Curvature Deformations on Complete Manifolds with Boundary

Abstract

This paper investigates conformal deformations of the scalar curvature and mean curvature on complete Riemannian manifolds with boundary. We establish sufficient conditions for the existence of conformal deformations to complete metrics with positive scalar curvature and mean convex boundary. Building upon these results, we explore further deformation scenarios, including those that increase the mean curvature. Finally, we consider the case of deforming complete manifolds with negative scalar curvature on manifolds with noncompact boundary.

This work represents the second part of the initial version of arXiv:2204.03582, which has been subsequently divided into two parts. In this revised version, we have significantly improved the removed section and have also included the case of negative curvature. All comments are welcome!