted into Russian by V.A.Zalgaller. How ever, the s tudy o f co n vex surface was b e yond the scop e of this bo o k, and mo s t probably , outside of the ar ea o f interest o f resear chers at that time. 1 In trinsic geometry of con ve x surfaces. Early pap ers of A.D.Aleksandrov were focus ed on the same questions. Starting with the classical results of Minko wsk i, Alexander Danilovich Aleksandrov e s tablished ne w inequalities for the mixed volumes o f con vex bo dies. Curiously enough, forty y ear s later, the algebraic a nalogues of his inequali- ties o btained a s a b ypro duct of his study , were success fully applied to the well-known V an-der-W aerden problem o n the estimation of the p erma nent , p os e d in 1926. Aleks a ndrov’s inequalities for the mixed volumes also hav e in teresting g eneralizatio ns and applications in algebraic geometry , in the theory of nonlinear elliptic equations and even in sto chastic pro cesses. Sim ultaneously , A.D.Aleksandrov applied the too ls fr om the mea s ure theory and functional ana lysis to the theory of conv ex b o dies. He introduced a functional spa c e ge ner ated by the supp o rt functions and sp ecial measures on them, the “surfac e functions” a nd the re la ted “ curv ature functions”. He prov ed that a co nv ex bo dy is determined uniquely , up to translation, by the cur v ature function. This theor em includes, a s the limiting cases, the theorems of Christoffel and of Minkowski. In the course of pro o f, Alexander Danilovich int ro duced the concept o f the gener alized differential equa tion in measures and of the corres po nding gener alized s o lution. In 1 941, A.D.Aleksandrov beg a n the study of the intrinsic geometry of conv ex surfaces. He widened the cla ss of regular conv ex s urfaces t o the class of arbitrary conv ex s urfaces (defined as domains on the bo undary of a co n vex bo dy). The problems within that wider c la ss required new techniques, beyond the Gaussia n geometry of r egular surface s . It was necessary to understand the in trinsic geometry o f an arbitrar y con vex surface (the prop er ties dep ending on the mea surements on the s ur face itself ) a nd to develop the to o ls for studying these prop erties , and then to find the connection betw een the intrinsic and the ex trinsic geometry of an arbitr a ry conv ex surface. A.D.Aleksandrov constructed the in trinsic geometry of c o nv ex (and o f arbitr ary) surfaces starting from the general concept of the metric space. Let R b e a metric space, a set such that for each pair of element s X, Y ∈ R there defined a n umber ρ ( X , Y ), the distanc e , s atisfying the following axioms: 1. ρ ( X , Y ) > 0 and ρ ( X , Y ) = 0 if and o nly if X = Y . 2. ρ ( X , Y ) = ρ ( Y , X ). 3. ρ ( X , Y ) + ρ ( Y , Z ) > ρ ( X , Z ) (the tr iangle inequality). F or example, the Euclidean space with the usual distance b etw een the p o ints is a metric space. A curve γ in a metric spa ce R is a co nt in uous image of a segment [ a, b ] consider ed together with the contin uous mapping F : [ a, b ] → R . Just as in the for the Euclidea n space, one can introduce the concept of the length of a curve in a metric spa ce by defining l γ = sup Σ ρ ( F ( t k − 1 ) , F ( t k )) , a = t 0 6 t 1 6 t 2 · · · 6 t n = b, where ρ ( F ( t k − 1 ) , F ( t k )) is the distance b etw een F ( t k − 1 ) and F ( t k ) in R , a nd the supre mum is taken ov er all finite partitions of the segment [ a, b ] by the t k ’s. The length so defined is additive: if a curve γ is comp osed by the curves γ 1 and γ 2 , then l ( γ ) = l ( γ 1 ) + l ( γ 2 ). Similar to the Euclidea n space, in a metric space, the set of curves of b o unded length in a compact domain is compact, that is, a ny infinite sequence of such curves contains a conv er ging subsequence. Moreover, the length o f the limiting cur ve is not gre ater than the low er limit o f the lengths of cur ves o f the s ubsequence. Suppo se that any t wo p oints X, Y in R can be c onnected b y a rectifiable curv e. Then the intrinsic distance ρ ∗ ( X, Y ) 2 betw een X , Y in R can b e defined as the infimu m of the leng ths of all the re c tifia ble curves connecting X and Y . It is easy to see that ρ ∗ satisfies the axioms 1,2,3. The metr ic ρ ∗ is ca lled the intrinsic metric on R . If ρ ( X , Y ) = ρ ∗ ( X, Y ) for any X and Y , then R is called a sp ac e with an intrinsic met ric (the length space). F or a metric space R to b e isometric to a surfac e in the E uclidean space, its metric m ust necessar ily b e intrinsic. Let R be a manifold with an intrinsic metric. A curve γ in R is ca lled a shortest p ath if its length is equal to the dis tance b etw een the endp oints, hence b eing no t gre a ter than the leng th of any o ther curve joining the same p oints. Each segment of a s ho rtest path is also a shor test path. The limiting curve for a conv er ging sequence of sho rtest paths is again a shor test pa th. In general, not every tw o p oints o n a manifo ld ca n be c o nnected by a shortest path, but every po int of a manifold has a neighbor ho o d such tha t an y tw o points from it can b e joined by a shortest path. If a ma nifo ld R is metric complete, that is, if any closed b ounded subse t of it is compact, then any tw o p oints of R can b e connected by a shortes t path. In R , one can define a triangle, a po lygon, a polygo nal line, etc. in the usual w ay . The definition of the a ngle b etw e e n shortest paths is fundament al for the theory . The idea is to compare a triangle in R to a triangle with the s ame sides on the plane. Let O ∈ R and let O A and OB b e the shortest paths. Cho o s e arbitrary points X ∈ O A and Y ∈ O B . Let O ′ X ′ Y ′ be a triangle on the plane with the same sides as O X Y . Let x = ρ ( O, X ) , y = ρ ( O , Y ) and let α ( x, y ) b e the angle at O ′ of O ′ X ′ Y ′ . The upp er angle b etwe en the shortest p aths OX and O Y is defined as the upper limit of α ( x, y ) when x, y → 0. Using the notion of the angle, one can define the exces s of a tr iangle as α + β + γ − π , wher e α, β , γ are the upper angles b etw een the sides. At this p oint, o ne can a lready define t wo-dimensional s pa ces of non-neg ative curv ature. Na mely , R is called a sp ac e of n on-ne gative curvatur e if any p oint o f R has a neighborho o d such tha t excess of each triangle lying in it is non-nega tive. E ach co n vex sur face is a space of non-negative curv ature. The space R ha s non-negative cur v a ture if and only if the function α ( x, y ) is non-increa sing: if x 1 6 x 2 and y 1 6 y 2 , then α ( x 1 , y 1 ) > α ( x 2 , y 2 ). F r om the mo no tonicity of α it follo ws that the upper limit can be replaced b y the usua l limit, which g ives the angle b et we en the s hortest p aths OX and OY . The next step is to define the curv ature. In the sense of A.D.Aleksandrov, the curv ature is an additive function o f sets. The ext r insic curvatur e of a set M o n a con vex surface is the area of the spherical imag e of M . The intrins ic curvatur e , as an o b ject of the intrinsic geometry o f a co nv ex surface, is firs t defined for the three “basic sets”: 1 ) the curv ature of an op en tria ng le is its excess; 2) the c ur v ature of a n op en shortest path is zero ; 3) the c ur v ature of a p oint equals 2 π − θ , where θ is the full a ng le a round the p o int. Then the curv ature is (uniquely) defined by additivity for all Borel sets. A.D.Aleksandrov prov ed that the cu rvatur e of any Bor el set on a c onvex su rfac e e quals the ar e a of its spheric al image , the Gauss’s Theor ema Egreg ium for a rbitrar y convex surfaces. He established the fundamental connection b etw een the int rinsic geo metry and the extrins ic pro p erties of a surfa c e , which implies a series o f imp or tant re s ults. In par ticular, the r elative curvatur e of a domain on a conv e x sur face (the ratio of the area of the spherical ima g e to the area of the doma in) is an isometric inv aria nt . So the Aleksandrov’s cla ss o f conv ex surfaces of b ounded rela tive curv ature admits an in trinsic-geometric definition. A.D.Aleksandrov pro v ed that a c omplete c onvex surfac e of b ounde d r elative curvatur e whose met r ic is not everywher e Euclide an is smo oth, and an inc omplete surfac e c an b e n on-smo oth only along str aight e dges with the endp oints on the b oundary . This was the first result on the dep endence of the reg ula rity of a conv ex surface on the regular ity of its intrinsic metric. In contrast to the Gauss theory of surfa c es, where the analytic metho ds dominate, in the Aleks a n- drov’s theor y , the central role is play ed by the geometr ic metho ds. The main to o l is the approximation of the surface b y so-called manifolds with polyhedr al metric (the co rresp onding method in extrinsic geometry is the approximation o f a g e neral convex surface by conv ex p olyhedr a ). A s imple and nat- ural idea o f po lyhedral a pproximation enabled one to prov e the results first fo r polyhedr a, and then, passing to the limit, in the general case. These a re the main to o ls of the curv ature theo ry of gener al conv ex surfaces which was briefly sketched a b ove. Let P b e a closed c onv ex po lyhedron. Supp ose that it is cut by p olygo nal lines into n parts, each 3 of which is then unfolded fla t to a planar p oly gon G i , 1 ≤ i ≤ n . The sys tem o f p olyg ons G 1 , . . . , G n , with a given identification of their vertices and side s , is called t he net of the polyhedro n P . Clear ly , each net sa tisfies the following conditions: 1 ) the complex F i G i / identification is homeomorphic to the sphere; 2) the identified edges a re e qual; 3) the s um of the angles a t the identified v ertices is at most 2 π . Now s uppo se that we are g iven planar p oly gons G i and an identification of their sides and vertices, which satisfies the conditions 1 ),2) a nd 3 ). Then, by the Aleksandrov’s “p olyhedro n gluing theorem”, it is p ossible to glue a closed conv ex p oly hedr on (p ossibly b ending the p olyg ons G 1 , . . . , G n along straight lines). This theorem g ives the answer to the W eyl problem fo r p olyhedra l metrics. Its proof uses r ather g eneral metho ds based on the top olog ical theorem on the inv aria nce of domain. Using the same metho d A.D.Aleksandrov solved the Minkowski problem of the existence o f a conv ex po lyhedron with the prescr ib ed areas a nd directio ns o f the fac es, the pro blem of the ex istence o f a poly hedron with the prescribe d curv atures and ma ny others. This method proved to b e a v er y effective general to ol to a wide class of problems in the are a . Using the “p olyhedro n gluing theorem” A.D.Aleksandrov gav e a surprisingly simple solution to the W eyl problem in the most g eneral settings: a two-dimensional metric sp ac e of p ositive curvatur e home omorphi c to the spher e is isomeri c to a close d c onvex surfac e . This illustrates the pow er of the direct geometric metho ds. Alexander Danilovich used the p olyhedr on gluing theorem as the first step in the deep development of the whole theory of deformatio ns of conv ex surfaces. He in tr o duced the “gluing metho d” based o n his “gener al gluing up theorem”. Below we brie fly explain the main ideas of his method. Suppo se w e are given a finite n um b er of domains in tw o-dimensio nal spaces of p o sitive cur v ature. Imagine that these domains are cut out from their spaces and some parts of their b oundaries are ident ified (“g lued together ”) in such a way that the r esulting spa ce is a tw o-dimensio nal ma nifo ld M . If the iden tified parts of the b oundar ies ha ve the sa me lengths, then we can define in a na tural wa y an intrinsic metric o n M . A.D.Aleksandrov gav e necessar y and sufficient co nditions for M to be a spac e of p ositive curv ature. These conditions ar e simple, natural and effective. In par ticula r, if M is home omorphic to the spher e, then it c an b e r e alize d as a close d c onvex surfac e . This is an extremely general theore m on the existence of a conv ex s ur face glued from the pieces o f abstra ctly given manifolds or from the pieces o f c o nv ex surfaces. Applying the gluing metho d A.D.A leksandrov prov ed the fo llowing lo cal theo rem: every p oint of a two-dimensional sp ac e R has a neighb orho o d isomeric to a c onvex surfac e if and only if R is a sp ac e of p ositive curvatu r e . This s olves the main problem of the in trinsic geo metry o f conv e x surfac es: the metr ics of conv ex surfaces a re characterized in a purely intrinsic way . Using the gluing metho d one ca n pr ov e that an ov aloid with a piece r emov ed admits deformations by gluing in different piece s to close the ov a loid up. One ca n pr ov e, for example, that a half of an e llips oid can b e defor med to a closed conv e x surfac e, a nd many other similar res ults. In es sence, the gluing metho d c ombined with the theorems on re alizability of a metr ic of p os itive curv ature by a co nv ex surfa ce allowed to solve, in the most genera l fo r m, all the main pro blems of the theory of deformation of conv ex surface s [22, 23, 32]. It will not b e an exa ggera tion to say that A.D.Aleksandrov ha s created a whole new Universe, the geometry of g eneral conv ex surfaces (a p o pular joke at the Conference on his 7 5 th birthday in Nov osibir sk was “A.D.Aleksandrov is not the Go d, but is Go dlike”, whic h als o was a referenc e to his large b ear d). But the new Universe has to b e inha bited, a nd this is where the difficulties and pro blems started. One of the difficult op en pr oblems was the problem of c o ngruency of clo sed isomer ic conv ex surface s and of co mplete noncompact isometric conv ex surfaces. Ano ther pr o blem concer ned the reg ularity of the con vex surfaces: the iso metr ic immersion o f a n analytic metr ic of a p os itive Gaussian curv ature given b y the Aleksandr ov’s theo r em pro duce s only a C 1 -regular surface. The “ g luing” metho d a pplied to a re g ular con vex sur face o f a p os itive Gauss curv ature only gua rantees the con vexity and, b y the Aleksandrov’s theore m o n limited rela tive curv ature, the C 1 -regular ity . The Minko wski pro ble m asks whether ther e exists a closed conv ex s ur face whos e Gauss curv ature K ( n ) is a given function of the outer norma l n . Minko wski himself proved that if the integral of 4 n/K ( n ) ov er the unit sphere is zero, then there ex is ts a unique (up to translation) close d co n vex surface with the Ga uss curv atur e K ( n ). How ever, o ne ca n say nothing ab out the regular ity of the surface, even when K ( n ) is a nalytic. La ter Levi show ed tha t if K ( n ) is analytic, then the res ulting surface is also ana lytic. In co nnection with these results one may ask several ques tions on the reg ularity of the r esulting surface. Namely , is it tr ue that for a regular function K ( n ) the Mink owski problem has a regula r so lution? More precisely , if the metric is o f the class C k , what is the re g ularity class of the s ur face? Is it true that the conv ex surface (not nec essarily clo sed) is regula r provided the function K ( n ) is regular? Without the answers to these questions the new theor y w as not complete. And the p erson who found the answers was A.V.Pogorelov. 2 Early ye ars of A.V.P ogorelo v A.V.Pogorelov was b or n on March 3, 191 9, in Koro cha town near Belgoro d (Russia ). O n his fa r ther’s “farm” there was just one cow a nd one hors e. During the colle ctivization they were taken from him. Once his father came to the collectiv e- farm stable a nd found his horse exhausted, dying from thirs t, while the gro om w as drank. V as ily Stepanovic h hit the gro om, a former pauper . This inciden t was rep orted as if a kulak has b e aten a peasant, and V a sily Stepanovic h w as forced to escap e the to wn, with wife Ek a terina Iv anovna, without even ta king the children. A week later E k aterina Iv anovna has secretly returned for the c hildren. This is how A.V.Pogorelov came to Khar kov, where his father bec ame a constructio n worker on the building of the tractor factor y . A.V.Pogorelov told me the story of how his pa rents hav e suffered during the collectivization I hav e heard from him only in 2000 . In m y opinion, these even ts had a strong influence on his life a nd on the way of his public b ehavior. He was a lwa ys very cautious in express io ns and liked to quo te his mother who kept s aying: silence is gold. Howev er, he never did the things co nt radicting his p o litical view s. Several times, he successfully escap ed b ecoming a mem ber of the Communist Part y (which was almos t c ompulsory for a per son of his scale in the USSR). As far as I kno w, he never signed an y letters of condemnation o f dissiden ts, but, again, a n y letters in their suppo r t, as well. Several times he was elected to the Supreme Soviet of Ukraine (although, as he said later, against his will). The ma thematical abilities of A.V.Pogorelov b eca me appar ent alr eady a t schoo l. His scho o l nick- name was Pascal. He be c a me the winner of one of the first school mathema tica l comp etitions orga nized by the K harko v Universit y , a nd then of several All-Ukrainia n Ma thematical Oly mpiads. Another tal- ent of A.V.Pogorelov was the painting. The par ents did not know, which profess ion to choo se for him. His mother a sked the son’s mathematics tea cher for advic e . He ha d a lo ok at the paintings and said that the boy has brilliant abilities , but in the time of industrialization the pa in ting will not give the reso urce for life. This advice determined their choice. In 1937, Alexey V asilyevic h b ecame a student of the Depar tment of Mathematics at the F acult y of Physics and Mathematics o f the Kharkov Univ ersity . His passion to mathematics immediately drew the attention o f the teachers. Pro fessor P .A.Solovjev gav e him the b o ok by T.Bonnezen and V.F e nch el “Theory o f co nv ex b o dies”. F r om that moment and for the r est o f his life, geometry beca me the main interest of Alexey V asilyevic h. His s tudy was int errupted by the W ar. He was co nscripted and sent to study at the Air F orce Zhuk ovski Acade my . But he s till thinks ab out geometry . In Augus t 1 943, in a letter to Pro fessor Y a.P .B la nk he says: ”V ery m uch I regret, that I left in Kharkov the abstracts of Bonnezen a nd F enchel on the co nv ex b o dies. There ar e many interesting problems in geo metr y “in the lar ge”... Do you hav e any interesting problem of geometry “in the large” or of geometry in general in mind?” After gra dua tion from the Academy in 194 5, A.V.Pogorelov starts his w ork as a designer engineer a t the Central Aero-Hydro dynamic Institute. But the desir e to finish his university education (he finished four out of five years) and to work in g eometry brings him to Moscow University . A.V.P ogorelov asks academician I.G.Petro vsky , the hea d of the Department of Mec hanics and Mathematics, whether he can finish his educa tio n. When Petrovski learnt that Alexe y V asilyevic h has a lr eady graduated 5 from the Zhuk ovski Aca de my , he decided tha t there was no need in the formal completion of the universit y . When A.V.Pogorelov express e d his in ter est in geometr y , I.G.Petro vski advised him to contact V.F.K a gan. V.F.Kagan asked, what area of geometry w as Alexey V a silyevic h int erested in, and the answer was: conv ex geometry . Kag an said that this is not his field of exp ertise and sugges ted to contact A.D.Aleksandr ov who was in Moscow at that time prepar ing to a mount climbing exp edition at the B.N.Delone a partment (A.D.Aleksandrov was a Ma ster of Sp orts on mount climbing, and B.N.Delone was the pioneer of Soviet mount climbing). The