Volume Growth and Curvature Decay of Complete Positively Curved K"{a}hler Manifolds

V OLUME GR O WTH AND CUR V A TURE DECA Y OF COMPLETE POSITIVEL Y CUR VED K ¨ AHLER MANIF OLDS XIAO YONG FU A ND ZHENGLU JIAN G Abstra ct. This pap er constructs a class of complete K¨ ahler metrics of p ositive holomorphic sectional curv ature o n C n and finds that the constructed metrics satisfy the follo wing prop erties: As the geod esic distance ρ → ∞ , the volume of geodesic b alls gro ws like O ( ρ 2( β +1) n β +2 ) and the Riemannian scalar curv ature deca ys like O ( ρ − 2( β +1) β +2 ) , where β ≥ 0 . 1. In troduction W e are conce rned with the v olume gro wth and c urv ature dec a y of complex n -dimensional ( n ≥ 2) complete noncompact K¨ ahler manifolds (d enoted by M ) with p ositiv e holomorphic sectional curv ature. F or conv enience, throughout this pap er, ρ = ρ ( x 0 , x ) repr esen ts the geo desic distance from a fixed p oin t x 0 to x in M , V ( B ( x 0 , ρ )) d enotes the vo lume of the geodesic ball B ( x 0 , ρ ) cen tered at x 0 with radius ρ and R ( x ) the Riemann ian s calar c urv ature at x. When M is assumed to b e of p ositiv e holomorphic b isectional cur v a ture, it is kno wn b y th e classical Bishop vo lume comparison theorem that V ( B ( x 0 , ρ )) ≤ ω ( n ) ρ 2 n for an y x 0 ∈ M , w here ω ( n ) is the v olume of the s tand ard u nit ball B 2 n in R 2 n . Also, if V ( B ( x 0 , ρ )) ≥ C ρ 2 n for some p ositiv e constan t C, then th e scalar curv at ure deca ys quadratically in th e av erag e sense. This is conjectured b y Y au [8] and confirmed by Chen and Zhu [3]. On th e other h and, it has b een sho wn by C hen and Zhu [3] that V ( B ( x 0 , ρ )) ≥ C ρ n for an y x 0 ∈ M , where C is a p ositiv e constan t dep end ing only on x 0 and M , and that the scalar cur v ature deca ys at least linearly in the av e rage sense. In view of th e ab ov e results, a complex n -dim en sional K¨ ahler manifold w ith p ositive holomorphic bisectional curv ature is s aid to b e of maximal volume gro wth if V ( B ( x 0 , ρ )) = O ( ρ 2 n ) and of minimal v olume gro wth if V ( B ( x 0 , ρ )) = O ( ρ n ) . There ha v e b een some examples of p ositive ly curved complete noncompact K¨ ahler manifold with maximal (resp ectiv ely , min imal) v olume gro wth and quadratic (resp ective ly , linear) curv a- ture deca y . In [5], K lem b ec k constru cted a complete, rotationally symm etric K¨ ahler metric g of p ositiv e holomorphic sectional curv ature on C n with m inimal v olume gro wth and linear curv a- ture deca y . More precisely , th e v olume of the geo desic ball B (0 , ρ ) with resp ect to the metric g gro ws lik e O ( ρ n ) and the scalar cur v a ture R ( x ) of the metric g d eca ys lik e O ( ρ − 1 ) . Rotat