Prescribing the curvature of Riemannian manifolds with boundary

TL;DR

This paper characterizes which curvature functions can be realized on surfaces with boundary by flat or scalar-flat metrics, establishing necessary and sufficient conditions and topological restrictions, and provides a classification of such manifolds.

Contribution

It extends curvature prescription results to manifolds with boundary, providing necessary and sufficient conditions and topological constraints for realizability.

Findings

Gauss-Bonnet condition is necessary and sufficient for flat metrics on surfaces with boundary.

Topological restrictions imply any negative function can be realized as mean or scalar curvature.

Classification theorem for manifolds with boundary based on curvature prescriptions.

Abstract

Let $M$ be a compact connected surface with boundary. We prove that the signal condition given by the Gauss-Bonnet theorem is necessary and sufficient for a given smooth function $f$ on $\partial M$ (resp. on $M$) to be geodesic curvature of the boundary (resp. the Gauss curvature) of some flat metric on $M$ (resp. metric on $M$ with geodesic boundary). In order to provide analogous results for this problem with $n\geq 3,$ we prove some topological restrictions which imply, among other things, that any function that is negative somewhere on $\partial M$ (resp. on $M$) is a mean curvature of a scalar flat metric on $M$ (resp. scalar curvature of a metric on $M$ and minimal boundary with respect to this metric). As an application of our results, we obtain a classification theorem for manifolds with boundary.