Compact K$\"a$hler manifolds homotopic to negatively curved Riemannian manifolds

TL;DR

This paper proves that compact Kähler manifolds homotopic to negatively curved Riemannian manifolds admit Kähler-Einstein metrics of general type and that certain holomorphic curves cannot exist on related symplectic manifolds.

Contribution

It establishes the existence of Kähler-Einstein metrics on manifolds homotopic to negatively curved Riemannian manifolds and rules out non-constant holomorphic curves under specific conditions.

Findings

Existence of Kähler-Einstein metrics of general type.

Non-existence of non-constant J-holomorphic entire curves.

Results apply to manifolds homotopic to negatively curved Riemannian manifolds.

Abstract

In this paper, we show that any compact K$\"a$hler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K$\"a$hler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure $J$ compatible with the symplectic form, there is no non-constant $J$-holomorphic entire curve $f:C \rightarrow X$.