ionally symmetric, complete gradien t K ¨ ahler-Ricci solitons of p ositiv e Riemannian sectional curv ature on C n ha v e been found by Cao ([1], [2]) and it has b een kno wn th at su c h solitons ca n b e divided in to t w o br anc hes: one is of m aximal v olume gro wth and qu adratic curv ature d eca y , and the other of minimal v olume gro wth and linear curv ature deca y . By the ab o v e analysis, it is n atur al to ask w hether th ere exists a complete noncompact K¨ ahler manifold of p ositiv e holomorphic sectional cu rv at ure satisfying V ( B ( x 0 , ρ )) = O ( ρ n (1+ ǫ ) ) and R ( x ) = O ( ρ − (1+ ǫ ) ) for any fixed ǫ ∈ (0 , 1) . T o our b est kno wledge, examples of such manifolds seem to b e elusive from the literature. In Date : No ve mber 20, 2018. 2000 Mathematics Subje ct Classific ation. 53C21. Key wor ds and phr ases. K¨ ahler metrics, H olomorphic sectional curv ature, V olume growth, Curv ature decay . 1 2 XIAO YON G FU AN D ZHENGLU JIANG this pap er, we shall construct a class of complete K¨ ahler metrics g of p ositiv e holomorphic sectional cur v ature o n C n and fi nd th at the co nstructed metrics satisfy the f ollo wing prop erties: As th e geo desic distance ρ → ∞ , the v olume of geo desic balls gro ws lik e O ( ρ 2( β +1) n β +2 ) and th e Riemannian s calar curv atur e deca ys lik e O ( ρ − 2( β +1) β +2 ) , where β ≥ 0 . Finally , w e should men tion th e w ork of Ni et al. [7] for some r esults on the relation b et wee n v olume gro wth a nd curv ature deca y of complete noncompact K¨ ahler manifold with p ositiv e holomorphic b isectional curv ature , and r efer to the b o ok of Greene and W u [4] and Kobay ashi and Nomizu [6] for bac kgroun d on differen tial geometry . 2. Me thod of Construction In this section, we shall giv e the conditions u n der wh ic h there exists a complete rotationally symmetric K¨ ahler metric on C n with p ositiv e h olomorphic sectional cur v ature. Recall that a complex n -dimensional complete K ¨ ahler manifold M is of p ositiv e holomorphic sectional cur v ature if P R i ¯ j k ¯ l a i ¯ a j a k ¯ a l > 0 for all nonzero n -tuples ( a 1 , a 2 , · · · , a n ) of complex n umbers , where R i ¯ j k ¯ l is the comp onents of the cur v ature tensor. Let r 2 = n P i =1 z i ¯ z i on C n . As in [5], we consider only rotationally symmetric metrics g i ¯ j = ∂ 2 f ( r 2 ) ∂ z i ∂ ¯ z j on C n , deriv ed fr om a global p oten tial function, where r → f ( r 2 ) ∈ C ∞ ( R ) . Clearly , g = f ′ ( r 2 ) n X i =1 dz i d ¯ z i ! + f ′′ ( r 2 ) n X i =1 ¯ z i dz i ! n X i =1 z i d ¯ z i ! . (2.1) Th us g is a complete metric on C n if f satisfies the follo wing tw o conditions: (i) f ′ ( r 2 ) > 0 and f ′ ( r 2 ) + r 2 f ′′ ( r 2 ) > 0 for all r , (ii) R ∞ 0 p f ′ ( r 2 ) + r 