Prescribing the curvature of Riemannian manifolds with boundary
Abstract
Let $M$ be a compact connected surface with boundary. We prove that the signal condition given by the GaussBonnet 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 DOI:
 10.48550/arXiv.1810.01311
 arXiv:
 arXiv:1810.01311
 Bibcode:
 2018arXiv181001311C
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs
 EPrint:
 21 pages