first audition lasted for ten minutes. Sitting on a backpack, A.D.Aleksandrov ask ed Alex e y V asilyevic h the fo llowing ques tion: is it true, that on a close d c onvex surfac e of the Gauss curvatur e K 6 1 , any ge o desic se gment of lengths at most π is minimizing? It to o k A.V. a year to a nswer this questio n (in affirmative) and to publish the result in 1 946 in [1 ]. The m ultidimensio na l ge ner al- ization of his theorem is a well-known theorem of Riemannian ge o metry , which was pro ved in 1959 by W.Klingenberg: on a c omplete simply c onn e cte d Riemannian manifold M 2 n of se ctional curvature satisfying 0 < K σ 6 λ , a ge o desic of the length 6 π / √ λ is m inimizing . In the odd-dimensional case, one needs a t wo-sided b ound for the curv atur e to obtain the same re sult, namely 0 < 1 4 λ 6 K σ 6 λ (and the inequality cannot b e improved). F ew years ago, I asked Alexey V a s ilyevic h, why the Soviet mathematicians at that time show ed no t m uch in terest to the global Riemannian geometry . He answered: “W e had enough interesting problems to think ab out” . Howev er, as V.A.T op onogov told me la ter , the first p erson who appr eciated his compariso n theo rem for triang les in a Riema nnian space was A.V.Pogorelov (in my o pinio n, it would be more correct to c a ll this theorem the Aleksandrov-T op onog ov theorem, since A.D.Aleksandrov discov er ed and prov ed it for genera l conv ex surfaces in the three-dimens io nal Euc lidea n space). Alexey V asilyevich became a po stgraduate- in-corre s po ndence at Moscow State Univ ersity under the sup er vision o f pro fessor N.V.Efimov. Having r ead the manuscript of the A.D.Aleksandrov’s b o ok “Intrinsic geo metry of co nv ex surfaces” , he s tarts his work in the g eometry of ge neral conv ex surfaces. One of the main r oles of a sup er v isor, in the opinion o f N.V.Efimov, was to inspire a p ost- g raduate student to solving difficult and challenging pr oblems. I gav e numerous talks b oth at the N.V.Efimov’s and the A.V.P ogorelov’s seminars . They were very differe nt by style. The N.V.Efimov’s seminar was lo ng ga thered, then the talk lasted for tw o hours or more, and the talk w as alwa ys prais e d very warmly , so it was almost imp oss ible to under stand the real v alue of the result. A.V. alwa ys started on time, very punctually . The rep ort lasted for at most a n ho ur. A.V. did not like to g o through the details of the pr o of (probably b eca use in many cas es, after the theor em was stated, he co uld prove it immediately). In the estimation of the results he was strict and even sev ere. F or example, in 196 8, three applicants for the Do ctor deg r ee presented their theses at the Pogore lov’s seminar in Kharkov. He supp o rted only one of them, V.A.T op onog ov, and rejected the other tw o , who went to Novosibirsk to A.D.Aleksandrov. All three theses were later successfully defended. A.V. pra ised ra rely , but when he did – that mea nt that the result was re a lly go o d. He had a very fast thinking, an e normous geometric intuition, and g r asp ed the essence of the r e sult very fast. Many seminar participants were afr a id to as k questions not to lo ok fo olish. In 19 47, A.V.Pogore lov defended his Candidate thesis. The main result of his thesis w a s the following theorem: every gener al close d c onvex su rfac e p ossesses thr e e close d qu asi-ge o desics [2]. This theorem gener alizes the Luster nik - Shnirelman theo rem on the existence of three closed g eo desic on a clo sed regula r conv ex sur face (a quasi-geo desic is a generaliza tion of a geodes ics; b oth the left and the right “turns” of a quasi-geo desic are nonnegative; for instance, the union of tw o genera tr ices of a round cone dividing the cone angle in tw o halves is a quasi-g eo desic). After defending his Candidate thesis, A.V. discharges from the military ser vice and mov es to Kharkov (pro bably , this was not an ea sy thing to do a t that time: he was dis charges by the s a me Order o f the Defence Minister, a s the so n of M.M.Litvinov, the for mer Soviet Minister for F oreign Affairs). In one year, he defends his Doctor Thesis o n the unique determination of a con vex surface of bounded rela tive curv ature. So o n after that, he prov es the theorem on the unique determina tio n in the most general settings [9]. 6 3 Rigidit y of Closed Con v ex Su rfaces. In 1813 , Cauch y prov ed the following rema r k able uniquenes s theor em. Cauc h y Theorem. Two close d c onvex p olyhe dr a, which ar e e qual ly-c omp ose d (t hat is, whose fac es ar e c ongruent and arr ange d in the same or der) ar e c ongruent A.D.Aleksandrov prov ed that it is p ossible, witho ut changing the C a uch y’s pro of, to replace the assumption of b eing equally comp osed with a weaker assumption of being isometric (it is easy to see that the convexit y assumption cannot b e dropp ed: co nsider, for example, a cubic hous e with a four-ch ute ro of and the same house with the ro of “pushed ins ide”). The fact that a conv ex surfa c e is uniquely determined by its metric was proved for C 3 -regular clos ed conv ex surfaces of po sitive Gauss curv ature by S.Kohn-F o s sen in 1923 and for C 2 -surfaces of non-nega tive Gauss curv a tur e by Herg lotz in 1942 . A.V.Pogorelov proved the generalizatio n of the Cauch y Theorem to the cas e of arbitrar y co nv ex surfaces: Theorem 1 (A.V.Pogorelov, 1 949, [5 ]) . Two close d isomeric c onvex su rfac es in the thr e e- dimensional Euclide an sp ac e ar e c ongruen t . The incr edible p ow e r of this theorem lies in the fact that it imp os es no reg ula rity a ssumptions on the surfaces whatso e ver. The surfaces can have edges, conic p oints, e tc. The o nly extrinsic hypothesis is the conv exity (which ca nno t b e dr opp ed: consider the sphere a nd the same s phere with a small cap cut out and r eflected inside). The pr o of of the Cauch y Theorem in these most general settings required mor e than a century , and even now, after more than half-a -century after its publication, no simpler or shorter pro of is known. A.V.Pogorelov’s pro o f go es as follows. Suppo se tha t there ar e t wo non-co ng ruent isomeric c lo sed conv ex surfa c es F 0 and F . Then, using the Aleksandrov’s gluing theorem and his gener alization of the Cauch y theor em, it is p ossible to s how that in an ar bitrarily small neighbour ho o d of F 0 , there exists a conv ex surfac e F 1 isometric a nd non-cong ruent to F 0 . The next step in the pro of is the “mixing” of sur faces. The mixing o f F 0 and F 1 is a family o f sur faces F λ , λ ∈ [0 , 1], where the surface F λ consists of the p oints in the spa ce which divide the segments, connecting the p oints o f F 0 and F 1 corres p o nding b y the is ometry in the ra tio λ : (1 − λ ). Then for some λ clo s e to 1 2 the surfa ces F λ and F 1 − λ app eare to be isomer ic. A conv ex surface is sa id to b e in a canonical p os itio n if it is a gr aph o f a conv ex function ov er the xy -plane a nd all its supp ort planes form a n angle le ss than π / 2 with that plane. Tw o curves lying on t wo isometric surfaces in a ca nonical p osition, ar e called normal e qu idistant curves , if they corres p o nd to ea ch other by the isometr y a nd the corres p o nding po int s of the cur ves are o n the same distance from the xy -pla ne. The contradiction, which pr ov es the theorem, is as follows: on one hand, a s it can be prov ed, non-congruent isometric con v ex surfaces in a ca nonical p os itio n canno t contain nor mal eq uidistant curves; on the other hand, such curves can be explicitly pro duced on F λ and F 1 − λ . This requir e s the theory of curves of the bo unded rotation v ariation, developed by A.D.Aleksandr ov and V.A.Zalgaller, and also n umerous geometric syn thetic constructions in tro duced b y A.V.P og orelov. The need for these constructions arose fr o m the lack of analytic to o ls to study co nic and edge p oints on a co nv ex surfa ce. All that makes the pro of extremely difficult to comprehend (in 1970, when I was giving m y first talk on a seminar in Leningrad, one of the questions was, whether I manag e d to g o through all the deta ils of the pro of of the Pogorelov’s theorem). Another pro of of this theorem follo ws from the r igidity theorem o f closed conv ex surfac es, that was a lso pr ov ed b y A.V.Pogorelov [9 ]. Y et another pro of can b e deduce d from the estimation of a defor mation of a c lo sed conv e x surface under a deforma tio n of its metric found by Y u.A.V olkov [4 2]. F or solv ing the pro ble m o f the unique determination of a conv ex surface by its metric Alexey V asi- lyevic h was aw a rded the Stalin prize of the sec o nd degree. Once, in 19 5 1, an unexp ected telegr am from Kiev has arrived saying that the Academy of Sciences nominated A.V.Pogorelov as a Corresp ondent Member. N.I.Ahiezer sa id: “ Let us pretend that we did not receive the telegram. The Universit y itself will nominate you”. 