On convex to pseudoconvex mappings

ON CONVEX TO PSEUDOCONVEX MAPPINGS S. IV ASHK OVICH Abstract. In the works of Darb oux and W alsh, see [D, W], it w as remar ked that a one to one self mapping of R 3 which sends conv ex sets to c o n vex ones is affine. It ca n b e remarked a lso that a C 2 -diffeomorphism F : U → U ′ betw een tw o do mains in C n , n > 2, which sends pseudo co n vex hypersurfa ces to pseudoconvex ones is either holo morphic or antiholomorphic. In this note we ar e in terested in the s elf mapping s of C n which send conv ex h yper - surfaces to pseudo convex ones. Their characteriz ation is the following: A C 2 - diffe o- morphism F : U ′ → U (wher e U ′ , U ⊂ C n ar e domai ns) sends c onvex hyp ersurfac es to pseudo c onvex ones if and only if t he inverse map Φ : = F − 1 is we akly plurih armonic, i.e., it satisfies some nic e se c ond or der PDE very close to ∂ ¯ ∂ Φ = 0 . In fact all plur iharmonic Φ-s do satisfy this equation, but there are also other solutions. 1. Formula tion Let U ′ , U b e domains in C n , n > 2 and let F : U ′ → U b e a C 2 -diffeomorphism. Co ordinates in the source w e denote b y z ′ = x ′ + iy ′ , in the target by z = x + iy . It will b e con v enien t for us to supp ose tha t U ′ is a conv ex neigh b orho o d of zero and that F (0 ′ ) = 0. The, somewhat un us ual c hoice to put primes on the ob jects in the source (and not in the target) is explained by the fact that in the statemen ts a nd in the pro ofs w e shall w ork more with the in v ers e map Φ then with F . Theorem 1. L et F : U ′ → U b e a C 2 -diffe o m orphism. Then the fol lowing c o n ditions ar e e quivalen t: i) F or every c onvex hyp ersurfac e M ′ ⊂ U ′ the image M = F ( M ′ ) is a pse udo c onvex hyp ersurfac e in U . i i) The inverse map Φ : = F − 1 : U → U ′ satisfies the fol lo wing se c ond or der PDE System ∂ ∂ Φ = ( d Φ − 1 (∆Φ) , dz ) ∧ ∂ Φ + ( d z , d Φ − 1 (∆Φ)) ∧ ∂ Φ . (1.1) i i i) The e quation (1.1) has the fol lowing ge ometric me a ning: for every z ∈ U and every ζ ∈ T z C n ∂ ∂ Φ z ( ζ , ¯ ζ ) ∈ span { d Φ z ( ζ ) , d Φ z ( iζ ) } . (1.2) Here we use the follo wing notation: for a v ector v = ( v 1 , ..., v n ) ∈ C n and dz = ( dz 1 , ..., d z n ) w e set ( dz , v ) = ¯ v j dz j and ( v , dz ) = v j d ¯ z j . Throughout this note w e shall use the Einstein summation conv en tion. Remark 1. P luriharmonic Φ-s clearly satisfy (1.1) (or (1.2)) and let us remark tha t this geometric c haracterization of pluriharmonic diffeomorphisms p erfectly agrees with an analytic one: The class P of pluriharmonic diffe omo rp hisms C n → C n is stable under Date : Nov em ber 19, 2018 . 