# Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary

## TL;DR

This paper establishes rigidity results for certain static spaces on compact manifolds with boundary, characterizing when they are isometric to standard models like the hemisphere, based on curvature conditions.

## Contribution

It introduces a general B"ochner type formula and applies it to classify $V$-static spaces under specific curvature and boundary conditions.

## Key findings

- Positive static triples with connected boundary and positive scalar curvature are isometric to the hemisphere under certain conditions.
- $V$-static spaces with non-negative sectional curvature are classified.
- A new rigidity result for $V$-static spaces with specific curvature assumptions.

## Abstract

We provide a general B\"ochner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero radial Weyl curvature and satisfies a suitable pinching condition. Moreover, we classify $V$-static spaces with non-negative sectional curvature.