7 An infinitesimal b en ding of a s urface is a n infinitesimal isometric defor mation. The corresp o nding deformation field is ca lled the b en ding field . A bending field is called trivial if it is the deriv ativ e at zero of (a different iable) rig id motion of a surface. A surface is called rigi d , if every its bending field is trivial. W.Blaschke proved that a clo sed regular conv ex surface of a p ositive Gauss is rigid. Let F b e a regular surface with the p osition vector r = r ( u, v ) and let τ ( u, v ) b e a smo oth vector field. The deformatio n r = r ( u, v ) + tτ ( u, v ) is an infinitesimal b ending if and only if h r u , τ u i = 0 , h r u , τ v i + h r v , τ u i = 0 , h r v , τ v i = 0 . F or the z -comp onent ζ of the field τ we obtain the e q uation z xx ζ y y − 2 z xy ζ xy + z y y ζ xx = 0 , (1) which implies that the surface z = ζ ( x, y ) ha s non-p ositive Gauss curv ature provided the curv ature of F is po sitive. F or an infinitesimal b ending o f a gene r al co nv ex sur fa ce, equatio n (1) holds a lmost everywhere and we hav e the following lemma. Main Lemma. If z = z ( x, y ) is a c onvex surfac e c ont aining no plane domains and ζ ( x, y ) is the z -c omp onent of its infinitesimal b ending field, t hen the surfac e z = ζ ( x, y ) has n on- p ositive cu rvatur e, in the sense t hat it c ontains no p oints of the strict c onvexity. In the regular case, this means that the Gauss curv a tur e is no n-p ositive. Using this fundamental fact A.V.Pogorelov prov e d the following theor em. Theorem 2. Every close d c onvex surfac e without plane domains is rigid, t hat is, the only p ossible infinitesimal b ending field ar e of the form τ = a × r + b , wher e r is t he p osition ve ctor of the surfac e, and a, b ar e c onstant ve ctors. A close d c onvex su rfac e c ontaining plane do mains is rigid out side these domains. A.V.Pogorelov’s r esults on the unique determination and o n the rigidity of co nv ex surfa ces for med the basis o f the geo metric theory o f shells. As far a s I know, the physicists first disa g reed with his theory , so me in quite a n aggr esive way . As E.P .Senkin (m y PhD sup er v isor) was telling, when A. V. show ed them the res ults o f the exp eriments which confir med his theory on the b ending of shells they said that the shells are “press ed by a finge r”. A.V.Pogorelov’s moving to Kharkov was really successful. N.I.Ahiezer drawn A.V.’s attention to the S.N.Bernstein’s pap ers on the Dirichlet problem for elliptic equatio ns [24]. Combining the a nalytic results of S.N.Bernstein w ith the synthetic geometric methods A.V.Pogorelov managed to solve the problem of the regular it y of a conv ex surface with a regular metric of positive curv a ture and of the regular ity o f a conv ex surface obtained as the solution of the Minko wski problem, with a regular po sitive Gauss curv ature K ( n ). Up to 1934, S.N.Bernstein w orked in Kharko v, but then, after publishing a pape r ag ainst the groundless usage of Mar x ism in mathematics, he w as for ced to leav e. After that, a public pe rsecution of S. N. Bernstein b egan. It worse saying that the fir st All-Union Mathematical Congre ss was held in Kharkov thanks to the fact that S.N.Bernstein worked her e. One may reg a rd A.V.Pogorelov as an S.N.Berns tein’s successor in the field of differential equations. Quite often, A.V. used the Bernstein’s r emark able theorem which says that a non-parametr ic s urface of non-p os itive Gauss cur v ature defined ov er the whole plane and with a s lower than a linear growth, is a cylinder. 4 Regularit y of con v ex surfaces with a regular metric the and Mink o wski problem. The regular ity of a surface is equiv alent to the reg ularity of the solutions of the Darb oux equation ( z uu − Γ 1 11 z u − Γ 2 11 z v )( z vv − Γ 1 22 z u − Γ 2 22 z v ) − ( z uv − Γ 1 12 z u − Γ 2 12 z v ) 2 = K ( E − z 2 u )( G − z 2 u ) − ( F − z u z v ) 2 , 8 where E , F, G a re the co e fficient s of the first fundamental form, Γ k ij are the Christoffel sy m bo ls, a nd z is the z -co or dinate of the p osition v ec to r. Its coefficients are determined only by the metric of the surface. This nonlinear equa tion is the Monge- Amp ` ere e lliptic eq uation provided the Gauss curv a ture is p ositive. The q ue s tion is how the regularity of solutions, for a conv ex surface F , dep ends on the regular ity o f co efficients. A.V.Pogorelov split the solution into the three s tages. At the first s tage, he considered a ca p C , the intersection of F with a closed half-space b ounded by a pla ne L . He obtained the estimates dep ending only on the metr ic for the angle b etw een the tangent plane to C and the plane L , a nd als o for the no rmal curv a tur es o f C . The estimates for the nor mal cur v atures a t the int erior p oints of C dep end only on the metric and on the distance to L . T o estimate the norma l curv atures, he used the metho d o f auxiliar y functions going back to S.N.Berns tein. The main difficulty , which was s uc c essfully ov er came by A.V., is to cho ose the co rrect auxilia r y function for every sp ecific problem. These estimates cor resp ond to the a pr iori estimates o f the fir st and the second deriv a tives of a solution of the Darb o ux equation depe nding on the co efficients o f the equation and their deriv atives. This means that, assuming the regula rity o f a solution of the Darb o ux equation, one gives the estimates of the deriv atives of that so lution dep e nding on the co e fficie n ts and their der iv atives, the distance from the boundar y and the deriv ativ es of the solution o f lo w er orders. The key p oint is that the a pr iori estimates for the first and the s e c ond deriv atives ar e obtained from the g e o metric arg uments (similar estimations were obtained by A. V. in ma ny other problems, as well). F r om this p oint on, one obtains the a priori estimates for the higher deriv atives using only analy tic metho ds. Let a function z = z ( x, y ) in a domain G satisfies an elliptic PDE F ( x, y , z , z x , z y , z xx , z y y , z xy ) = 0 . (2) Then the estimates for the third deriv a tives of z at a p oint ( x, y ) dep end on the distance fr om ( x, y ) to the b ounda ry of G , and also on the supr e ma of the mo dules of the function z a nd its deriv atives up to the second order , the suprema of the mo dules of the deriv atives of the function F up to the third order, and the suprema of the modules of ( F r ) − 1 , ( F t ) − 1 , ( F r F t − 1 4 F 2 s ) − 1 , where r = z xx , s = z xy , t = z y y . The e s timates for the fo ur th a nd the subsequent deriv a tives of z are based on the Schauder theory o f the linear elliptic PDE’s. If G is the dis c x 2 + y 2 6 ε 2 , then the estimates on the k -th der iv atives of z when k ≥ 3 depend on the suprema of the mo dule of z and its deriv atives up to the seco nd order in G , the suprema o f the modules of ( F r ) − 1 , ( F t ) − 1 , ( F r F t − 1 4 F 2 s ) − 1 , and the suprema of the modules of the deriv atives of F up to order s , where s = 3 fo r k = 3 a nd s = k − 1 for k > 3. Mo reov e r , the same v alues allow one to obta in the estimates for the least H¨ older’s consta n ts fo r the k -th deriv atives of z , with an a rbitrary exp onent α (0 < α < 1 ). On the s econd stage, an ana ly tic metric g o f p ositive Ga uss curv a ture with p os itive geo desic curv ature of the b oundary defined in a dis c is realized as a conv ex analytic cap, which can then be analytically extended over the b oundary . T o do that, A.V.Pogorelov use s the metho d of prolonga tion ov er pa rameter, the idea of which go es back to S.N.Bernstein, and which then w as brillia nt ly applied by Alexey V asilyevich in many o ther problems. F ollowing this metho d one has to (I) prov e that it is p os sible to include the metric g into a one-par ameter family of analytic metrics g t , 0 6 t 6 1 , g 1 = g , wher e the metric g 0 is realized as a conv ex analy tic ca p (in fact, g 0 is the metric of a spherical cap); (II) prove that if the metric g t 0 can b e r ealized as a co nv ex analytic ca p, then the same is true for close metr ics g t ; (II I) prove that if the metrics g t n can b e realized and t n → t 0 , then the metric g t 0 can also b e realized. This will imply that the metr ic g = g 1 can b e also realized as an analytic con vex ca p. Pro blem (I I) is equiv alent to so lving the bo undary v a lue problem for the Dar b o ux equa tio n inside a unit disk with the zer o b oundary co nditions. Let G be a b ounded domain with an analytic bo undary and let φ b e a function analytic o n ∂ G with res p ect to the arc- length par ameter. By a theorem of S.N.Ber nstein, the 9 bo undary v alue problem for an elliptic eq uation F = 0 in G with z | ∂ G = φ has a solution if and only if the equatio n can b e included into a family of equatio ns F t = 0 dep ending on a parameter 0 6 t 6 1 such that (1) F 1 = F ; (2) the equation F 0 = 0 with the same b oundary conditions has an a na lytic solutio n; (3) for all 0 6 t 6 1, the existence of a solution z t to the boundar y v alue problem for the e q uation F t = 0 with the same boundar y conditions implies the uniform b o undedness of z t and all its partial deriv atives up to the second order . F rom the a priori es timates for the first and the second deriv a tives it follows that condition (3) of the Bernstein theorem is satisfied, which solves (II). Problem (I I I) can be solved us ing the a prio r i estimates o n the deriv atives up to the fourth order. This shows that the limiting surface is C 3 -regular . The analy ticity of the limiting cap now follows from the Bernstein’s theorem on analy ticit y of so lutions of an elliptic equa tio n. The third sta ge of the pro of of the regula rity o f a co nv ex sur face with a regular metric g o es as follows. A theo r em of A.D.Aleksa ndrov, the re g ularity of the metric and the fact that the curv a tur e is p ositive imply that any sur face re a lizing the given metric is smo o th a nd strictly convex. The r efore, at any p o in t of the sur face, one can cut off a cup F 0 by a pla ne parallel to the tang ent plane at tha t po int . Then one approximates the metric of a ca p by ana ly tic metrics in domains b ounded by analytic curves of p ositive geo des ic curv atur e . These metrics can be realized a s analytic ca ps, which conv er ge to a limiting cap F 1 isomeric to F 0 . The a prio ri estimates imply the regular it y of the cap F 1 . F or the C 2 and the C 3 -regular ity , o ne uses the Heintz’s a prior i estimates for the fir st and the second deriv atives of the p ositio n vector of the surfa ce. If, instead of the abov e a priori e stimates for hig he r deriv atives, one applies the Nire n ber g’s theo rem on the reg ularity of a twice