Ricci pinching on complete Kahler manifolds

Abstract

We study Ricci pinching problem on complete Kähler manifolds. For complete noncompact manifolds with Euclidean volume growth, we prove flatness under nonnegative bisectional curvature and Ricci pinching, and Ricci flatness under nonnegative Ricci curvature, quadratic curvature decay, and the same pinching condition. In both cases, the pinching constant may tend to zero at a logarithmic rate. Examples show that the logarithmic exponent is sharp in complex dimension two. Second part of the paper proves that a complete Kähler manifold of complex dimension at least two with positive holomorphic bisectional curvature is compact if its sectional curvature is nonnegative and its Ricci curvature is uniformly pinched outside a compact set. No volume-growth assumption or curvature upper bound is needed. Heat smoothing of the Busemann function supplies a strictly plurisubharmonic exhaustion with bounded gradient, so the general top Ricci-mass theorem of Datar--Pingali--Seshadri applies. The new ingridents are integral estimates for the Ricci form. The analytic arguments use scalar curvature gap theorems and a critical-$L^n$ vanishing theorem for closed real $(1,1)$-forms under nonnegative Ricci curvature and nonnegative quadratic orthogonal bisectional curvature. The Hodge-Laplace heat eqution deformation conserves the signed top-degree integral, including its initial value, while norm domination and scalar heat-semigroup decay force this integral to vanish. For semipositive initial forms this gives pointwise vanishing of the top wedge power.