2 f ′′ ( r 2 ) dr d iv erges. Since g is a rotationally symmetric K¨ ahler metric, withou t loss of generalit y , we m a y restrict the computation of the cur v ature tensor R j ¯ kl ¯ m of g to the complex line L = { z i = 0 | i > 1 } , th us obtaining R j ¯ kl ¯ m = − ∂ 2 g j ¯ k ∂ z l ∂ ¯ z m − X p,q g ¯ pq ∂ g j ¯ p ∂ z l ∂ g ¯ k q ∂ ¯ z m ! = − f ′′ ( r 2 )( δ j ¯ k δ l ¯ m + δ j ¯ m δ l ¯ k ) − r 2 [ f ′′′ ( r 2 ) − ( f ′′ ( r 2 )) 2 f ′ ( r 2 ) ]( δ j ¯ k 1 δ l ¯ m + δ j ¯ m 1 δ l ¯ k + δ l ¯ m 1 δ j ¯ k + δ l ¯ k 1 δ j ¯ m ) − [ r 4 f ′′′′ ( r 2 ) − r 2 (2 f ′′ ( r 2 ) + r 2 f ′′′ ( r 2 )) 2 f ′ ( r 2 ) + r 2 f ′′ ( r 2 ) + 4 r 2 ( f ′′ ( r 2 )) 2 f ′ ( r 2 ) ] δ j ¯ kl ¯ m 1 . (2.2) Let ( a 1 , a 2 , · · · , a n ) b e any complex n -tuples of complex num b ers. Then X R i ¯ j k ¯ l a i ¯ a j a k ¯ a l = − (2 A + 4 B + C ) | a 1 | 4 − 4( A + B ) | a 1 | 2 ( n X j =2 | a j | 2 ) − 2 A ( n X j =2 | a j | 2 ) 2 where A = f ′′ ( r 2 ) , B = r 2 [ f ′′′ ( r 2 ) − ( f ′′ ( r 2 )) 2 f ′ ( r 2 ) ] , VOLUME GRO WTH AND CUR V A TU RE DECA Y 3 C = r 4 f ′′′′ ( r 2 ) − r 2 (2 f ′′ ( r 2 ) + r 2 f ′′′ ( r 2 )) 2 f ′ ( r 2 ) + r 2 f ′′ ( r 2 ) + 4 r 2 ( f ′′ ( r 2 )) 2 f ′ ( r 2 ) . Th us, in addition to (i) and (ii), the follo wing conditions for g i ¯ j = ∂ 2 f ( r 2 ) ∂ z i ∂ ¯ z j to b e a complete K¨ ahler metric on C n of strictly p ositiv e holomorphic sectional curv ature are r equired: (iii) A < 0 , i.e., f ′′ ( r 2 ) < 0 for all r, (iv) 2 A + 4 B + C < 0 , i.e., 1 4 r ∂ ∂ r  r ∂ ∂ r ln( f ′ ( r 2 ) + r 2 f ′′ ( r 2 ))  < 0 , (v) A + B < 0 , i.e., f ′′ ( r 2 ) + r 2 f ′′′ ( r 2 ) − r 2 ( f ′′ ( r 2 )) 2 f ′ ( r 2 ) < 0 . Therefore a complete metric o f p ositiv e holomorphic sectional curv ature on C n can b e generated from a fu n ction f ∈ C ∞ ( R ) satisfying (i)-(v). In the next sectio n, w e shall in tr o duce a class of functions in C ∞ ( R ) satisfying (i)-(v). 3. Metrics of Pos itive Cur v a tur e In this section, we shall sho w a family of complete K¨ ahler metrics of p ositiv e h olomorphic sectional curv ature on C n and discuss the curv ature deca y and v olume gro wth of these metrics. Let us consider the follo win g f unction f ( r 2 ) = 1 ( β + 1) α β Z r 2 0  [ α + ln(1 + x )] β +1 − α β +1 x  dx ∈ C ∞ ( R ) (3.1) where α > β ≥ 0 . Then f ′ ( r 2 ) = 1 ( β + 1) α β  [ α + ln(1 + r 2 )] β +1 − α β +1 r 2  , f ′′ ( r 2 ) = − [ α + ln(1 + r 2 )] β +1 ( β + 1) α β r 4 + [ α + ln(1 + r 2 )] β α β r 2 (1 + r 2 ) + α ( β + 1) r 4 and f ′ ( r 2 ) + r 2 f ′′ ( r 2 ) = [ α + ln(1 + r 2 )] β α β (1 + r 2 ) . It follo ws that f satisfies (i), (ii) and (iii) (see App end ix A). Also, 1 4 r ∂ ∂ r  r ∂ ∂ r ln( f ′ ( r 2 ) + r 2 f ′′ ( r 2 ))  = − α ( α − β ) + β r 2 + (2 α − β ) ln (1 + r 2 ) + ln 2 (1 + r 2 ) (1 + r 2 ) 