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Scalar-mean rigidity beyond warped product spaces

Jinmin Wang, Zhizhang Xie

TL;DR

This paper addresses extending scalar-mean extremality and rigidity from nonnegative curvature operator and warped-product settings to a broad class of conformal metrics. It develops a spin-geometry approach using Dirac operators and boundary analysis to compare scalar curvature and boundary mean curvature under conformal deformations, deriving an extremality result and, under extra positivity, a rigidity result. The work yields new families of scalar-mean extremal and rigid manifolds beyond warped products, including conformal deformations of Euclidean balls and torus-based constructions. These results connect index theory, conformal geometry, and boundary curvature to expand the catalog of extremal 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.

Scalar-mean rigidity beyond warped product spaces

TL;DR

This paper addresses extending scalar-mean extremality and rigidity from nonnegative curvature operator and warped-product settings to a broad class of conformal metrics. It develops a spin-geometry approach using Dirac operators and boundary analysis to compare scalar curvature and boundary mean curvature under conformal deformations, deriving an extremality result and, under extra positivity, a rigidity result. The work yields new families of scalar-mean extremal and rigid manifolds beyond warped products, including conformal deformations of Euclidean balls and torus-based constructions. These results connect index theory, conformal geometry, and boundary curvature to expand the catalog of extremal 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.

Paper Structure

This paper contains 4 sections, 3 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Scalar-mean extremality
  3. Scalar-mean rigidity
  4. Examples of the model conformal metrics

Key Result

Theorem 1.3

Let $(N^n,\partial N,g)$ be a compact spin Riemannian manifold with boundary. Suppose that $u\in C^\infty(N)$ and $\widetilde{g}=u^2 g$ is a conformal metric. If then $(N,\partial N,\widetilde{g})$ is spin scalar–mean extremal.

Theorems & Definitions (12)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • proof : Proof of Theorem \ref{['thm:comparison']}
  • proof : Proof of Theorem \ref{['thm:rigidity']}
  • Theorem 4.1
  • proof
  • Example 4.2
  • Example 4.3
  • ...and 2 more