# Conformal scalar curvature rigidity on Riemannian manifolds

## TL;DR

This paper investigates conformal scalar curvature rigidity on Riemannian manifolds, proving that under certain boundary conditions, the conformal metric must coincide with the original metric on specific domains, extending previous results.

## Contribution

It extends existing scalar curvature rigidity results to more general Riemannian manifolds with boundary conditions, showing conformal metrics are uniquely determined on certain domains.

## Key findings

- Conformal scalar curvature rigidity holds on some smooth domains.
- The original metric is uniquely determined under boundary conditions.
- Extension of previous rigidity results to broader Riemannian settings.

## Abstract

Let $(M, \bar g)$ be an $n$-dimensional complete Riemannian manifold. In this paper, we considers the following conformal scalar curvature rigidity problem: Given a compact smooth domain $\Omega$ with $\partial \Omega$, can one find a conformal metric $g$ whose scalar curvature $R[g]\ge R[\bar g]$ on $\Omega$ and the mean curvature $H[g] \ge H[ \bar g]$ on $\partial \Omega$ with $\bar g = g$ on $\partial \Omega$? We prove that $\bar g = g$ on some smooth domains in a general Riemannian manifold, which is an extension of the previous results given by Qing and Yuan, and Hang and Wang.