1991 Mathematics Subje ct Classific ation. Primar y - 32 F10, Secondary - 5 2A20, 32U15. Key wor ds and phr ases. Convex, pseudo co n v ex, pluriharmonic. 1 2 S. IV ASHKO VICH biholomorphic p ar ametrization of the sour c e and R -line ar of the tar get . Really , these parametrization preserv e accordingly pseudo con v ex it y and conv exit y of h ypersurfaces. 2. The item ( i) of the Theorem is cle arly equiv a len t to the fo llo wing one: F or every s trictly c onvex quadric M ′ ∩ U ′ 6 = ∅ the image M = F ( M ′ ∩ U ′ ) is a pseudo c onvex hyp ersurfac e in U . I.e, it is enough to chec k this condition only for quadrics. 3. The f a ct that (1.1) a dmits other solutions then just pluriharmonic mappings is v ery easy t o see from t he form of its linearization at iden tit y ∂ ∂ Φ = (∆Φ , dz ) ∧ dz . (1.3) Remark that an y map of the fo rm Φ( z ) = ( ϕ 1 ( z 1 ) , ..., ϕ n ( z n )) satisfies (1.3) pro vide d all ϕ j , except for some j 0 , are ha rmonic. And this ϕ j 0 can b e then an arbitrary C 2 -function. 2. An auxiliar y comput a tion Denote b y ζ = ξ + iη a tangent v ector at p o int z ∈ C n . Recall that the real Hessian of a real v a lued function ρ in C n = R 2 n is H R ρ ( z ) ( ζ , ζ ) = ∂ 2 ρ ( z ) ∂ x i ∂ x j ξ i ξ j + ∂ 2 ρ ( z ) ∂ y i ∂ y j η i η j + 2 ∂ 2 ρ ( z ) ∂ x i ∂ y j ξ i η j . (2.1) A hypersurface M = { z ∈ U : ρ ( z ) = 0 } , with ρ is C 2 -regular, ρ (0) = 0 and ∇ ρ | M 6 = 0, is strictly con v ex if the defining f unction ρ can b e c hosen with p ositive definite Hess ian, i.e., H R ρ ( z ) ( ζ , ζ ) > 0 for all z ∈ M and all ζ 6 = 0. One readily chec ks t he following expression o f the real Hessian of ρ in complex co ordinat es H R ρ ( z ) ( ζ , ζ ) = ∂ 2 ρ ( z ) ∂ z i ∂ z j ζ i ζ j + ∂ 2 ρ ( z ) ∂ ¯ z i ∂ ¯ z j ¯ ζ i ¯ ζ j + 2 ∂ 2 ρ ( z ) ∂ z i ∂ ¯ z j ζ i ¯ ζ j . (2.2) Recall that the Hermitian part L ρ ( z ) ( ζ , ¯ ζ ) = ∂ 2 ρ ∂ z i ∂ ¯ z j ζ i ¯ ζ j of the Hessian is called the L evi form of ρ (and of M ). M is strictly pseudo con v ex if its Levi form is p ositiv e definite on the complex tang en t space T c z