A topological rigidity theorem on noncompact Hessian manifolds

TL;DR

This paper establishes a topological rigidity result for noncompact Hessian manifolds with nonnegative Hessian sectional curvature, showing they are diffeomorphic to Euclidean space under certain volume growth conditions, extending previous results.

Contribution

It introduces a new geometric flow approach to construct Hessian metrics and proves a rigidity theorem for noncompact Hessian manifolds with maximal volume growth.

Findings

Constructed Hessian metrics with nonnegative bounded Hessian sectional curvature.

Proved that such manifolds with maximal volume growth are diffeomorphic to ^n.

Extended previous rigidity theorems to a broader class of Hessian manifolds.

Abstract

In this work, we obtain a short time solution for a geometric flow on noncompact affine Riemannian manifolds. Using this result, we can construct a Hessian metric with nonnegative bounded Hessian sectional curvature on some Hessian manifolds with nonnegative Hessian sectional curvature. Our results can be regarded as a real version of Lee-Tam \cite{LT20}. As an application, we prove that a complete noncompact Hessian manifold with nonnegative Hessian sectional curvature is diffeomorphic to $\mathbb{R}^n$ if its tangent bundle has maximal volume growth. This is an improvement of Theorem 1.3 in Jiao-Yin \cite{JY25}.