2 [ α + ln(1 + r 2 )] 2 and f ′′ ( r 2 ) + r 2 f ′′′ ( r 2 ) − r 2 ( f ′′ ( r 2 )) 2 f ′ ( r 2 ) = − β α β +1 e y − α − β α β +1 − y β +2 + y β +1 e y − α − y β +1 + α β +1 y y − β r 2 (1 + r 2 ) 2 { [ α + ln(1 + r 2 )] β +1 − α β +1 } (3.2) whic h yield (iv) and (v) (see App endix A). Here, y = α + ln(1 + r 2 ) . Therefore g i ¯ j = ∂ 2 f ( r 2 ) ∂ z i ∂ ¯ z j with f ( r 2 ) d efined by (3.1) is a class of complete K¨ ahler metrics of p ositiv e sectional cur v atures on C n . No w we turn to the computation of th e vo lume gro wth and scalar cu rv ature deca y of C n equipp ed with the metric g i ¯ j = ∂ 2 f ( r 2 ) ∂ z i ∂ ¯ z j with f ( r 2 ) d efined by (3.1). First, let us estimate th e 4 XIAO YON G FU AN D ZHENGLU JIANG v olume gro wth of C n . As b efore, our computation is restricted on L. Ins er tin g (3.1) into (2.1), w e h av e g =  [ α + ln(1 + r 2 )] β α β (1 + r 2 )  dz 1 d ¯ z 1 +  [ α + ln(1 + r 2 )] β +1 − α β +1 ( β + 1) α β r 2  n X i =2 dz i d ¯ z i ! . (3.3) And the vo lume form ω n of (3.3) is giv en by ω n =  √ − 1 2  n  [ α + ln(1 + r 2 )] β α β (1 + r 2 )  ×  [ α + ln(1 + r 2 )] β +1 − α β +1 ( β + 1) α β r 2  n − 1 dz 1 ∧ d ¯ z 1 ∧ dz 2 ∧ d ¯ z 2 ∧ · · · ∧ dz n d ∧ ¯ z n . Since the geod esic distance function ρ from the origin of C n is give n by ρ = Z r 0 s [ α + ln(1 + t 2 )] β α β (1 + t 2 ) dt and satisfies ρ = O (ln β +2 2 r ) ( r → ∞ ) , (3.4) the volume gro wth V ( B (0 , ρ )) of geodesic ball B (0 , ρ ) of C n equipp ed with (3.3) is V ( B (0 , ρ )) = Z B E (0 ,r ) ω n = Z S 2 n − 1 (1) " Z r 0  √ − 1 2  n +1  [ α + ln(1 + t 2 )] β α β (1 + t 2 )  ×  [ α + ln(1 + t 2 )] β +1 − α β +1 ( β + 1) α β t 2  n − 1 t 2 n − 1 dt # dθ = O (ln ( β +1) n r ) as r → ∞ = O ( ρ 2( β +1) n β +2 ) as ρ → ∞ , where B E (0 , r ) is the Euclidean ball corresp onding to the geod esic b all B (0 , ρ ) . T o determine the rate of curv ature deca y , without loss of generalit y , we ma y restrict the computation of the s calar cur v ature R = P i,j g i ¯ j R i ¯ j of g to the complex line L = { z i = 0 | i > 1 } . Then the Ricci curv ature, denoted b y Ric , is as follo ws: Ric = − √ − 1  β ∂ ¯ ∂ ln  α + ln(1 + r 2 )  − ∂ ¯ ∂ ln(1 + r 2 )  − √ − 1( n − 1) n ∂ ¯ ∂ ln  [ α + ln(1 + r 2 )] β +1 − α β +1  − ∂ ¯ ∂ ln r 2 o = − √ − 1  β α + ln(1 + r 2 ) − 1  dz 1 ∧ d ¯ z 1 + (1 + r 2 )( n P i =2 dz i ∧ d ¯ z i ) (1 + r 2 ) 2 + √ − 1 β r 2 [ α + ln(1 + r 2 )] 2 (1 + r 2 ) 2 dz 1 ∧ d ¯ z 1 − √ − 1( n − 1) ( β + 1)[ α + ln(1 + r 2 )] β [ α + ln(1 + r 2 )] β +1 − α β +1 n P i =1 dz i ∧ d ¯ z i 1 + r 2 − √ − 1( n − 1)  ( β + 1)[ α + ln (1 + r 2 )] β − 1 [ β − α − ln (1 + r 2 )] [ α + ln(1 + r 2 )] β +1 − α β +1 VOLUME GRO WTH AND CUR V A TU RE DECA Y 5 − ( β + 1) 2 [ α + ln(1 + r 2 )] 2 β ([ α + ln(1 + r 2 )] β +1 − α β +1 ) 2  r 2 dz 1 ∧ d ¯ z 1 (1 + r 2 ) 2 + √ − 1( n − 1) n P i =2 dz i ∧ d ¯ z i r 2 . This imp lies that R 1 ¯ 1 =  β α + ln(1 + r 2 ) − 1  1 (1 + r 2 ) 2 + ( n − 1)( β + 1)[ α + ln(1 + r 2 )] β [ α + ln(1 + r 2 )] β +1 − α β +1 1 1 + r 2 − β r 2 [ α + ln(1 + r 2 )] 2 (1 + r 2 ) 2 + ( n − 1)( β + 1)[ α + ln(1 + r 2 )] β − 1 [ β − α − ln(1 + r 2 )] [ α + ln(1 + r 2 )] β +1 − α β +1 r 2 (1 + r 2 ) 2 − ( n − 1)( β + 1) 2 [ α + ln(1 + r 2 )] 2 β ([ α + ln(1 + r 2 )] β +1 − α β +1 ) 2 r 2 (1 + r 2 ) 2 and that for i ≥ 2 , R i ¯ i =  β α + ln(1 + r 2 ) − 1  1 1 + r 2 − n − 1 r 2 + ( n − 1)( β + 1)[ α + ln(1 + r 2 )] β [ α + ln(1 + r 2 )] β +1 − α β +1 1 1 + r 2 . Also, R i ¯ j = 0 for i 6 = j. As a consequence, the scalar curv ature R = n P i =1 g i ¯ i R i ¯ i of C n equipp ed with the rotationally symmetric metrics g i ¯ j = ∂ 2 f ( r 2 ) /∂ z i ∂ ¯ z j with f ( r 2 ) defin ed by (3.1) deca ys lik e R = O  1 ln β +1 r  as r → ∞ . (3.5) By (3.4), (3.5) can b e written as R = O  ρ − 2( β +1) β +2  as ρ → ∞ . (3.6) App end ix A. Put G ( x ) = [ α + ln(1 + x )] β +1 (1 + x ) − ( β + 1) x [ α + ln(1 + x )] β − α β +1 (1 + x ) . Then f ′′ ( r 2 ) = − G ( r 2 ) ( β +1) α β r 4 (1+ r 2 ) . Since lim r 2 → 0 f ′′ ( r 2 ) = − α − β 2 α < 0 for α > β ≥ 0 , (iii) holds if G ( x ) > 0 for all x > 0 . S in ce G (0) = G ′ (0) = 0 , it suffices to sho w th at G ′′ ( x ) > 0 when α > β ≥ 0 . Ind eed, if α > β ≥ 0 and x ≥ 0 , then G ′′ ( x ) = [(2 α − β ) + 2 αx ] ln (1 + x ) ( β + 1) − 1 (1 + x )( α + ln (1 + x )) − β + α ( α − β ) + [ α 2 − β 2 + β ] x ( β + 1) − 1 (1 + x )( α + ln (1 + x )) − β + ln 2 (1 + x ) ( β + 1) − 1 ( α + ln(1 + x )) − β > 0 . T o pro v e (v), let H ( y ) = β α β +1 e y − α − β α β +1 − y β +2 + y β +1 e y − α − y β +1 + α β +1 y . Since lim r 2 → 0  f ′′ ( r 2 ) + r 2 f ′′′ ( r 2 ) − r 2 ( f ′′ ( r 2 )) 2 f ′ ( r 2 )  = − α − β 2 α < 0 , it is easily known from (3.2) that (v) holds if H ( y ) > 0 for all y > α. Since H ( α ) = H ′ ( α ) = 0 , it suffi ces to pro v e that H ′′ ( y ) = β α β +1 e y − α + y β [ y e y − α − β ( β + 1)] + y β − 1 [ e y − α − 1] β ( β + 1) + 2 y β [ e y − α − 1]( β + 1) > 0 for all y > α when α > β ≥ 0 . 6 XIAO YON G FU AN D ZHENGLU JIANG Let I ( y ) = β α β +1 e y − α + y β [ y e y − α − β ( β + 1)] . Then we need to sho w that I ( y ) > 0 for all y ≥ α wh en α > β ≥ 0 . Put α > β ≥ 0 and I n ( y ) = y I ′ n − 1 ( y ) ( n = 1 , 2 , 3 , · · · ) and I ′ 0 ( y ) = I ( y ) . Then I n ( y ) = − y β β n +1 (1 + β ) + e y − α y [ y n + β + α 1+ β β P n − 1 ( y ) + y β Q n − 1 ( y )] with P n − 1 ( y ) = 1 + n − 1 P i =1 p n − 1 i y i and Q n − 1 ( y ) = (1 + β ) n + n − 1 P j =1 q n − 1 n − j ( β ) y i , where p n − 1 i are p ositiv e in tegers, q n − 1 j ( β ) are p olynomials of degree j whith resp