Complex Monge-Ampère Operators in Analysis and Pseudo-Hermitian Manifolds

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1 Complex Monge-Ampère Operators in Analysis and Pseudo-Hermitian Manifolds Song-Ying Li October 3, 2007 Abstract: The paper is a short survey around the author s recent works on topics related to complex Monge-Ampère equations and strictly pseudoconvex pseudo-hermitian manifolds. 1. Invariant differential operators In complex analysis of one variable, the fact that the invariant property for Laplace operator under holomorphic change of coordinates plays an important role. Namely, Let φ : D 1 D 2 be a holomorphic mapping, and let u be a smooth function on D 2. If v(z) = u φ(z) for z D 1, then (1.1) v(z) = ( u) φ(z) φ (z) 2 For example, this property couple with the existence, uniqueness and regularity theory of the Dirichlet problem of Laplacian was used beautifully by Painlevé [50] to prove the smooth extension for a proper holomorphic map between two smoothly bounded domains in the complex plane. However, Laplace operator is no longer invariant under holomorphical change of variables in C n when n > 1. Instead, the complex Monge-Ampère operator is invariant under holomorphic change of coordinates. Let u be a real-valued function on a[ domain D in C n, we let H(u)(z) denote the complex Hessian matrix 2 u(z) z i z j ], which is an n Hermitian matrix at each z D. Then the complex Monge-Ampère operator is defined as follows: [ 2 u(z) ] (1.2) M[u](z) = det H(u)(z) = det. z i z j Primary subject 32V05, 32V20. Secondary subject 53C56. pseudoconvex CR manifold, pseudo-hermitian geometry Keywords: Strongly 1

2 If φ : D 1 D 2 is a holomorphic mapping, and if u C 2 (D 2 ) and v(z) = u φ(z), then (1.3) M[v](z) = M[u] φ(z) det φ (z) 2, z D 1, where det φ (z) is the Jacobian of the holomorphic mapping φ. It was pointed out by Kerzman [27] that if one can prove the existence, uniqueness and regularity for the Dirichlet problem of degenerate complex Monge-Ampère equation on a smoothly bounded strictly pseudoconvex domain in C n, then one may use the Painlevé s [50] approach to reprove the Feffereman mapping theorem [13]: Any biholomorphic mapping between two smoothly bounded strictly pseudoconvex domains in C n can be extended as a diffeomorphism between their closures. However, the regularity of a degenerate complex Monge-Ampère equation M[u] = f may not have smooth solution, thus the idea of Painlevé and Kerzman above may not work. Along this line, there are some ideas and related works can be found in Krantz and Li [31] and Li [37]. 2. Boundary value problems Boundary value problems for existence, uniqueness and regularity for plurisubharmonic solutions for complex Monge-Ampère equations have received considerable studies by many authors. We consider Monge- Ampère equation: (2.1) M[u](z) = f(z, u, u) 0 in D, associated to either Dirichlet boundary value: (2.2) u = φ(z), on D or Neumann boundary: (2.3) D ν u = φ(z, u), φ u γ 0 on D, where D is a bounded pseudoconvex domain in C n Dirichlet boundary value problem In [2], Bedford and Taylor proved the following theorem. THEOREM 1 Let D be a bounded strictly pseudoconvex domain in C n with C 2 boundary. For 0 < α 1, if φ C 2α ( D) and f = f(z) 0 with f(z) 1/n C 0,α (D), then the Dirichlet boundary value problem (2.1) and (2.2) has a unique plurisubharmonic solution u C 0,α (D), where (2.1.1) u C 0,α = sup{ u(z) u(w) z w α : z, w D, z w} and C 0,α (D) = C α (D) when 0 < α < 1. 2

