On perimeter minimizing sets in manifolds with quadratic volume growth

TL;DR

This work proves a rigidity theorem for perimeter-minimizing sets in manifolds with non-negative sectional curvature under quadratic volume growth, showing global isometric splitting $M \cong N \times \mathbb{R}$ and $E \cong N \times [0,\infty)$. The method uses a blow-down analysis in the synthetic setting of $\mathsf{RCD}(0,N)$ spaces, examining tangent cones at infinity that are metric cones, and leverages the Splitting Theorem in this setting. It also identifies the boundary of $E$ as a slice in the product, i.e. $\partial E \cong N \times \{0\}$, providing a clear geometric description. The results extend rigidity phenomena for minimal perimeters beyond stable minimal hypersurfaces, under weaker curvature assumptions and quadratic volume growth, and highlight $\mathsf{RCD}(0,N)$ technology as a robust framework for geometric analysis.

Abstract

This paper studies whether the presence of a perimeter minimizing set in a Riemannian manifold $(M,g)$ forces an isometric splitting. We show that this is the case when $M$ has non-negative sectional curvature and quadratic volume growth at infinity. Moreover, we obtain that the boundary of the perimeter minimizing set is identified with a slice in the product structure of $M$.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.4
• Definition 2.5: Sets of locally finite perimeter
• Definition 2.6
• Proposition 2.7
• Proposition 2.8
• Definition 2.9: Perimeter minimizing sets