differentiable solution of an elliptic equa tion with regula r co efficients to the Dar b o ux equation, the results will b e mor e precis e. It follo ws from the Nir e n ber g’s theorem that a C 2 -surface F 1 with a C n -metric b e longs to the class C n − 1 ,α , 0 < α < 1. As F 0 and F 1 are c ongruent, the cap F 0 is also C n − 1 ,α -regular . This establishes the following theore m. Theorem 3 (A.V.Pogorelov, [3, 4, 10, 9]) . A c onvex su rfac e with a C n -r e gular metric, n > 2 , of p ositive Gauss cu rvatur e is C n − 1 ,α -r e gular, for al l α ∈ (0 , 1) . If t he metric is analytic, then the surfac e is also analytic . A.V. published his fir s t result on the regularity of conv ex surfaces in 194 9 [3]. Later, in 1950, he prov ed Theorem 3 for n ≥ 4 [4]. The final r esult fo r n = 2 , 3, was published in [1 0]. Later, in 195 3, L.Nir e nber g (b eing familiar with the res ults of A.V.Pogorelov) prov ed the following regular ity theo rem [36]. A C m, α -metric of positive Gaussia n cur v ature, with m > 4 , 0 < α < 1 , can be realized by a C m, α -regular surface; a C 4 -metric of p ositive Gaussia n curv ature can b e rea lized b y a C 3 ,α -regular surface, with 0 < α < 1. In the classes C k , the Nirenberg ’s results on the regularity of the surface are the same as the Pogorelov’s ones, but in the H¨ older classes they are more precise. They are base d o n the a priori estimates of the H¨ o lder cla ss of the seco nd der iv atives o f a solutions of an elliptic equation (2) obtained by L.Nirenber g in [3 7]. Note tha t the Pogorelov’s regula rity theorem is fundamentally mor e general than the Nirenberg’s one, in the following sense. The fact that a tw o-dimensio nal manifold with an in trinsic metric of non-negative curv ature c a n b e lo cally isometrically immer sed as a conv ex surface follows from the A.D.Aleksandrov’s theo rem. Riemannian metrics of p ositive Gauss cur v ature are of tha t t yp e. The Pogorelov’s theorem g uarantees, that al l these surfaces are regular pr ovided the metr ic is regula r, while the Niren ber g’s theorem s ays that amo ng them, one c an find a regular surfac e . This mo re general result was o bta ined with the help of the Pogore lov’s theo rem on the unique determination of a conv ex cap, which was not used by Nirenber g . If one considers metrics in a H¨ older clas s , ther e is no loss of regular ity at all: 10 Theorem (I.H.Sabitov [39]) . A c onvex surfac e with a C n,α -r e gular metric of p ositive curvatu r e, wher e n > 2 , 0 < α < 1 is C n,α -r e gular. A na tur al ques tio n is whether the co nv erse is tr ue, more precisely , wha t is the co nnection b et ween the regular it y class of a submanifold in a Riemannian space and the regular it y class o f the induced metric. At first g lance, the r egularity of the metr ic sho uld b e low er. How ever, using the har monic co ordinates I.H.Sabitov and S.Z.Shefel proved the following theo rem. Theorem ([40]) . Eve ry C k,α ( k > 2 , 0 < α < 1) re gular submanifo ld F l in a R iemannian sp ac e M n of the r e gularity not lower than that of F l , is a C k,α isometric al ly immerse d Riemannian manifold M l of the class C k,α . The Pogorelov’s r egularity theor em implied new results o n the reg ularity of solutions of the Monge- Amp ` ere equatio n, which b ecame a foundation of the geo metr ic theory of b oth the tw o -dimensional and the multidimensional theor y of the Monge-Amp` ere equatio n (which we will discuss in Section 7). Once I sa id A.V. that in m y opinion, the reg ularity theor e m is his b est result, but he ans wered that he rega rds the theorem on the unique determination as the be st. Sim ultaneously with the re gularity theorem, in 19 5 2 A.V. published the solution to the Minko ws ki problem [6]. Theorem 4 . A c onvex surfac e whose Gauss cu rvatur e is p ositive and is a C m function of the outer normal ( m > 3) , is C m +1 -r e gular. Note that this is a lo cal theorem on a s urface domain whose Gauss image is a s mall disc on the unit sphere. This theorem combined with the Mink owski uniqueness theorem implies the regularity of the solution to the Minko wsk i problem. Theorem 5. L et K ( n ) b e a p ositive C k function on the u nit spher e Ω , k > 3 , such that Z Ω n dω K ( n ) = 0 , wher e dω is t he ar e a density on Ω . Then ther e ex ists a C k +1 -r e gular s u rfac e F whose Gauss cu rvatur e at the p oint with the outer normal n is K ( n ) . The surfac e F is unique up to a p ar al lel t r anslation. In 195 3, L.Nirenberg prov e d a similar result (using the pr olongatio n over par ameter and the a priori es timates of Miranda): if K ∈ C k,α , k > 2 , 0 < α < 1, then the sur fa ce is C k +1 ,α -the re g ular. If K ∈ C 2 , then the s urface is C 2 -regular [36]. T he Nirenberg ’s theorem is g lobal, it req uires the function K to b e defined over the whole sphere, as he did not prove a n y lo c al theo r em. Note how ever that the regularity is a lo c a l prop erty . A.V.Pogorelov told me that b y the R.Courant’s o pinio n, his results on the problem of reg ularity of a surface with a re g ular intrinsic metric and the proble m of regular ity of a solution to the Mink owski problem are more ge neral and more natural than thos e o f L.Nirenberg. 5 Con v ex surfaces in Riemannian space. Perhaps the greatest achiev e men t o f A.V.Pogorelov in the area of application of the ana lytic metho ds to the theory of conv ex surfaces is the following theorem. Theorem 6 ([9 , 8]) . L et R b e a c omplete t hr e e-dimensional Ri emannian sp ac e whose curvatur e is less than some c onstant C , and let M b e a R iemannian manifold home omorphic to the spher e whose Gauss curvatur e is gr e ater t han C . Then M admits an isometric immersion in R . If the metrics of R and M ar e C n -r e gular, n > 3 , then all such immersio ns ar e C n − 1 ,α -r e gular, with any α ∈ (0 , 1) . 11 Mor e over, the isometric immersion is unique in the fol lowing sense: given any two p oints x ∈ M , y ∈ R , a two-dimensional su bsp ac e L ⊂ T y R and a unit normal n to L at y , ther e ex ists a u nique isometric immersion f : M → R su ch that f ( x ) = y , d f ( T x M ) = L , and a neighb orho o d of y on the immerse d surfac e f ( M ) lies fr om the side of L define d by n (that is, the curvatur e ve ctors of al l ge o desic of f ( M ) at y ar e n onne gative multiples of n ). The pr o of of this theorem uses the prolong ation ov e r para meter and co nsists of three steps. (I) First, one pr ov es the exis tence of a con tinuous family of Riemannian manifolds M t , t ∈ [0 , 1], each of the Gauss c ur v ature greater than C , such that M 1 = M and that M 0 is isometrically immersible in R . T he manifold M 0 is a geo des ic s phere of a small ra dius in R . Using the Aleksandrov’s immersion theor e m and the Pogorelov’s theor em o n the regular ity of a c onv ex surface with a reg ular metric of the Gauss curv ature grea ter than C , one c a n immerse b oth M 0 and M in the space o f cons ta nt curv ature C as closed r e g ular conv e x surfaces F 0 and F resp ectively . Then it is p oss ible to include them in a co ntin uous family of regular closed conv ex surfaces F t , t ∈ [0 , 1] , F 1 = F , of Gaus s curv ature grea ter than C . Then for every t , the manifold M t is F t , with the induced metric. (II) The next step is to show that if a manifold M t 0 is isometrically immersible in R , then the nearby manifolds M t also are. First, Pogorelov consider s infinitesimal b ending of a regular sur face in a Riemannian s pace. A vector field ξ on a sur face F in a Riemannian spa c e R is a fie ld of infinitesimal b ending if D i ξ j + D j ξ i = 0 , i, j = 1 , 2 , where D i is the cov a riant der iv ative in R , a nd ξ i are the cov a riant co mpo nent the of ξ . If F is parameterize d in suc h a w ay that its second fundamental form is ν (( du 1 ) 2 + ( du 2 ) 2 ), then the equations of the infinitesimal b ending are of the form    ∂ ξ 1 ∂ u 1 − ∂ ξ 2 ∂ u 2 − ( ˜ Γ i 11 − ˜ Γ i 22 ) ξ i = 0 , ∂ ξ 1 ∂ u 2 + ∂ ξ 2 ∂ u 1 − 2 ˜ Γ i 12 ξ i = 0 , where ˜ Γ i j k are the Christoffel symbols of F . The following theorem generalizes Theor em 2 for an arbitrary am bien t space. Theorem 7 ([9, 8]) . L et F b e a c onvex su rfac e home omorphic t o t he sphe r e in a Riemannian sp ac e. Su pp ose F has p ositive ext rinsic curvatur e. Then any fi eld of infinitesimal b ending, which vanishes at some p oint of F to gether with its c ovariant derivatives at that p oint, vanishes identic al ly. Let F b e a surface homeo morphic to the sphere, w ith po sitive extrinsic cur v a ture in a Rieman- nian space R . Let F t , t ∈ [0 , 1], b e a r egular deformation of F = F 0 , a nd let ds 2 t = ds 2 + t dσ 2 t , b e the induced metric on F t , wher e ds 2 is the metr ic o n F . When t → 0 , dσ 2 t tends to some limit, which is uniquely determined by the deforma tion. W e a r e interested in the inv ers e problem: given the limit lim t → 0 dσ 2 t = dσ 2 = σ ij du i du j , find the corr esp onding field o f deformation ξ . The c o mpo nents of ξ satisfy the following system o f differential equations    ∂ ξ 1 ∂ u 1 − ∂ ξ 2 ∂ u 2 − ( ˜ Γ i 11 − ˜ Γ i 22 ) ξ i = σ 11 − σ 22 2 , ∂ ξ 1 ∂ u 2 + ∂ ξ 2 ∂ u 1 − 2 ˜ Γ i 12 ξ i = σ 12 , which is precisely the sy stem o f differential equations for the genera liz ed analytic functions studied b y I.N.V ekua . Using his theory Pogorelov proved that the solutio n ξ exists and is regular . The field ξ is then used in the iterative metho d to find isometric immersions of metrics M t close to the immersed one M t 0 . 