3 Smooth plurisubharmonic solution was obtained by Caffarelli, Kohn, Nirenberg and Spruck [7], they proved the following theorem: THEOREM 2 Let D be a bounded pseudoconvex domain in C n with C boundary. If φ C ( D) and f = f(z) C (D) is positive on D, then the Dirichlet boundary value problem (2.1) and (2.2) has a unique plurisubharmonic solution u C (D). When f = f(z) 0 allows to have zero in D, (2.1) and (2.2) may not have C 2 (D)-plurisubharmonic solution. For example, a counterexample was constructed by Gamelin and Sibony[22] that there is a plurisubharmonic function u(z) C 1,1 (B n ) \ C 2 (B n ) such that (2.1.2) M[u] = 0, in B n ; u Bn C 4 ( B n ), where B n is the unit ball in C n. More general example was given by Bedford and Fornaess in [1]. When f 1/n C 1,1 (D), we have the following results: (i) It was proved by Caffarelli, Kohn, Nirenberg and Spruck [7] that when φ constant on D, then (2.1) and (2.2) has a unique plurisubharmonic solution u C 1,1 (D). (ii) The C 1,1 estimate for a general φ C 3,1 ( D) was obtained by Krylov [32]. When D is not strictly pseudoconvex, B. Guan [14] was able to obtain the Caffarelli-Kohn-Nirenberg-Spruck type theorem (Theorem 2.2) by proposing a plurisubharmonic sub-solution condition: (2.1.3) det H(u 0 ) f(z), z D, u 0 = φ(z) on D to replace the strictly pseudoconvex condition on D. The idea and result of B. Guan [14] were proved very useful by P. Guan[15] and Li [37]. For D being a weakly pseudoconvex domain of finite type, C α regularity theorem was studied by Li [37]. We say that a bounded pseudoconvex domain D in C n has plurisubharmonic type m if there is a plurisubharmonic function ρ C 2 (D) C 2/m (D) such that ρ(z) < 0 on D with ρ(z) = 0 on D and H(ρ) I n is positive semi-definite for all z D. The following theorem was proved by Li in [37]. THEOREM 3 Let m > 2 be a real number, and let D be a bounded pseudoconvex domain of plurisubharmonic type m in C n with C 2 boundary. For any 0 < α 2/m, if φ C mα ( D) and f 0 with f(z) 1/n C α (D), then the Dirichlet problem of the complex Monge- Ampère equation (2.1) and (2.2) has a unique plurisubharmonic solution u C α (D) in weak sense. 3

4 Remark: When m = 2, i.e., D is strictly pseudoconvex, the above result (Theorem 1) was due to Bedford and Taylor [2]. In particular, we consider a complex ellipsoid type domain: (2.1.4) E 2m = {(z 1, z 2 ) C 2 : z z 2 2m < 1} C 1/m (E 2m ) C (E 2m ). If one let (2.1.5) ρ(z) = m( z 2 2 (1 z 1 2 ) 1/m ), then ρ(z) < 0 on E 2m, ρ(z) = 0 on E 2m. Moreover, H(ρ) I 2 is positive semi-definite. Therefore E 2m is a pseudoconvex domain of plurisubharmonic type 2m. The following example in [37] shows that there is a gap phenomenon for best regularity for the solution for (2.1) and (2.2) between m = 1 and m > 1 (here E 2m has type is 2m). EXAMPLE 1 For any m > 1, there is a plurisubharmonic function u C 1/m (E 2m ) C (E 2m ) satisfies (i) u E2m = φ C ( E 2m ); (ii) det H(u)(z) = f(z) C (E 2m ); (iii) u C β (E 2m ) for any β > 1/m. Recently, many works have been done for the Dirichlet problem with non-smooth function f in (2.1). We refer the readers to papers written by Blocki [5], Cegrell and S. Kolodzej [6], S. Kolodziej [28, 29] and references therein Kähler-Einstein metric Let ρ be a C 2 negative function on a domain D in C n, and let (2.2.1) u(z) = log( ρ(z)). Let J(ρ) be the Fefferman operator defined as follows: [ ] ρ ρ (2.2.2) J(ρ) = det ( ρ). H(ρ) Then a relation between J(ρ) and M(u) was given in [11] and [38] as follows: (2.2.3) M(u) = J(ρ)e (n+1)u. Then (2.2.4) J(ρ) = 1 in D, ρ = 0 on D 4