M = { ζ ∈ T z C n :  ∂ ρ ( z ) , ζ  = 0 } for ev ery z ∈ M . Here ( · , · ) stands for the standard Hermitian scalar pro duct in C n . Let F : C n z ′ ⊃ U ′ → U ⊂ C n z b e a C 2 -diffeomorphism. Let further z ′ = z ′ ( z ) b e the co or dina t e represen ta t ion of the in v e rse mapping z ′ = Φ( z ) : = F − 1 ( z ) and let M = F ( M ′ ) ⊂ U b e the image o f a hypersurface M ′ ⊂ U ′ . Then M = { z : ρ ( z ) = 0 } , where ρ ( z ) := ρ ′ ( z ′ ( z )). Lemma 2.1. The L evi form of ρ at p oint z de c om p oses as L ρ ( z ) ( ζ , ¯ ζ ) = L 0 ρ ( z ) ( ζ , ¯ ζ ) + L 1 ρ ( z ) ( ζ , ¯ ζ ) , (2.3) wher e L 0 ρ ( z ) ( ζ , ¯ ζ ) = 1 4 H R ρ ′ ( z ′ ) ( d Φ z ( ζ ) , d Φ z ( ζ )) + 1 4 H R ρ ′ ( z ′ ) ( d Φ z ( iζ ) , d Φ z ( iζ )) (2.4) and L 1 ρ ( z ) ( ζ , ¯ ζ ) = 2  ∇ ρ ′ ( z ′ ) , ∂ ¯ ∂ Φ z ( ζ , ¯ ζ )  = 2 Re  ∂ ρ ′ ( z ′ ) , ∂ ∂ Φ z ( ζ , ¯ ζ )  . (2.5) Pro of. H ere w e denote b y d Φ z is the differential of the in v erse map Φ : = F − 1 at po in t z , ∇ ρ ′ ( z ′ ) the real gradient of ρ ′ at z ′ , h · , ·i = Re ( · , · ) - the standa r d Euclidean scalar pro duct in C n . ON CONVEX TO PSEUDOCON VEX MAPPINGS 3 Denote by ν the ve ctor with compo nents ν j = ∂ z ′ j ∂ z α ζ α and by µ with µ j = ∂ z ′ j ∂ ¯ z α ¯ ζ α , i.e., ν = ∂ Φ z ( ζ ) and µ = ∂ Φ z ( ζ ) . Remark that ν + µ = d Φ z ( ζ ) and i ( ν − µ ) = d Φ z ( iζ ) . (2.6) W rite L ρ ( z ) ( ζ , ¯ ζ ) = ∂ 2 ρ ∂ z α ∂ ¯ z β ζ α ¯ ζ β = ∂ ∂ z α  ∂ ρ ′ ∂ z ′ i ∂ z ′ i ∂ ¯ z β + ∂ ρ ′ ∂ ¯ z ′ i ∂ ¯ z ′ i ∂ ¯ z β  ζ α ¯ ζ β = = ∂ 2 ρ ′ ∂ z ′ i ∂ z ′ j ∂ z ′ i ∂ ¯ z β ∂ z ′ j ∂ z α ζ α ¯ ζ β + ∂ 2 ρ ′ ∂ ¯ z ′ i ∂ ¯ z ′ j ∂ ¯ z ′ i ∂ ¯ z β ∂ ¯ z ′ j ∂ z α ζ α ¯ ζ β + ∂ 2 ρ ′ ∂ z ′ i ∂ ¯ z ′ j  ∂ z ′ i ∂ ¯ z β ∂ ¯ z ′ j ∂ z α + ∂ z ′ i ∂ z α ∂ ¯ z ′ j ∂ ¯ z β  ζ α ¯ ζ β + +  ∂ ρ ′ ∂ z ′ i ∂ 2 z ′ i ∂ z α ∂ ¯ z β + ∂ ρ ′ ∂ ¯ z ′ i ∂ 2 ¯ z ′ i ∂ z α ∂ ¯ z β  ζ α ¯ ζ β = ∂ 2 ρ ′ ∂ z ′ i ∂ z ′ j µ i ν j + ∂ 2 ρ ′ ∂ ¯ z ′ i ∂ ¯ z ′ j ¯ ν i ¯ µ j + ∂ 2 ρ ′ ∂ z ′ i ∂ ¯ z ′ j [ µ i ¯ µ j + ν i ¯ ν j ] + +  ∂ ρ ′ ∂ z ′ i ∂ 2 z ′ i ∂ z α ∂ ¯ z β + ∂ ρ ′ ∂ ¯ z ′ i ∂ 2 ¯ z ′ i ∂ z α ∂ ¯ z β  ζ α ¯ ζ β = L 0 ρ ( z ) ( ν, µ ) + L 1 ρ ( z ) ( ζ , ¯ ζ ) with L 0 ρ ( z ) ( ν, µ ) = ∂ 2 ρ ′ ∂ z ′ i ∂ z ′ j ν i µ j + ∂ 2 ρ ′ ∂ ¯ z ′ i ∂ ¯ z ′ j ¯ ν i ¯ µ j + ∂ 2 ρ ′ ∂ z ′ i ∂ ¯ z ′ j [ µ i ¯ µ j + ν i ¯ ν j ] (2.7) and L 1 ρ ( z ) ( ζ , ¯ ζ ) =  ∂ ρ ′ ∂ z ′ i ∂ 2 z ′ i ∂ z α ∂ ¯ z β + ∂ ρ ′ ∂ ¯ z ′ i ∂ 2 ¯ z ′ i ∂ z α ∂ ¯ z β  ζ α ¯ ζ β . (2.8) W e need to get more information ab out t he structure of b ot h terms L 0 ρ and L 1 ρ of the Levi form. Let’s prov e that the f ollo wing relation holds L 0 ρ ( z ) ( ν, µ ) = 1 4 H R ρ ′ ( z ′ ) ( ν + µ, ν + µ ) + 1 4 H R ρ ′ ( z ′ ) ( i ( ν − µ ) , i ( ν − µ )) . (2.9) T o see this we mak e the following c hange in (2.9): µ j = V j + iW j , ν j = V j − iW j . or V = 1 2 ( ν + µ ) = 1 2 d Φ z ( ζ ) , W = i 2 ( ν − µ ) = 1 2 d Φ z ( iζ ) . (2.10) Then L 0 ρ ( z ) ( ν, µ ) = ∂ 2 ρ ′ ∂ z ′ i ∂ z ′ j ( V i − iW i ) ( V j + iW j ) + ∂ 2 ρ ′ ∂ ¯ z ′ i ∂ ¯ z ′ j  V i + iW i   V j − iW j  + + ∂ 2 ρ ′ ∂ z ′ i ∂ ¯ z ′ j  ( V i + iW i )  V j − iW j  + ( V i − iW i )  V j + iW j  = = ∂ 2 ρ ′ ∂ z ′ i ∂ z ′ j ( V i V j + W i W j ) + i ∂ 2 ρ ′ ∂ z ′ i ∂ z ′ j ( V i W j − W i V j ) + ∂ 2 ρ ′ ∂ ¯ z ′ i ∂ ¯ z ′ j  V i V j + W i W j  + + i ∂ 2 ρ ′ ∂ ¯ z ′ i ∂ ¯ z ′ j  W i V j − V i W j  + 2 ∂ 2 ρ ′ ∂ z ′ i ∂ ¯ z ′ j  V i V j + W i W j  = H R ρ ′ ( z ′ ) ( V , V ) + H R ρ ′ ( z ′ ) ( W , W ) . W e used t he ob vious relations ∂ 2 ρ ′ ∂ z ′ i ∂ z ′ j ( V i W j − W i V j ) = 0 = ∂ 2 ρ ′ ∂ ¯ z ′ i ∂ ¯ z ′ j  W i V j − V i W j  and the complex expression o f the real Hessian ( 2 .2). Therefore 4 S. IV ASHKO VICH L 0 ρ ( z ) ( ν, µ ) = H R ρ ′ ( z ′ ) ( V , V ) + H R ρ ′ ( z ′ ) ( W , W ) . (2.11) F rom (2.10) and (2.11) w e get the fo rm ula (2.4) of the Lemma. Remark 2. If t he real Hess ian of ρ ′ at z ′ is p ositive (resp. non-negativ e) definite then the compo nen t L 0 ρ ( z ) ( ν, µ ) of the Levi form of ρ at z = F ( z ′ ) is also p ositiv e (resp. non- negativ e) definite fo r any C 2 -germ of a diffeomorphism F . No w w e turn to L 1 ρ . Note that in complex notations ∇ ρ = ∂ ρ as we ll as that standard Euclidean scalar pro duct h· , ·i in C n is equal to the real part o f the Hermitian one ( · , · ). Therefore from (2.8) w e get L 1 ρ ( z ) ( ζ , ¯ ζ ) =  ∂ ρ ′ ∂ z ′ i ∂ 2 z ′ i ∂ z α ∂ ¯ z β + ∂ ρ ′ ∂ ¯ z ′ i ∂ 2 ¯ z ′ i ∂ z α ∂ ¯ z β  ζ α ¯ ζ β = =  ∂ ρ ′ , ∂ 2 z ′ ∂ ¯ z α ∂ z β ¯ ζ α ζ β  +  ∂ ρ ′ , ∂ 2 z ′ ∂ ¯ z α ∂ z β ¯ ζ α ζ β  =  ∂ ρ ′ , ∂ ∂ Φ z ( ζ , ¯ ζ )  +  ∂ ρ ′ , ∂ ∂ Φ z ( ζ , ¯ ζ )  = 2 Re  ∂ ρ ′ , ∂ ∂ Φ z ( ζ , ¯ ζ )  = 2  ∇ ρ ′ , ∂ ∂ Φ z ( ζ , ¯ ζ )  , whic h prov es (2.5).  