Scalar curvature deformation and mass rigidity for ALH manifolds with boundary

TL;DR

This paper investigates scalar curvature deformation on ALH manifolds with boundary, characterizing mass minimizers and establishing positive mass theorem rigidity through boundary data and metric deformation analysis.

Contribution

It introduces new results on local surjectivity of the scalar curvature map and characterizes mass minimizers for ALH manifolds with boundary.

Findings

Scalar curvature map is locally surjective under boundary conditions

Characterization of ALH manifolds minimizing mass integrals

Rigidity results for positive mass theorems

Abstract

We study scalar curvature deformation for asymptotically locally hyperbolic (ALH) manifolds with nonempty compact boundary. We show that the scalar curvature map is locally surjective among either (1) the space of metrics that coincide exponentially toward the boundary, or (2) the space of metrics with arbitrarily prescribed nearby Bartnik boundary data. Using those results, we characterize the ALH manifolds that minimize the Wang-Chru\'sciel-Herzlich mass integrals in great generality and establish the rigidity of the positive mass theorems.