The existence of negatively curved metrics on locally conformally flat manifolds with boundary

TL;DR

This paper constructs negatively curved conformal metrics on locally conformally flat manifolds with boundary using Morse functions, and also creates positive Einstein tensor metrics without the conformally flatness assumption.

Contribution

It introduces new methods to produce negatively curved and positive Einstein tensor conformal metrics on manifolds with boundary, expanding the class of such geometries.

Findings

Constructed negatively curved conformal metrics on locally conformally flat manifolds.

Developed a method to produce conformal metrics with positive Einstein tensor without conformal flatness.

Extended the understanding of metric existence under boundary conditions.

Abstract

We use certain Morse functions to construct conformal metrics with negative sectional curvature on locally conformally flat manifolds with boundary. Moreover, without conformally flatness assumption, we also construct conformal metric of positive Einstein tensor.