3. Proof of the Theorem ( i) ⇐ ⇒ ( i i i) W e start with the pro of of the geometric characterization of con v ex to pseu do con v e x mappings giv en in ( i i i) of the Theorem. By a complex (real) line in C n w e mean a n 1- dimensional complex (real) subspace of C n . The same for complex (real) plain. T ak e a complex line l = span { ζ } in T z C n and let Π ′ ⊂ T z ′ C n b e the real plain - image of l under d Φ z , i.e., Π ′ = span { d Φ z ( ζ ) , d Φ z ( iζ ) } . Let l ′ := ∂ ∂ Φ z ( l ) denotes the real (!) line - image of l under the mapping ∂ ∂ Φ z : C n z → C n z ′ , defined a s ζ 7→ ∂ ∂ Φ( ζ , ¯ ζ ) : = ∂ 2 Φ( z ) ∂ z α ∂ z β ζ α ¯ ζ β . W e consider l ′ as a real line in T z ′ C n . Lemma 3.1. Supp ose that given a diffe omorphism F : U ′ → U . Then F sen ds c onve x quadrics to pseudo c onvex hyp ersurfac es if and onl y if f or ev ery z ∈ U and for al l l ′ := ∂ ∂ Φ z ( l ) and Π ′ = d Φ z ( l ) as a b ove one has l ′ ⊂ Π ′ . Pro of. Let us prov e the “ only if ” assertion first. W e ma y supp ose that z ′ = 0 ′ . T ak e an y strictly con v ex M ′ = { ρ ′ ( z ′ ) = 0 } defined b y a C 2 -function ρ ′ with positive defined Hessian suc h that T 0 ′ M ′ ⊃ Π ′ . By M denote the image F ( M ′ ). Consider the following family of h yp ersurfaces in U ′ : M ′ t = { z : ρ ′ t ( z ′ ) := ρ ′ ( z ′ ) + t h ∇ ρ ′ (0 ′ ) , z ′ i = 0 } , t ∈ R , t 6 = − 1. All M ′ t are strictly con v e x (they ha v e the same quadratic part as M ′ ), all path through zero and M ′ 0 = M ′ . In addition all M ′ t are smo oth at zero with T ′ 0 M ′ t = T 0 ′ M ′ for all t 6 = − 1, b ecause ∇ ρ ′ t (0 ′ ) = (1 + t ) ∇ ρ ′ (0 ′ ). Moreo v er, if w e take some ζ ∈ T c 0 M then ζ will stay to b e complex t a ngen t to all M t := F ( M ′ t ) at zero b ecause T 0 M t = dF 0 ′ ( T 0 ′ M ′ t ) is the same f or all t . F rom Lemma 2 .1 w e see that ON CONVEX TO PSEUDOCON VEX MAPPINGS 5 L ρ t (0) ( ζ , ¯ ζ ) = L 0 ρ (0) ( ζ , ¯ ζ ) + 2(1 + t )  ∇ ρ ′ (0 ′ ) , ∂ ∂ Φ(0)( ζ , ¯ ζ )  , (3.1) b ecause L 0 ρ t (0)( ζ , ¯ ζ ) = L 0 ρ (0)( ζ , ¯ ζ ) for a ll t due to the fact that coefficien ts of L 0 ρ dep end only on the second deriv ativ es of ρ ′ at 0 ′ and o n d Φ 0 . Supp ose  ∇ ρ ′ (0 ′ ) , ∂ ∂ Φ(0)( ζ , ¯ ζ )  6 = 0. Then taking an appropriate t 0 w e can mak e L ρ t (0) ( ζ , ¯ ζ ) = 0 b ecause L 0 ρ (0) ( ζ , ¯ ζ ) do not dep end on t . Remark tha t t 0 6 = − 1 b ecause L 0 ρ (0) ( ζ , ¯ ζ ) > 0 . Now we can deform M ′ t letting t run o v e r