12 (II I) The last step is to pr ov e that if every M t n is iso metrically immers ible , and t n → t 0 , then M t 0 also is. T o prove that, P o gorelov obtained the estimates on the normal curv atur es of a co nv ex surface homeomorphic to the sphere o f p ositive extrinsic curv ature in a r e gular Riema nnian s pace. These estimates dep end only on the metric of the surface and the metric of the s pa ce and, in turn, allow one to estimate the second deriv atives of the position vector of the sur face. Then the estimates for the hig her der iv atives follow fro m the eq uation of iso metric immersio n, which is elliptic, as the extrinsic curv ature is p ositive. Combining these three s teps A.V.Pogorelov g ives a solution to the generalize d W eyl problem for an isometric immersion in a Riemannian space. When a difficult problem is solv ed, then fir st every one admires the solution, then g ets used to it, and then, if the theorem is not a n instr umen tal to ol, p e ople b egin forg etting it. But this is not what happ ened to Theo rem 6 : in 1997, in his talk on receiving the AMS pr ize for “Pseudo-ho lomorphic curves in symplectic manifolds”, M.Gromov said that the idea of the pro of of the ex istence of pseudo - holomorphic curves came to him when he was reading the A.V.Pogorelov’s pr o of. As far as I know, the problem of the reg ularity of a co nv ex surface with a regula r metric in Riemannian spa ce, when the Gauss cur v ature of the surface is greater than the sectional cur v ature o f the ambien t Riemannian space, is still op en. A.V.Pogorelov liked concrete pro ble ms . A.L.V erner told that when A.V. was giving a talk on the A.D.Aleksandrov’s seminar in Leningrad, he rep eated s everal times ”Y ou Alex a nder Danilovic h is who pos es the problems, but I am who solves them”. After Pogore lov received the Lenin’s prize, A.D.Aleksandrov said joking that “we prove theor ems together, but receive prizes separa tely”. Alexey V a silyevic h did not ha ve many p ostgr aduate students. He started to w o rk w ith the post- graduates studen ts when the Department of Geometry of the Institute of Low T emp eratures was op ened and the v acancies needed to b e filled in. Often, he w as g iving to his student a problem, the answer to whic h (and the metho d of obtaining the answer), was alre ady known to him. Usually , the problem concerned the improv ement of s o me o f his results. The last A.V.’s p ostgra duate defended his thesis in 1970 . The p erso n who r eally supe rvised the A.V.’s p ostgra dua tes was E .P .Senkin (he mov ed fro m Leningrad to Kharko v that time). He w as a versatilely talented per s on who p oss e s sed a gift of prais ing the students, unlike A.V.P o gorelov. Unfortunately , Eug e n y Polik ar povich was ill by that time and could not help me with the choice the problem for my thesis. In 1 970, when I was the first year p ostgra duate, after one of the seminars, I asked A.V., if he has a “g o o d” question in mind for me. He answered: “I would be happy if you gave me the sa me advice” . In 19 79, on the 60th anniversary of A.V., I reminded this answer to him and thanked for b elieving in me and no t giving me a problem for the thesis which leads to nowhere. I never was a p ostg raduate student of A.V.Pogorelov, but was learning from him on his seminar s what a g o o d problem a nd what a g o o d theorem is. During mo r e than 30 years, I gave talks a t his seminar and always exp ected the A.V.’s e v a luation with thr ill. In my 5-th year, I constructed a n example of an infinite conv ex p olyhedron and a ray on it who se spherical image ha d a n infinite length. There was a co njecture that an interior p oint of a sho rtest ge o desic has a neighbor ho o d whose spherical image has a finite length. My example provided a partial counterexample to this conjecture and I wan ted to give a talk on it on the conference on “ geometry in the large” in Petrozav o ds k (Russia) in 1969. How ever, the or ganizers rejected my applica tion o n the ground that a similar example w as constructed by V.A.Zalgaller , a nd they susp ected me in plagia rism. Finally , I was given a time for my rep ort. Probably , I r ep o rted badly , so A.V. just repeated m y rep or t c o mpletely . On that confer ence, I first met A.D.Aleksandrov who s aid to me that in his o pinion, conv ex geometry is already “ closed”. His words made a great impres s ion on me, s o I chose a co mpletely different area for my p os tg raduate resear ch. Actually , fr o m tha t time, A.D.Aleksandrov stopp ed his res e arch in this field a nd started to do chronogeometry , s cho ol textb o oks, ethics, philo s ophy , etc. It was alwa ys easier to talk and even to deba te with A.D.Alexandrov rather than with A.V.Pogorelov, he was more libe r al. At the 80th 13 anniversary o f A.V.P ogorelov I s aid that he w a s more acces sible at the conferences, but was g rander in Kharkov. 6 Surfaces of b ounded extrinsic curv ature Perhaps the most conceptual r esults of A.V.Pogorelov ar e contained in his series of pap ers on smo o th surfaces of bounded extrinsic curv ature. In my o pinion, now adays, after half a cen tury , these w or ks are his most cited ones. A.D.Aleksandrov founded the theory o f the gener al metr ic manifolds, whic h a re natural g e neral- izations of Riemannian manifolds. In particular, he introduced a class o f tw o- dimensional manifolds of b ounded curv a ture. Every metric t wo-dimensional manifold which lo cally is a unifor m limit o f Riemannian manifolds whose total abso lute cur v atures (the integrals of the mo dule of the Ga us s curv ature) are uniformly b ounded is an Alexandrov’s ma nifold o f bo unded curv ature. There is a natur al question, which sur faces in R 3 carry such a metric? A partial ans wer was obtained by Pogorelov, who introduced the notion of surfac e s of bo unded curv ature. A surfac e of b ounde d curvatu r e is a C 1 -surface, the area of the spherical imag e of which (counting the m ultiplicity of the cov e r ing) is lo cally b ounded. He in tro duced the concept of a regula r p oint of a C 1 - surface. A regula r p oint can b e elliptic, hyperb olic, parab olic, o r a p oints of inflation de p ending on the t yp e of the intersection o f the surface with the tangent plane. An y p o in t on a s urface of b ounded curv ature ca n b e joined to any other po int in a sufficiently small neighbo ur ho o d by a r ectifiable cur ve lying on the surfac e . This defines a n int rinsic metric o n the underlying manifold. Pogorelov prov e d that the manifold with this metric is o f bo unded in trinsic cur v ature and found connectio ns betw een the in trinsic and the extrinsic curv ature of the s urface. F o r sur faces o f b ounded cur v a ture, the ana lo gue of the Gauss theorem holds: for e very Borel set G , the area o f the spher ical image equa ls the intrinsic cu rvatur e ω ( G ) of the se t G . The latter is defined by ω ( G ) = ω + ( G ) − ω − ( G ), wher e ω + ( G ) (resp ectively ω − ( G )) is the supremum (resp ectively the infimum) of the sums of the po sitive (re sp e ctively the negative) excesses of sets of pairwise disjoint triangles in G . A very close connection was found b et ween the intrinsic and the extrinsic geometry of a sur face: a complete surface of b ounded extrinsic curv ature and non-negative not identically v anis hing in trinsic curv ature is either a closed conv ex surface or an infinite co n vex surface. A complete surface of b ounded extrinsic curv ature whose intrinsic cur v a ture v anishes iden tically is a cylinder. The firs t P ogorelov’s pap er o n surfa ces of bounded extrinsic curv ature w a s published in 1953 [7]. On the other hand, in 19 54, J.Nas h published a pap er on C 1 isomeric immers io ns, which w as improv ed by N.Kuiper in 1 9 55. They prov e d that a tw o -dimensional Riemannian ma nifo ld, in rather general settings, a dmits a C 1 isometric immersio n (embedding ) in R 3 . Moreover, this immers io n (embedding) is, in a sense, as fre e as is a top olog ic al immer sion (embedding) of the under lying manifold. In particular, these results have some “co unterintuit ive” co rollar ies: a unit sphere can b e C 1 -isometrically embedded in an arbitra r ily small ball in R 3 ; there exists a closed C 1 -embedded lo cally E uclidean surface in R 3 homeomorphic to a torus, etc. [38]. These results s how that for a C 1 -surface, even with a “ go o d” intrinsic metric, there is in general no apparent connection betw een the int rinsic and the extrinsic geometry . F o r instance, a C 1 -regular surface whose metric is C 2 -regular and is of p ositive Gauss curv ature, do es not hav e to b e convex even lo c a lly . Perhaps the class of s urfaces of bounded extrinsic curv atur e in tro duced b y P og orelov is the mo st natural and the widest p ossible one, in which the connection b etw een the extrinsic and the intrinsic geometry is preserved under the weakest smo othness as sumptions p ossible. In m y opinion, this is the deep e st and the most difficult series of results of P o gorelov. The pro ofs are the alloy of the measure theory and brilliant synthetic g e o metric co nstructions. 