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Scalar-mean rigidity theorem for compact manifolds with boundary

Jinmin Wang, Zhichao Wang, Bo Zhu

TL;DR

The paper proves a scalar-mean rigidity theorem for compact $n$-manifolds with boundary in dimensions $n\in\{2,3,4\}$, extending Schoen--Yau dimension reduction to control mean curvature alongside scalar curvature. By developing capillary $\mu$-bubble methods and a dimension-reduction argument, the authors show that nonnegative scalar curvature together with positive boundary mean curvature, plus a nonzero-degree, distance-non-increasing boundary map to $\mathbb{S}^{n-1}$, forces rigidity: the manifold is isometric to the Euclidean disk and the boundary map is an isometry; Shi--Tam inequalities are employed to certify equality. They also derive sharp extremality results for spherical radius and best $\mathrm{NNSC}$ fill-ins, establish a Listing-type comparison, and prove a Lipschitz rigidity theorem, with the latter requiring an oriented-trace formulation of the boundary map. The results hold without spin assumptions in these dimensions and highlight how mean curvature and scalar curvature interact under dimension reduction, while noting higher-dimensional extensions require improved regularity of capillary hypersurfaces. Overall, the work advances non-spin rigidity theory in low dimensions and clarifies limits and connections with existing spin-based results.

Abstract

We prove a scalar-mean rigidity theorem for compact Riemannian manifolds with boundary in dimension less than five by developing a dimension reduction argument for mean curvature, which extends Schoen-Yau's dimension reduction argument for scalar curvature. As a corollary, we prove the sharp spherical radius rigidity theorem and best NNSC fill-in in terms of the mean curvature. Moreover, we prove a Lipschitz Listing type scalar-mean rigidity theorem for these dimensions.

Scalar-mean rigidity theorem for compact manifolds with boundary

TL;DR

The paper proves a scalar-mean rigidity theorem for compact -manifolds with boundary in dimensions , extending Schoen--Yau dimension reduction to control mean curvature alongside scalar curvature. By developing capillary -bubble methods and a dimension-reduction argument, the authors show that nonnegative scalar curvature together with positive boundary mean curvature, plus a nonzero-degree, distance-non-increasing boundary map to , forces rigidity: the manifold is isometric to the Euclidean disk and the boundary map is an isometry; Shi--Tam inequalities are employed to certify equality. They also derive sharp extremality results for spherical radius and best fill-ins, establish a Listing-type comparison, and prove a Lipschitz rigidity theorem, with the latter requiring an oriented-trace formulation of the boundary map. The results hold without spin assumptions in these dimensions and highlight how mean curvature and scalar curvature interact under dimension reduction, while noting higher-dimensional extensions require improved regularity of capillary hypersurfaces. Overall, the work advances non-spin rigidity theory in low dimensions and clarifies limits and connections with existing spin-based results.

Abstract

We prove a scalar-mean rigidity theorem for compact Riemannian manifolds with boundary in dimension less than five by developing a dimension reduction argument for mean curvature, which extends Schoen-Yau's dimension reduction argument for scalar curvature. As a corollary, we prove the sharp spherical radius rigidity theorem and best NNSC fill-in in terms of the mean curvature. Moreover, we prove a Lipschitz Listing type scalar-mean rigidity theorem for these dimensions.
Paper Structure (10 sections, 18 theorems, 135 equations, 2 figures)

This paper contains 10 sections, 18 theorems, 135 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preparations on the capillary $\mu$-bubble
  3. Scalar-mean extremality
  4. The proof of the main theorems
  5. Scalar-mean rigidity theorem
  6. Listing type scalar-mean comparison theorem
  7. Lipschitz scalar-mean rigidity
  8. Capillary mu-bubble and its variation
  9. Maximum principal of the capillary mu-bubble
  10. Shi-Tam inequality and its extension

Key Result

Theorem 1.1

Suppose that $(M^n,g)$ is a closed, smooth, spin Riemannian manifold and $F\colon (M,g)\to(\mathbb{S}^n,g_{\mathbb{S}^n})$ is a smooth map of $\deg(F) \neq 0$$f_*([M]) = \deg(F) [\mathbb{S}^n]$.

Figures (2)

  • Figure I: $\mu$ -bubble setup
  • Figure II: Capillary $\mu$-bubble setup

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.5: Listing-type scalar-mean rigidity theorem
  • Remark 1.6
  • Theorem 1.7
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 27 more