ect to β j whose coefficients are p ositiv e in tegers, where n = 1 , 2 , 3 , · · · , i = 1 , 2 , · · · , n − 1 , j = 1 , 2 , · · · , n − 1 . It can b e sho wn that I n ( α ) > 0 ( n = 1 , 2 , 3 , · · · ) and that I n ( y ) > − y β β n +1 (1 + β ) + y β +1 (1 + β ) n > y β β (1 + β )[(1 + β ) n − 1 − β n ] for all y ≥ α > β ≥ 0 . Hence there exists a p ositiv e integ er n 0 whic h dep end s only on β , su c h that I n ( y ) > 0 for all y ≥ α > β ≥ 0 as n ≥ n 0 . By the definition of I n ( y ) , we kno w that I ′ n 0 − 1 ( y ) > 0 f or all y ≥ α when α > β ≥ 0 . It f ollo ws that I n 0 − 1 ( y ) > 0 for all y ≥ α when α > β ≥ 0 . Repeating the ab ov e analysis, w e conclude that I ′ n ( y ) > 0 and I n ( y ) > I n ( α ) ( n = 0 , 1 , 2 , 3 , · · · ) for all y ≥ α wh en α > β ≥ 0 . In particular, I ( y ) > 0 for all y ≥ α when α > β ≥ 0 . Ackno wledgement This pap er is completed under the directio n o f Profess or Zhu Xi-Ping . The authors would like to express their gra titude for his contin uous guidance and muc h v alua ble advice. XF is also grateful for suppo rt from NSF C 1 0171 1 14. ZJ is supp orted by NSF C 10 27112 1 and spo nsored by SRF fo r R OCS, SEM. The authors w ould also like to thank Dr Chen Binglong for his helpful comments on this pap er. Finally , the author s would lik e to thank the referee of this pap er for his helpful c o mment s and sugg estions. References [1] Cao H. D., Existence of gradient K¨ a hler-Ricci solutions, Elliptic and P arab olic Metho ds in Geometry , Min- nesota, pp . 1-16, 1994. [2] Cao H. D ., Limits of Solutions to the K¨ ahler-Ricci flow, J. Differential Geom., 45 , pp. 257-272, 19 97. [3] Chen B. L., Z hu X. P ., V olume growth and curv ature decay of p ositiv ely curved K¨ ahler manifolds, Q. J. Pure Appl. Math., 1 , pp. 68-108, 2005 . [4] Greene R. E., W u H., F unction Theory on Manifolds whic h p ossess a p ole, V ol. 66 9, Springer-V erlag, Berlin- Heidelb erg-NewY ork, 1979. [5] Klembeck P . F., A complete K¨ ahler metric of p ositiv e curv ature on C n , Proc. A mer. Math. So c., 64 , pp. 313- 317, 1977. [6] Kobaya shi S., N omizu K., F oundation of d ifferentia l geometry , V ol. I I , Interscience, New Y ork, 1969. [7] N i L., Sh i Y.-G ., T am L.-F., Poisson equation, P oincar ´ e-Lelong equation and curv ature deca y on complete K¨ ahler manifolds, J. Differen tial Geom., 57 , pp. 339-388, 200 1. [8] Y au S.-T., A review of complex d ifferenti al geometry , Pro c. Symp. Pure Math., V ol. 52, Pa rt 2, Amer. Math. Soc., 199 1. Dep ar tm ent of Ma thema tics, Zhongshan University, Guangz hou 510 275, C hina E-mail addr ess : mcsfxy@mail .sysu.edu.cn Dep ar tm ent of Ma thema tics, Zhongshan University, Guangz hou 510 275, C hina E-mail addr ess : mcsjzl@mail .sysu.edu.cn