A uniqueness theorem in Kähler geometry

Abstract

We consider compact Kähler manifolds with their Kähler Ricci tensor satisfying F(Ric) = constant. Under the nonnegative bisectional curvature assumption and certain conditions on F, we prove that such metrics are in fact Kähler-Einstein.

Parabolic complex Monge–Ampère type equations on closed Hermitian manifolds

“…As in [8,10,18], differentiating S α twice and applying the strong concavity of S α [11], we have at point p,…”

On the maximal rank problem for the complex homogeneous Monge–Ampère equation

“…The most famous result along these lines, and in the positive direction, is that of Lempert [17] who proves that on a convex domain in C n the solution to the complex HMAE with prescribed singularity at an interior point (the pluri-complex Green function) is smooth and of maximal rank. The maximal rank problem for other partial differential equations in the complex case has also been taken up by Guan-Li-Zhang [12] and by Li [18].…”

A microscopic convexity principle for nonlinear partial differential equations

“…In higher dimensions, there is extensive literature devoted the sphere theorem of immersed hypersurfaces (e.g., [11,13]). We prove the following sphere theorem, we refer to [17,18,8] for applications in classical and conformal geometry, and refer to [15] for applications in Kähler geometry. Theorem 1.6.…”