Scalar curvature rigidity on locally conformally flat manifolds with boundary

Abstract

Inspired by the work of F. Hang and X. Wang and partial results by S. Raulot, we prove a scalar curvature rigitidy result for locally conformally flat manifolds with boundary in the spirit of the well-known Min-Oo conjecture. Our results imply that Min-Oo’s conjecture is true provided the considered manifold is locally conformally flat. In exchange, we require less knowledge on the geometry of the boundary than in the original statement of Min-Oo’s conjecture. Furthermore, our result can be extended to yield a similar rigidity result for geodesic balls in a hemisphere.<br /> Applications of our techniques include rigidity results for more general domains in a hemisphere and geodesic balls in Euclidean space as well as an extension of our result to locally conformally symmetric manifolds. To that end, we additionally establish that our results are valid for manifolds with parallel Ricci tensor, under slightly stronger assumptions on the geometry of the boundary.