B-sub-manifolds and their stability

TL;DR

This paper introduces B-minimal sub-manifolds in Riemannian manifolds, explores their stability, and connects these concepts to mean curvature flow solitons, providing new insights into geometric analysis.

Contribution

It defines B-minimal sub-manifolds via warped products, analyzes their stability, and relates them to known solitons like the grim reaper, extending the understanding of minimal sub-manifolds.

Findings

The grim reaper is stable under symmetric stability.

B-minimal sub-manifolds relate to solitons of mean curvature flow.

Discussion of graphic B-minimal sub-manifolds in Euclidean space.

Abstract

In this paper, we introduce a concept of B-minimal sub-manifolds and discuss the stability of such a sub-manifold in a Riemannian manifold $(M,g)$. Assume $B(x)$ is a smooth function on $M$. By definition, we call a sub-manifold $\Sigma$ {\em B-minimal} in $(M,g)$ if the product sub-manifold $\Sigma\times S^1$ is a {\em minimal} sub-manifold in a warped product Riemannian manifold $(M\times S^1, g+e^{2B(x)}dt^2)$, so its stability is closely related to the stability of solitons of mean curvature flows as noted earlier by G. Huisken, S. Angenent, and K. Smoczyk. We can show that the "grim reaper" in the curve-shortening problem is stable in the sense of "symmetric stable" defined by K. Smoczyk. We also discuss the graphic B-minimal sub-manifold in $R^{n+k}$.