A C-estimate for the Parabolic Monge-ampère Equation on Complete Non-compact Kähler Manifolds

In this article we study the Kähler Ricci flow, the corresponding parabolic Monge Ampère equation and complete noncompact Kähler Ricci flat manifolds. In our main result Theorem 1 we prove that if (M, g) is sufficiently close to being Kähler Ricci flat in a suitable sense, then the Kähler-Ricci flow (1) has a long time smooth solution g(t) converging smoothly uniformly on compact sets to a complete Kähler Ricci flat metric on M . The main step is to obtain a uniform C-estimates for the corresponding parabolic Monge Ampère equation. Our results on this can be viewed as a parabolic version of the main results in [10] on the elliptic Monge Ampère equation. 1. Let (M, g0) be a complete non-compact Kähler manifold with complex dimension n and consider the following Kähler-Ricci flow on M : (1) { ∂gi̄ ∂t = −Ri̄ gi̄(x, 0) = (g0)i̄ We are interested in studying when (1) admits a long time solution g(t) converging smoothly on M to a complete Kähler metric g(∞). Such a limit g(∞) must be Kähler Einstein with zero scalar curvature by (1). We are thus interested in studying when (M, g0) converges to a Kähler Ricci flat metric under (1). When M is compact Cao [3] established that a necessary and sufficient condition for such convergence is that: (2) (R0)i̄ = (f0)i̄ Date: July 2008. Research partially supported by NSERC grant no. #327637-06. Research partially supported by Hong Kong RGC General Research Fund #GRF 2160357. 1 2 Albert Chau and Luen-Fai Tam for a smooth potential function f0 on M , thus re-establishing the famous Calabi Conjecture first proved by Yau [11]. In Theorem 1 we establish a non-compact version of Cao’s result. We prove that when (M, g0) is complete, non-compact with bounded curvature, with volume growth Vx0(r) ≤ Cr 2n for some x0 and C for all r, and satisfyies a Sobolev inequality, then: Under the above conditions, the Kähler-Ricci flow (1) has a long time solution g(t) converging smoothly on M provided (2) is satisfied and |f0|(x) ≤ C 1+ρ 0 (x) for some C, ǫ > 0 and all x. See Theorem 1 for details. The result is motivated by the work of Tian-Yau [10] who proved the existence of a Kähler Ricci flat metric in the compliment of a divisor in a projective variety under certain conditions. They first constructed a Kähler metric satisfying the condition in Theorem 1 and then solve the elliptic complex Monge Ampère equation to obtain a Kähler Ricci flat metric. Our results here can be viewed as a parabolic version of the results on the elliptic Monge Ampère equation in [10]. We also refer to [1, 3] for results on convergence of the Kähler-Ricci flow to Kähler Eienstein metrics with negative scalar curvature. In [3], it was proved that (1) converges after rescaling to a Kähler Eienstein metric with negative scalar curvature provided (R0)i̄ + (g0)i̄ = (f0)i̄ for smooth f0. A non-compact version of this result was proved in [1]. 2. Let (M, g0) be a complete non-compact Kähler manifold with complex dimension n such that (2) holds for some smooth potential f0 on M . We want to study the long time behavior of the KählerRicci flow (1), when f0 is sufficiently close to zero in a suitable sense. Note that (M, g0) is Kähler Einstein with zero scalar curvature when f0 = 0. We are thus interested in the behavior of the Kähler-Ricci flow on complete Kähler manifolds which are close to being Kähler Einstein. We will prove the following: Theorem 1. Let (M, g0) be a complete non-compact Kähler manifold with bounded curvature and n ≥ 3. Assume the following: (a) Condition (2) holds for smooth f0 satisfying: (3) |f0|(x) ≤ C1 1 + ρ 0 (x) for some C1, ǫ > 0, and all x ∈ M where ρ0(x) is the distance function from a fixed p ∈ M C -estimate for the parabolic complex Monge-Ampère equation 3 (b) The following Sobolev inequality is true: (4) ( ∫ M |φ| 2n dV0 ) n−1 n ≤ C2 ∫ M |∇0φ| 2 for some C2 > 0 and all φ ∈ C ∞ 0 (M). (c) There exists constant C3 > 0 such that (5) V0(r) ≤ C3r 2n for some C3 > 0 and all r where V0(r) is the volume of the geodesic ball with radius r centered at some p ∈ M . Then (1) has long time solution g(t) converging uniformly on compact sets in the C∞ topology to a complete Kähler Ricci flat metric g∞ on M . We will also consider the following parabolic Monge Ampère equation corresponding to (1):