5 5 if and only if (2.2.5) M(u) = e (n+1)u in D, u = on D. When D is a smoothly bounded strictly pseudoconvex domain in C n. Question about whether the equation (2.2.4) has a unique solution with u is plurisubharmonic in D was studied by C. Feffereman in [12]. He proved the uniqueness, and provided a formal approximation for a solution. Cheng and Yau [11] (later, for more general domain, see [46]) proved the existence of such a solution, which is real analytic in D, for all smoothly bounded weakly pseudoconvex domains D in C n. Moreover, the potential function u given by (2.2.5) defines a complete Kähler-Einstein metric u ij dz i dz j on D. In particular, when D is a smoothly bounded strictly pseudoconvex domain in C n, they proved that ρ C n+3/2 (D). More precise result was obtained by Lee and Melrose[34], who gave an asymptotic expansion for the solution of (2.2.4) with u plurisubharmonic in D, which is: (2.2.6) ρ(z) = a j (δ(z) n+1 log δ(z)) j j=0 with a j C (D) and δ(z) is the distance function from z to D. In particular, ρ C n+2 ɛ (D) for any ɛ > 0. It is very natural to ask the following question: What domains D, on which, (2.2.4) does have smooth solution ρ? Some related work on Bergman kernel was given by R. Graham [19]. More details of the Fefferman type approximation for solution of (2.2.4) will be presented in Tran [51] as well as the result for domains whose boundary is ellipsoid associated to the above question are also studied there Neumann boundary value problem Neumann boundary value problems for complex Monge-Ampère equation was studied by Li [40] while Neumann problem for existence, uniqueness and C regularity theorem for real Monge-Ampère equation on a smoothly bounded strictly convex domain in IR n was obtained by P. L. Lions, N. Trudinger and Urbas [45]. The complex case for Neumann problem, we need some the condition on the minimum λ 1 of the principle curvature of D. Notice that if D is convex then λ 1 0. The following theorem was proved in [40]. THEOREM 4 Let D be a bounded strictly pseudoconvex domain in C n with smooth boundary. Let φ C ( D IR) be such that (2.3.1) φ u γ 0, γ 0 + λ 1 > 0, γ 0 > 0.

6 Then the Neumann boundary value problem (2.1) and (2.3) has a unique plurisubharmonic solution u C (D). Remark: We would like to point out here that the oblique boundary value problem for complex Monge-Ampère equations on a smoothly bounded strictly pseudoconvex domain is still open. 3. Characterizations for balls The characterization for balls in C n+1 is always an interesting subject (see [30], [55], [47], [9], [23],[38], [39] and references therein). In this section, we give several characterizations for ball in C n by using complex Monge-Ampère operators as well as the Webster pseudo scalar curvatures Characterizations involved complex Monge- Ampère operators Let M be a complex manifold of dimension n. Let 0 < R, and assume that there is a biholomorphic map φ : M B(0, R), where B(0, R) is the ball in C n centered at 0 with radius R. Then (3.1.1) τ(z) = φ(z) 2 : M [0, R) is a strictly plurisubharmonic onto map such that (3.1.2) M[log τ] = 0, M \ τ 1 (0). Conversely, the following theorem was proved by Stoll [48] for R = ; by D. Burns [4] for the both R is finite and R is infinite. THEOREM 5 Let M n be an n-dimensional complex Manifold, and let τ : M [0, R) be an onto strictly plurisubharmonic function such that (3.1.2) holds. Then there is a biholomorphic mapping φ : M B(0, R). For the case R <, without loss of generality, we may consider R = 1. Let (3.1.3) ρ(z) = τ(z) 1 = φ(z) 2 1. Then ρ is a negative defining function for M and (3.1.4) J(ρ) = M[ρ] = det φ (z) 2 with log J(ρ) is pluriharmonic in D. Moreover, (3.1.5) log(ρ(z) + 1) = log φ(z) 2 6

