Remarks on minimal hypersurfaces in gradient shrinking Ricci solitons

TL;DR

This work investigates compact two-sided stable minimal hypersurfaces in gradient shrinking Ricci solitons with a lower scalar curvature bound. Using stability inequalities and the μ-bubble approach, it proves rigidity: for dimensions $3\le n\le 7$ and $R\ge (n-1)\lambda$, any such hypersurface is totally geodesic with vanishing normal Ricci curvature, and, in the area-minimizing case, yields a local (and under constancy of scalar curvature, global) splitting $M \cong \Sigma \times \mathbb{R}$ with $\Sigma$ Einstein. The explicit product example $\mathbb{S}^{n-1} \times \mathbb{R}$ with a suitable potential shows sharpness, and the work ends with an open question on higher codimension.

Abstract

In this paper, we prove that any compact 2-sided smooth stable minimal hypersurface in gradient Ricci soliton $(M^{n},g,f)$ with scalar curvature $R\geq(n-1)λ$ must have vanished second fundamental form and vanished normal Ricci curvature. For shrinking gradient Ricci solitons with scalar curvature $R\geq(n-1)λ$, the existence of an area-minimizing hypersurface would imply $M$ is splitting.

Theorems & Definitions (14)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3: Theorem 3.1WM2023
• Theorem 1.4: Theorem 3.2WM2023
• Definition 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2: Existence of $\mu$-bubble
• Lemma 2.3: First and second variation of $\mu$-bubble