14 7 Multidimensional M ink o wski problem and the m u ltidimen- sional M on ge-A mp` ere equation. Many problems of geometry “in the lar ge”, in the analytic form, are reduced to the existence a nd uniqueness problems for certain partial differential equations, in particular , for the Monge-Amp` ere equation, and conv ersely , the geometric methods and r e sults can b e used to pro ve the existence and uniqueness of solutions for differential equatio ns. As A.V.Pogorelov s a id ab out the Monge-Amp` ere equa tion, “This is a gr eat equation, which I ha d a priv ilege to s tudy”. He pioneer e d the study of the pro p e rties of s o lutions o f the gene r al multidimen- sional Mong e-Amp` ere equatio n in the se r ies of papers [15]– [19] published in 198 3-198 4, and later in the monogr aph [2 0]. Ea rlier, he obtained the results in the cas e when the r ight-hand side is a function of the indep endent v aria bles x 1 , . . . , x n , but not of the unknown function z and its deriv a tives [14]. The Mo nge-Amp` ere equation is a partial differential equatio n of the for m det( z ij ) = f ( z 1 , . . . , z n , z , x 1 , . . . , x n ) , f > 0 , (3) where z i = ∂ z ∂ x i , z ij = ∂ 2 z ∂ x i ∂ x j . On conv ex solutions z ( x 1 , . . . , x n ), this equation is of elliptic t ype. Rewrite equation (3) in the form θ ( z 1 , . . . , z n , z , x 1 , . . . , x n ) det( z ij ) = ϕ ( x 1 , . . . , x n ) . (4) Equation (4) can b e written in equiv alent for m: Z m θ ( z 1 , . . . , z n , z , x 1 , . . . , x n ) det( z ij ) dx 1 . . . dx n = Z m ϕ ( x 1 , . . . , x n ) dx 1 . . . dx n , (5) where m is an a r bitrary Borel subset of the domain G wher e the solution z is soug ht . If the so lution z ( x ) is a co nv ex function, we can make the change of v ariables p i = ∂ z ∂ x i , i = 1 , . . . , n , on the left-hand side. Then the resulting equa tion, in contrast to equatio n (4), makes sense for any conv ex but no t necessarily regula r function z ( x ). This enables one to define a gener alize d solution o f the Monge- Amp ` ere equation as follows. Let z ( x ) b e a con vex function giv en in a domain G a nd let m ⊂ G b e a Borel subset. Let m ∗ be the set of p = ( p 1 , . . . , p n ) such that the hyper plane z = p 1 x 1 + · · · + p n x n + c is a support hyperpla ne to the h y pe r surface z = z ( x ) at some point ( x, z ( x )) , x ∈ m . Such a p oint ( x ( p ) , z ( p )) is unique fo r almost all p ∈ m ∗ . The function z ( x ) is called a gener alize d solution of equation (4), if for any Bor el subset m ⊂ G , Z m ∗ θ ( p 1 , . . . , p n , z ( p ) , x 1 ( p ) , . . . , x n ( p )) dp 1 . . . dp n = Z m ϕ ( x 1 , . . . , x n ) dx 1 . . . dx n . The e x pression on the le ft-hand s ide (for a n a rbitrary conv ex function z ) is calle d the c onditional curvatur e of the set m . The concept of a generalized solution go es bac k to A.D.Aleksandrov. O ne of the main problems is to prov e the ex is tence o f a genera lized solution of (4) under certain na tur al as s umptions ab out the functions ϕ, θ . In the t wo-dimensional case with θ = 1, the existence w as pr ov ed b y A.V.Pogorelov, and in the general ca se, by A.D.Aleksandrov. Then Pogorelov prov ed the ex is tence of a s olution o f the Dirichlet pr oblem and the ma ximum principle for g eneralized solutions o f the Mo nge-Amp` ere equation with θ z 6 0, whic h implies the uniqueness of a solution o f the Diric hlet problem. He als o consider ed similar problems for the Monge- Amp ` ere equation on the sphere. The first step in the pro o f of the existence o f a generalized s olution of the Mong e-Amp` ere e quation is to show that there ex is ts a con vex p olyhedr on whose vertices pro ject to the given p oints B i in the x -space, with the giv en conditiona l cur v atures µ i > 0. Next, b y passing to the limit, one prov es that given a convex domain G and a b ounded measure µ on the Bo rel subsets of G , there ex ists a convex hypersurface z = z ( x ) , x ∈ G , such tha t for every Bor el subse t m ∈ G , the conditional curv ature 15 of m is µ ( m ). F or the Monge-Amp` ere equatio n, µ ( m ) = R m ϕ ( x ) dx . If the function θ ( p, z , x ), whic h defines the conditional curv atur e, strictly decreas es in z , then the conv ex hypersurfa ce z = z ( x ) is uniquely determined b y its boundary v alues and the measure µ . This implies the existence and the uniqueness of a genera liz ed solution of the Monge- Amp ` ere equation. The pr o of of the re gularity of the solution a ssuming the functions θ and ϕ to be reg ular, ca n be r educed to the pro of of the reg ularity of a convex hypersurfa ce with the given co nditional curv ature. This can b e done using the prolongatio n ov er par ameter. The most difficult part of the pr o of is finding the a pr iori estimates for the solution and its deriv atives up to the third or der (the a priori estimates for the hig he r or der der iv atives c a n b e then obtained from equation (4)). A.V.Pogorelov prov e d the following theorem. Theorem 8. A gener alize d solution of the Monge-A m p ` er e e quation (4) , with θ and ϕ b eing r e gular p ositive functions and θ z 6 0 , is r e gular in a neighb orho o d of every p oint of the strict c onvexity. I f θ, ϕ ∈ C k ( G ) , k > 3 , then t he solut ion is C k +1 ,α -r e gular fo r al l α ∈ (0 , 1) . The key r ole in this theorem (as in man y others) is play ed by the a pr iori estimates of a solution of an elliptic equa tio n, together with its deriv atives (up to the third order , fo r the Monge- Amp ` ere equation). These estimates do not directly follow from the equation. They are needed to g uarantee the C 2 ,α -regular ity of the limiting solution of a s equence of regula r so lutions. Then, if the co efficient s are regular , one can apply the standar d to ols of the theory of elliptic eq uations. Pogorelov’s a priori estimates for the third deriv atives of a solution z = z ( x ) of (3 ) are based on the idea of E.Calabi’s [26], who cons ide r ed a Riemannian metric ds 2 = z ij dx i dx j , computed the Laplacian of the scalar curv ature of it and obtained the estimates for the third deriv atives of z in the case f = const > 0. Earlier , Pogorelov solved the multidimensional Mink owski problem. In 1968 – 19 71, he published a ser ies of pap ers in “DAN”, the Dok lady Ak ademii Nauk (Pro cee ding s of the USSR Aca dem y of Sciences) where he found a pr io ri estimates for the second a nd the third deriv atives, and prov ed regular ity of a so lution of the Minko wski pro blem in the multidimensional case using the prolong ation ov er parameter . Theorem 9 ([14, 35]) . L et K ( n ) b e a p ositive C k function, k > 3 , on t he unit spher e S m − 1 ⊂ R m satisfying Z S m − 1 n K ( n ) dω = 0 . Then ther e exists a (un ique u p to p ar al lel t r anslation) C k +1 ,α -r e gular c onvex hyp ersurfac e with the Gauss curvatur e K ( n ) at the p oint with the outer n ormal n , for every 0 < α < 1 . If K ( n ) is analytic, then the c orr esp onding hyp ersurfac e is also analytic. Using this theorem Pogorelov proved the regular it y of the solutions of equation (4) with θ = 1 and the regular it y o f generalized solutions o f the Dirihlet pro blem [1 1 , 12, 14]. One of the implica tions is the following theorem: a unique conv ex solution z = x ( x 1 , . . . , x n ) of the equation det( z ij ) = const > 0 defined ov er the whole space x 1 , . . . , x n is a quadratic p olynomia l [14]. Note that A.V.Pogorelov did not publish long pap ers from the middle of 50-th. Usually he pub- lished a brief note in DAN, and la ter, a separate small b o ok. F or instance, the bo ok “Multidimensiona l Minko wski Problem” was published only in 19 75. In 1976 – 1 9 77, S.Y u.Cheng and S.T.Y au published the pap ers [2 7, 28]. They found small inacc ur a- cies and incompleteness in the Pogorelov’s br ie f notes in DAN (av oiding technicalities was a matter of the jour nal style; later a ll the details were given in [1 4]) and decla r ed that Pogorelov ha d no complete pro of. Then they gave their own pr o of of reg ularity o f a solution o f the multidimensional Minko wski problem and the Monge- Amp` ere equation det( z ij ) = F ( x 1 , . . . x n , z ) > 0 . The pro o f heavily relies on the Pogorelov’s a priori es timates on the second and the third der iv atives, and on the other res ults, in particula r, the Aleksa ndrov’s and Pogorelov’s theorems on the g eneralized 16 solutions o f the Mong e-Amp` ere equation. The results of these pa p er s can b e viewed as a restatement of the earlier results of Pogorelov, but by no means as the new r esults. The Eng lish translatio n o f “Multidimensiona l Mink owski problem” appeare d in 1978 with a very nice fo reword by L.Nire n ber g. Howev er, nowhere in the translatio n it is mentioned when the Russian original was publishe d . This is an (unfortunate) reaso n wh y o ften the authors first refer to [27] and then to the English translation of the Pogorelov’s b o ok. Let M be a compact K¨ ahler ma nifold with the K ¨ ahle r metric ds 2 = g j ¯ k dz j d ¯ z k and the K¨ ahler fo r m ω = i 2 g j ¯ k dz j ∧ d ¯ z k . In 1954, E.Ca labi conjectured that for a given (1 , 1) form σ = i 2 π ˜ R j ¯ k dz j ∧ d ¯ z k representing the first Chern clas s of M , there exists a K¨ ahler metric on M with the Ricci tenso r ˜ R j ¯ k whose K¨ ahler form belongs to the same cohomology cla ss a s ω . T o solv e the Calabi conjecture, one needs to solve the complex Monge-Amp` ere equatio n det  g i ¯ k + ∂ 2 ϕ ∂ z j ∂ ¯ z k  = det( g j ¯ k ) e F , c = 0 , 1 , for a real function ϕ , where F is a given function satisfying R M e F dz = V o l M in the ca se c = 0. S.T.Y au solved the Calabi conjecture using the prolong ation over parameter . A substantial part of the pr o of is finding the a prio r i estimates for the seco nd and the third deriv atives. In his pa per [43], Y au writes that the Pogorelov’s estimates for the r eal Mo ng e-Amp` ere equation were the basis for his estimates in the complex case but give no refere nce s on Pogor elovs articles. How ever, in the recen t pap er “Perspec tives on Geometric Analysis” (arXiv: math.DG/06 0 2363), by an inaccurate citing, Y a u ag ain lessens the Pogorelov’s r ole in the solving of tw o fundamental problems: the r egularity of a co nv ex surface with a r egular metric a nd the Minko wski pro blem. Concerning the firs t one, he re fer s only to the 19 61 pap er in DAN, but not to the 19 49 pa p e r [3] or a long pa per in the “ No tes of the Kha r ko v Mathematical So ciety” publishe d in 1950 . Also, he r efers to the 195 3 Nirenber g’s pap er, which was published la ter than the or iginal pa pe r of Pogor e lov. As to the t wo-dimensional Minkowski problem, he refers to the 1 953 Pogore lov’s pap er whic h ha s nothing to do with the sub ject (it follows even from the title). Me anwhile, the Pogorelov’s solution o f the Minkowski problem w as publishe d in 1952, [6], which is a y ear