7 is plurisubharmonic in M. Conversely, using the Maximum Principle for non-linear PDEs, in [38], Li was able to prove that partial conditions involved (3.1.4) and (3.1.5) is sufficient for characterizing the unit ball in C n. THEOREM 6 Let M n be an n-dimensional complex Manifold. Let ρ be a smooth defining function for M so that u(z) = log( ρ) : M [0, ) is a strictly plurisubharmonic onto map. If log J[ρ] is plurisubharmonic in M and log(1 + ρ) is plurisubharmonic in z near M, then M is biholomorphic to the unit ball B n in C n. In particular, when J(ρ) constant, then there is biholomorphic map φ : M B(0, 1) with a constant Jacobian det φ (z). Remark: In the previous theorem, we assume that u = log( ρ) is plurisubharmonic on M. Then u ij dz i dz j forms a Kähler metric on D. The proof of Theorem holds if we replace the condition: log J(ρ) is plurisubharmonic in M by the condition: log J(ρ) is subharmonic in the metric u ij dz i dz j Pseudo-Hermitian manifolds Let M = {z C n+1 : ρ(z) = 0} be a strictly pseudoconvex real hypersurface with a defining function ρ C 4 (C n+1 ). We use H(M) to denote the holomorphic tangent bundle of complex dimension n. Let (3.2.1) θ = 1 ( ρ(z) ρ(z)). 2i Then θ(x) = 0 for all X H(M). Let {θ 1,, θ n } be a local basis for the cotangent bundle H(M). Then (3.2.2) dθ = i n α,β=1 h αβ θ θ β. The Levi form L θ associated to θ is a bilinear form defined by (3.2.3) L θ (u, v) = idθ(u, v), u, v H(M). It was proved by Webster [52] that there are unique 1-form ωα β (0, 1)-form τ β so that the following hold: and (3.2.4) d θ β = θ α ω β α + θ τ β, (3.2.5) dh αβ h γβ ω γ α h αγ ω γ β = 0

8 8 and (3.2.6) τ α θ α = 0 or A αγ = A γα, where (3.2.7) τ α = n h αβ τ β = β=1 n A αβ θ β. β=1 The curvature 2-form (see [53]) is defined as follows: (3.2.8) Ω α β = dω α β ω γ β ωα γ iθ β τ α + iτ β θ α, where θ β = h βγ θ γ. Let (3.2.9) Ω α = dτ α τ β ω α β. Then (3.2.10) 0 = θ β Ω α β + θ Ω α. Thus (3.2.11) 0 = h γα Ω γ β + h βγω γ α = Ω βα + Ω αβ. Therefore, (3.2.12) Ω αβ = R αβγl θ γ θ l + θ λ αβ, where R αβγl is Webster pseudo curvature tensors. Then (3.2.13) λ αβ + λ βα = 0 and (3.2.14) R αβγl = R βαγl = R γβαl = R βαlγ. Thus, the Webster pseudo Ricci curvature and pseudo scalar curvature are (3.2.15) R αβ = R γl R αβγl, R = h αβ R αβ. For a Kähler manifold, one can easily compute the Ricci curvature through the metric. This is not the case for a pseudo-hermitian CR manifold. With a complex computation, Li and Luk in [43] were able to obtain the following result:

9 THEOREM 7 Let M be a smooth strictly pseudoconvex hypersurface in C n+1. Let U be a neighborhood of M, and let θ = (i/2)( ρ ρ) with ρ C 3 (U) being a defining function for M and J(ρ) > 0 in U. Then for any w, v T 1,0 (M), the Webster pseudo Ricci curvature is given by the formula: (3.2.16) Ric(w, v) = n+1 k,j=1 In particular, if J(ρ) 1 on D then 2 log J(ρ) z k z j w k v j det H(ρ) +(n+1) L θ (w, v). J(ρ) (3.2.17) Ric(w, v) = (n+1) det H(ρ)L θ (w, v), R = n(n+1) det H(ρ). We know that if φ : D B n+1 is a biholomorphic map with ρ(z) = φ(z) 2 1 then R = c n(n + 1). Using Formula (3.2.17) and the main theorem in [38] or Theorem 3.1, the following volume and Webster pseudo scalar curvature comparison theorem was proved in [43]. THEOREM 8 Let D be a bounded strictly pseudoconvex domain in C n+1 with smooth boundary. Let ρ C 3 (D) be a defining function for D with J(ρ) C 2 (D) being positive and log J(ρ) being pluriharmonic in D. Let M = D and θ = ( i/2)( ρ ρ). Let θ 0 = 1 2i ( ρ 0 ρ 0 ) with ρ 0 being the unique plurisubharmonic solution for the Monge-Ampère equation (3.2.18) det H(ρ 0 ) = J(ρ) in D; ρ 0 (z) = z 2 on D. Then (i) If the Webster pseudo scalar curvature R θ satisfies (3.2.19) R θ D θ n(n + 1) D (dθ)n θ 0 (dθ 0 ), n then D must be biholomorphically equivalent to the unit ball in C n+1. (ii) If J(ρ) 1 on D and the Webster pseudo scalar curvature R satisfies (3.2.19) with θ 0 = 1 n+1 2i j=1 (zj dz j z j dz j ) then D must be biholomorphically equivalent to the unit ball in C n+1 having a constant Jacobian biholomorphic map. The CR-Yamabe problem was solved by Jerison and Lee [24], Gamara and Yacoub [20] and Gamara [21]. Which states: Given a compact, integrable, strictly pseudoconvex pseudo-hermitian manifold (M, θ 0 ) in the sense of Webster [52], there is positive smooth function h on M so that (M, hθ 0 ) has constant pseudo scalar curvature. We here are interested in what special pseudo-hermitian form θ with constant pseudo scalar curvature on M will imply M is CR equivalent to S 2n+1. Along this line, the following results are proved by Li in [39]: 9

10 THEOREM 9 Let ρ C 3 (D) be a defining function for D so that u(z) = log( ρ(z)) is strictly plurisubharmonic in D. Let M = D and θ = 1 2i ( ρ ρ). Assume log J(ρ) is harmonic in Kähler metric 2 u z i z j dz i dz j, the following two statements hold: (a) If R θ c > 0, constant on D, then D is biholomorphic to the unit ball B n+1 in C n+1. (b) Webster pseudo scalar curvature 10 det H(ρ) (3.2.20) R θ = n(n + 1) J(ρ) on D. Notice that if u(z) = log( ρ(z)) be the potential function for Kähler-Einstein metric for D, then J(ρ) = 1 on D. By Theorem 1.1 and Theorem 3.1 in [37], we have the following corollary. Corollary 10 Let D be a smoothly bounded strictly pseudoconvex domain in C n+1. Assume that u(z) = log( ρ(z)) is the potential function for the Kähler-Einstein metric for D satisfying (2.5) and θ = 1 2i ( ρ ρ) on M = D. If R θ c > 0, constant on D, then there is a biholomorphic map φ : D B n+1 so that det φ (z) is a constant on D. References [1] E. Bedford and J. Fornaess, Counter examples to regularity for complex Monge-Ampère equation, Invent. Math., 50(1979), [2] E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., 37(1976), [3] M. S. Baouendi, P. Ebenfelt, L. P. Rothschild, Real submanifolds in complex space and their mappings. Princeton Mathematical Series, 47. Princeton University Press, Princeton, NJ, [4] D. Burns, Curvatures of Monge-Ampère foliations and parabolic manifolds, Ann. Math. 115 (1982), [5] Z. Blocki, The complex Monge-Ampère operator in hyperconvex domains, Ann. Scuola Norm. Sup. Pisa Cl. sci., 23(1996), [6] U. Cegrell and S. Kolodziej, The global Dirichlet problems for the complex Monge-Ampère equations, J. Geom. Anal., 9(1999), [7] L. Caffarelli, J. J. Kohn, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, II: complex Monge-Ampère, and uniformly elliptic equations, Comm. on Pure and Appl. Math. (1985), [8] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, Inc. New York, London, Toroto, Tokyo, 1984.

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