a neigh b orho o d of t 0 . M ′ t sta ys strictly conv ex while the Levi form o f M t c hanges its sign on the v ector ζ . Contradiction with assumed prop erty of F . Therefore  ∇ ρ ′ (0 ′ ) , ∂ ∂ Φ(0)( ζ , ¯ ζ )  = 0 , (3.2) for ev ery strictly conv ex M ′ = { z ′ : ρ ′ ( z ′ ) = 0 } suc h that T 0 ′ M ′ ⊃ Π ′ . F or an y v ector v ∈ T 0 ′ C n orthogonal to Π ′ w e can ta k e a strictly con v ex h yp ersurface M ′ = { z ′ : ρ ′ ( z ′ ) = 0 } suc h that ∇ ρ ′ (0 ′ ) = v . Therefore ∂ ∂ Φ(0)( ζ , ¯ ζ ) is orthogonal to ev ery suc h v . So l ′ = ∂ ∂ Φ(0)( l ) ⊂ Π ′ and t he “only if ” a ssertion of the lemma is pro v ed. T o pro v e the opp osite direction tak e a con ve x quadric M ′ = { z ′ : ρ ′ ( z ′ ) = 0 } and set M = F ( M ′ ). Let ζ ∈ T c z M . Use a g ain Lemma 2.1. The term L 0 ρ ( z ) ( ζ , ¯ ζ ) is clearly p o sitive. The term L 1 ρ ( z ) ( ζ , ¯ ζ ) is zero b ecause ∂ ∂ Φ z ( ζ , ¯ ζ ) ∈ d Φ z ( h ζ i ) ⊂ T z ′ M ′ .  Let us reform ulate the result obt a ined as follo ws (and r emark that the equiv alence o f ( i) and ( i i i) in Theorem is prov ed): Corollary 3.1. I f F sends c onvex quadrics to pseudo c onvex h yp ersurfac es if and only if for e v ery z ∈ U and every ve ctor ζ ∈ T z C n the fol lo w ing holds: ∂ ∂ Φ z ( ζ , ¯ ζ ) ∈ span { d Φ z ( ζ ) , d Φ z ( iζ ) } . (3.3) F or the con v enience of future references let us form ula t e the ab o v emen tioned statemen t ab out holo morphic mappings: Corollary 3.2. A C 2 -dife om orphism F : U ′ → U sends ps eudo c onvex quadrics to pseudo- c onvex hyp ersurfac es if and only if F is either ho l o morphic or antiholomorphic. Pro of. This is we ll known but still let us give a pro of. Supp ose, for example, that Φ is an tiholomorphic, then ν = 0 as defined in (2.6). Therefore (2.5) tells us that L 1 ρ ( z ) ( ζ ′ , ¯ ζ ′ ) ≡ 0 in the represen tation (2 .3 ). Now (2.7) sho ws that and ga v e us L ρ ( z ) ( ζ , ¯ ζ ) = L ρ ′ ( z ′ )  ∂ Φ z ( ζ ) , ∂ Φ z ( ζ )  for ev ery complex tangen t ζ . Conclusion follows. Supp ose that, vice v ersa, F sends pse udo con v e x quadrics to pseudo con v ex h yp ersur- faces. (3.3) show s that ∂ ∂ Φ z ( ζ , ¯ ζ ) b elongs t o the pla in span { d Φ z ( ζ ) , d Φ z ( iζ ) } for all ζ ∈ C n z . And therefore fo r ev ery ζ complex tangent to M = { ρ ( z ) = 0 } the v ector ∂ ∂ Φ z ( ζ , ¯ ζ ) is tangen t t o M ′ = { ρ ′ ( z ′ ) = 0 } . Consequen tly L 1 ρ ( z ) ( ζ , ¯ ζ ) ≡ 0 for an y ρ . Apply (2.9) to the quadric 6 S. IV ASHKO VICH ρ ′ ( z ′ ) = n X j =1 ( z 2 j + ¯ z 2 j + ε | z j | 2 ) + L ( z ) + L ( z ) (3.4) (where ε > 0 and L is a C - linear form) and get L 0 ρ ( z )  ζ , ¯ ζ  = n X j =1 ( ν j µ j + ¯ ν j ¯ µ j + ε | ν j | 2 + ε | µ j | 2 ) = 2 R e n X j =1 ν j µ j ! + ε ( k ν k 2 + k µ k 2 ) . (3.5) T aking differen t linear fo r ms L in (3.4) w e can deplo y an y ζ ∈ C n as a complex tangent and therefore, if Φ is neither holomorphic no an tiholomorphic, then w e see from (2.6) that ν and µ can b e tak en arbitr a ry . But for arbitrary ta k en ν and µ (3.5) cannot b e p ositiv e. Con tradiction.  ( i i i) ⇐ ⇒ ( i i) W e shall nee d the fo llowing line ar algebra lemma. Let V and W be C -linear spaces. W e supp ose that on V some Hermitian scalar pro duct ( · , · ) is fixed. Let B ( ζ , ¯ η ) : V × V → W b e a sesquilinear map. Its trace is defined as T r B = P α B ( e α , ¯ e α ) for an orthonormal frame in ( V , ( · , · )). Let, furthermore C : V → W b e an R -linear isomorphism. D enote b y C 1 , 0 (resp. C 0 , 1 ) the complex linear (resp. an tilinear) part of C . Lemma 3.2. The fol lowing pr op erties ob the p air ( B , C ) a r e e quivalent: B ( ζ , ¯ ζ ) ∈ span { C ( ζ ) , C ( iζ ) } for al l ζ ∈ V . (3.6) B ( ζ , ¯ η ) =  C − 1 ( T r B ) , η  C 1 , 0 ( ζ ) +  ζ , C − 1 ( T r B )  C 0 , 1 ( η ) for al l ζ , η ∈ V . (3.7) Pro of. D efine the induced quadratic map A : V → V as A ( ζ , ¯ ζ ) = C − 1 ◦ B ( ζ , ¯ ζ ). Note that A is not sesquilinear in general. Note that the image of ev ery complex line in V under a quadratic map is a real line. W rite (3.6) in the form A ( ζ , ¯ ζ ) = k ( ζ ) · ζ , where k is a complex v alued function. One readily sees that k ( λζ ) = ¯ λk ( ζ ). The p olarization equalit y for A A ( ζ + η , ¯ ζ + ¯ η ) + A ( ζ − η , ¯ ζ − ¯ η ) = 2 A ( ζ , ¯ ζ ) + 2 A ( η , ¯ η ) giv es k ( ζ + η )( ζ + η ) + k ( ζ − η )( ζ − η ) = 2 k ( ζ ) ζ + 2 k ( η ) η , or, for complex indep enden t v ectors k ( ζ + η ) + k ( ζ − η ) = 2 k ( ζ ) and k ( ζ + η ) − k ( ζ − η ) = 2 k ( η ) , whic h implies additivit y of k : k ( ζ + η ) = k ( ζ ) + k ( η ) for complex independent ζ , η and, therefore for all. So k is an an tilinear form on V and b y Ries represen tation w e obtain a vec tor v such that k ( ζ ) = ( v , ζ ) for all ζ ∈ V