Notes on conformal metrics of negative curvature on manifolds with boundary

TL;DR

This paper develops a method using Morse functions to construct conformal metrics with prescribed curvature properties, proving that any Riemannian metric on a compact 3-manifold with boundary can be conformally transformed into one with negative sectional curvature.

Contribution

It introduces a novel approach using Morse functions to achieve prescribed curvature conditions on conformal metrics of manifolds with boundary.

Findings

Any Riemannian metric on a compact 3-manifold with boundary is conformal to a metric of negative sectional curvature.

The method involves controlling the eigenvalues of the modified Schouten tensor within a specified cone.

The approach provides a new tool for conformal geometry on manifolds with boundary.

Abstract

We use certain Morse functions to construct conformal metrics such that the eigenvalue vector of modified Schouten tensor belongs to a given cone. As a result, we prove that any Riemannian metric on compact 3-manifolds with boundary is conformal to a compact metric of negative sectional curvature.