earlier than the Nirenberg’s paper cited by Y au. As to the multidimensional Minkowski problem, only the 1976 Cheng-Y au’s pap er is cited. There are no references to the Pogorelov’s pap ers and b o o ks on the sub ject (even those tra nslated into English) whatso ever. Another e x ample is the bo o k [3 3]. O n page 256 the author says: “The(Minko wsk i) Problem has be e n partia lly solved by Minkowski, Aleks androv, Lewy , Nirenberg , and Pogorelov” then referring to the 1952 pa p er [6], but without mentioning the pap ers [1 1, 12] and the 19 75 b o ok [14] (English translation 19 7 8). The Pogorelov’s results o n the multidimensional Mong e-Amp` ere equation were gener alized in v ar - ious directions, such as improving the regularity of the solution [29], proving the regularity up to the bo undary [30], studying the degener ate Mo nge-Amp` ere equations [34], and applying the res ults and the methods to other cla sses of completely nonlinear second order equations [41]. In par ticular, it is prov ed in [29] that a conv ex solution z = z ( x ) of the eq uation det( z ij ) = f ( x 1 , . . . , x n ) in a conv ex domain Ω, with f ∈ C α (Ω), belo ngs to C 2 ,α (Ω) (whic h is the best possible smo o thness). Note that the multidimensional Monge-Amp` ere equa tio n a pp ea rs also in s tatistical mechanics, meteorolog y , financia l ma thematics and in other ar e as [31]. Curiously enough, none of the Pogorelov’s studen ts in Kharkov work e d in differential equations, but his metho ds and r esults were activley developed by the L e ningrad mathematical school. It s eems that the mathematical influence of A.V.Pogorelov was pr op ortional to the distance from Kharkov. * * * 17 The mathematical legacy of A.V.Pogore lov is enormous. The mos t influential results not mentioned in the previo us sections include a complete solution of the F ourth Hilb ert Pro blem and obtaining the necessary and sufficient conditions for a G -space to b e Finsler [1 3, 21]. Un til 1970, A.V.Pogorelov lectured at Kha r ko v Univ er sity . B a sed on this lecture notes, he pub- lished a ser ies of brilliant textbo o k s on analytic and differential ge ometry and the foundatio n o f geometry . So metimes , during routine lectures, he was thinking ab o ut his re s earch. Anecdote s ays that on one of suc h lectures reflecting o n s omething co mpletely differen t he started improvising and bec ame lost. Then he op ened the textbo o k with the words: “What do es the author say on the topic? Oh, y es , it is o bvious . . . ”. In contrast, when le c turing o n a to pic in teresting to him, A.V.P o gorelov was very enth usia s tic and inspired (I remember one o f his top olog y courses for the 4th year students). But p er ha ps the b est of his le c turing brilliance was see n when he w as pr e s ent ing his own results. His talks w er e real fine art per formances. In his opinion, one of the most v aluable qualities of a mathe- matical res ult is its b eauty and naturalness . That is wh y he usually omitted technicalities, and for the sake of simplicity a nd b eauty was r eady to sacrifice the generality . F or many years, A.V.Pogorelov w a s the editor -in-chief of the “Ukrainsk ij Geometr icheskij Sbo rnik” (Ukrainian Geometrical, a geometry journa l published ann ually in Khar ko v Universit y . He was very jealous about the publicatio ns in it by the “lo c al” mathematicians . I remem ber , once I submitted a pap er to a different journal, but has not submitted one to the UGS. A.V. became v er y disapp ointed at me and sa id: “Wise peo ple s ay , the one who wants to be c ome famous, must b eco me famous on his own place”. A.V.Pogorelov was the author of one of the mos t p opular sc ho ol textb o oks in geometry . This beg an as follows. He was a member of the commission on the scho o l education whose head was A.N.Kolmogor ov. A.V. disa greed with the textbo o k wr itten by A.N.Kolmogorov a nd his coa uthors and wrote his own manual for teachers on element ary geometry , in which he built the whole schoo l geometry course starting with a set of na tur al and intuit ive axio ms. The manual was published in 196 9 a nd formed a basis for his schoo l textbo o k. A.V. used to say: “My textb o ok is the Kise- lyo v’s impr oved textb o ok” (“Elementary geometr y” by A.P .Kiselyo v is pro bably the most well-kno wn Russian-langua ge school g eometry textb o o k ; it was first published in 189 2 , with the last edition in 2002; many g enerations of students studied the Kiselyov’s “Geometry ”). The first v ersion o f the A.V.Pogorelov’s textb o ok s parked shar p cr iticism from A.D.Aleksandrov whom Pogorelov deeply r e- sp ected. This criticism was based on implementing the axiomatic approach as ea rly as in year six at school: “ What is the p oint to prov e ‘obvious’ statement s (from the student’s p oint of view)?”. After reworking of the textbo ok, these dis a greements were resolved, and they rema ined in s tr ong fr iendship till the last days of A.D.Aleksandrov. Alexey V asilyevich was a p erson of the highest decency . When a five y ear contract with the “Pros vesc heniye” P ublisher was coming to a n end, another publisher offered a very tempting contract to him. He refused on the unique gr ound that it will b e unfair to the editor of the textb o ok . It should be noted that the money for the school textbo ok republishing were the main sour ce of his living in the middle of the 90-th. A.V.Pogorelov told me that I.G.Petro vsk y invited him to the Moscow Universit y , I.M.Vinogradov invited to Moscow Mathematical Ins titute, A.D.Aleksandrov invited to Leningrad sev er al times. He even sp ent one year (1955-195 6) in Le ningrad, but then r eturned to K ha rko v . He prefer red to stay in Kharko v, far from the fuss a nd noise of the capita ls. In Khar ko v he pr oved his theorems, and to Moscow and Leningra d he wen t to shine. A.V.Pogorelov is a remar k able example of the mathematical longevity . I remember, in 19 92, on the A.D.Aleksandrov’s 8 0 -th anniversary , I asked M.Berg er a questio n o n geometry . He a nswered that he is too old for geometry . He w as 65 at that time. How ever, a t this very ag e A.V.P o gorelov received his final results on the multidimensional Monge- Amp ` ere equa tion. Only in 1 995, in the age of 76, he said “At my age, it is alrea dy not neces sary to do mathema tics ”. A.V.Pogorelov co mbin ed in himself a diligence of the peasa nt and a mathematica l brilliancy . He simply co uld not live without working. He inher ited this from his parents, V asily Stepanovic h and Ek aterina Iv a novna. On the 50-th anniv ersary of A.V.Pogore lov, N.I.Akhiezer b ow ed to his parents 18 thanking them for their son. A.V.Pogorelov was a v ery handso me man. He liked to be pho tographed. He often b ehav e ar tisti- cally and had a g o o d sense of humor. Once I vis ited A.V. and left a bag at his place. When i returned to pic k it up, A.V. said with a smile: “No need to ap olo gize. If y ou for got, then y ou probably were thinking ab out something”. I remember a s in 1982, on A.D.Aleksandrov’s 70-th anniv e rsary , s o me- bo dy ask ed his employ ee A.I.Medjanik to sing (he has a bea utiful v o ic e). After the s ong, A.V . said: “Hav e you heard him singing? Now imagine how the boss sings!” R.J.Bar ri, N.V.Efimov’s wife, told me that when Pogorelov was a Efimov’s pos t- graduate student, Efimov never in vited any one to his place on the day when they had co nsultations (on Thursday). This rule w as broken only once, when V.A.Rokhlin was leaving Moscow. On tha t party , Alexey V asilyevich sa ng the Ukrainian songs. A.V.Pogorelov was a mo dest p erso n, des pite of all his titles. In 19 72, when the Khar ko v g eometers flied throug h Mosc ow to Sama r k and to the All-Union g eometry conference, the flight w as delay ed in Moscow and w e had to spend a night in the waiting ha ll. B eing a Mem b er of the Supreme So viet of Ukraine, Pogorelov could go to a VIP-hall, but he has chosen to stay with us. In 2000, at the a ge of 81, A.V.Pogorelov mov ed to Moscow. He lived in No vokosino (one of the outer suburbs of Moscow), and when going to the Ins titute of Mathematics, used only the public transp ort, with several changes, instead of c a lling a ca r from the Academy of Science s . In Moscow he contin ued to work, to think o n the geometry problems and not only on them. He even br ought from Kharkov a drawing b o ard, on which he pro jected an elec tr ic gener ator based on the sup erconductivity . Alexey V asilyevic h Pogorelov was a p erson blessed by a n incredible natural talent co mbined with a constant tireless lab or. Ac knowledgemen t. I co rdially thank V.A.Marchenk o for inspiring me on this pa pe r and for many critical remar ks and J.G.Reshetnjak for thoroughly reading the manuscript and making numerous useful corrections . I am a lso thankful to A.L.Y amp olsky and to Y u.Nikolay evsky for the English translation. References [1] A.V.Po gor elov, One theorem a bo ut g eo desic on closed conv ex surface. — Math. sb. 18 (6) (1946 ), No. 1, 1 81 – 183. [2] A.V.Po gor elov, Quasigeo des ic lines on co nv ex surface. — Math. sb. 25 (67) (194 9), No. 2, 275 – 306. [3] A.V.Po gor elov, On the regular it y of a conv ex surfaces with a r egular metric.— Sov. Math. Dokl USSR, LXVI I (1949), No. 6 , 1 051 – 1053. [4] A.V.Po gor elov, Regularity o f convex surfaces.— Zapiski m atematichesko go otdelenija fiz-mat fakulteta of Kharkov university , XXI I (19 50), 5 – 49. [5] A.V.Po gor elov, Rigidity of gene r al co nv ex s urfaces. — Izd. A N of Ukr aine , (1951 ). [6] A.V.Po gor elov, Regularity o f a conv ex surface with given Gauss curv ature. — Matem. sb., 31 (73) (1952), No . 1, 88 – 103. [7] A.V.Po gor elov, Extrinsic c urv ature of smo oth s urfaces. — Dokl. Akad. Nauk U SSR , 89 (1953), No. 3, 4 07 – 409. 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