and therefore A ( ζ , ¯ ζ ) = ( v , ζ ) ζ and consequen tly B ( ζ , ¯ ζ ) = C (( v , ζ ) ζ ) for all ζ ∈ V . (3.8) F urthermore, B ( ζ , ¯ η ) + B ( η , ¯ ζ ) = B ( ζ + η , ¯ ζ + ¯ η ) − B ( ζ , ¯ ζ ) − B ( η , ¯ η ) = C (( v , ζ + η )( ζ + η )) − − C (( v , ζ ) ζ ) − C (( v , η ) η ) = C (( v , η ) ζ ) + C (( v , ζ ) η ) . ON CONVEX TO PSEUDOCON VEX MAPPINGS 7 and − iB ( ζ , ¯ η ) + iB ( η , ¯ ζ ) = B ( ζ + iη , ¯ ζ − i ¯ η ) − B ( ζ , ¯ ζ ) − B ( η , ¯ η ) = C (( v , ζ + iη ) ( ζ + iη )) − − C (( v , ζ ) ζ ) − C (( v , η ) η ) = C ( − i ( v , η ) ζ ) + C ( i ( v , ζ ) η ) . Therefore 2 B ( ζ , ¯ η ) = C (( v , η ) ζ ) − iC ( v , η ) ζ ) + C ( ( v , ζ ) η ) + iC ( i (( v , ζ ) η ) . So w e obtain B ( ζ , ¯ η ) = ( v , η ) C 1 , 0 ( ζ ) + ( ζ , v ) C 0 , 1 ( η ) . (3.9) Set in (3.9) ζ = η = e α . Then T r B = X α B ( e α , ¯ e α ) = C 1 , 0 X α ( v , e α ) ! + C 0 , 1 X α ( v , e α ) ! = C ( v ) . Therefore v = C − 1 ( T r B ) a nd (3.7) is established. The opp o site implication is easy , b ecause (3.7) tells, if η is take n to b e equal to ζ , that B ( ζ , ¯ ζ ) = aC 1 , 0 ( ζ ) + ¯ a C 0 , 1 ( ζ ) = a 1 2 ( C ( ζ ) − iC ( iζ )) + ¯ a 1 2 ( C ( ζ ) + iC ( iζ )) = = Re a · C ( ζ ) + Im a · C ( iζ ) ∈ span { C ( ζ ) , C ( iζ ) } .  W e apply this lemma for B = ∂ ∂ Φ z : T z C n → T z ′ C n , C = dF − 1 z ′ : T z ′ C n → T z C n and, as a result A = d F z ′ ◦ ∂ ∂ Φ z : T z C n → T z C n for ev ery z = F ( z ′ ) and get Corollary 3.3. A C 2 -diffe o m orphism F sends c on v e x quadrics to pseudo c onvex hyp ersur- fac es if and only if ∂ ∂ Φ z ( ζ , ¯ η ) =  dF z ′  T r ∂ ∂ Φ z  , η  ∂ Φ( ζ ) +  ζ , dF z ′  T r ∂ ∂ Φ z  ∂ Φ( η ) (3.10) for a l l z = F ( z ′ ) an d al l ζ , η ∈ T z C n . And this is equiv alen t to (1.1). Theorem is prov ed. Reference s [D] Darboux M.: Sur un th´ eor` eme fondamental de la g´ eom´ etrie pro jective (Extrait d’une lettre ` a M. Klein). Math. Ann. 17 , N 1, 55-61 (1880 ). [S] Sch ¨ opf P . : Konvexit¨ atstreue und Linearit¨ a t v on Abbildungen. Math. Z., 177 , No. 4, 533-5 40 (1981 ). [W] W alsh J. L.: On the T ransformation of Conv ex Point Sets. Annals of Ma th. 32 , No. 4, 262-2 66 (1921 ). Universit ´ e de Lille-1, UFR de Ma th ´ ema tiques, 59655 Villeneuve d’A scq, France. E-mail addr ess : i vachkov @math. univ-lille1.fr IAPMM Na t. Acad. Sci. Ukraine, L viv, Nauko v a 3b, 79601 Ukraine.