Existence and obstructions for the curvature on compact manifolds with boundary

TL;DR

This paper investigates the conditions under which certain curvature functions can be realized on compact manifolds with boundary, providing new existence and nonexistence results based on topological invariants.

Contribution

It establishes that the Gauss-Bonnet sign condition is both necessary and sufficient for prescribed curvature functions in conformal classes, advancing understanding of conformal deformation problems.

Findings

Gauss-Bonnet sign condition is necessary and sufficient

New existence and nonexistence results depend on Euler characteristic

Deep analysis of pointwise conformal deformations

Abstract

We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the geodesic curvature or the Gaussian curvature of some conformally equivalent metric. Our approach allows us to solve problems that are impossible to solve in the pointwise conformal case. Moreover, we obtain a deep and more delicate information on pointwise conformal deformations. We prove new existence and nonexistence results for metrics with prescribed curvature in the conformal setting, which depend on the Euler characteristic.