Scalar-mean rigidity beyond warped product spaces

TL;DR

This paper extends scalar-mean extremality and rigidity theorems to a broader class of Riemannian manifolds with boundary, specifically those conformal to manifolds with nonnegative curvature operator, surpassing previous warped product limitations.

Contribution

It introduces new scalar-mean extremality and rigidity results for conformally related manifolds, expanding the scope beyond warped product spaces with nonnegative curvature.

Findings

Established scalar-mean extremality for conformal manifolds.

Proved rigidity theorems beyond warped product spaces.

Identified new families of extremal and rigid manifolds.

Abstract

The main scalar-mean extremality and rigidity results in the existing literature concern manifolds whose curvature operators are nonnegative, or warped product spaces with a log-concave warping function whose leaves carry metrics of nonnegative curvature operator. In this paper, we establish scalar-mean extremality and rigidity theorems for a broad class of Riemannian manifolds with boundary whose metrics are conformal to ones with nonnegative curvature operator. In particular, our results extend these theorems beyond the warped product setting and yields new families of manifolds exhibiting scalar-mean extremality and rigidity.