Locally Maximizing Metric of Width on Manifolds with Boundary

TL;DR

This paper employs min-max theory to establish the existence of a sequence of properly embedded free boundary minimal hypersurfaces in compact manifolds with boundary, under specific metric maximization conditions.

Contribution

It extends previous results by demonstrating the existence of equidistributed FBMHs assuming the metric is a local maximizer of the width in its conformal class.

Findings

Existence of a sequence of properly embedded FBMHs.

Under the assumption of metric being a local maximizer of width.

Extension of Ambrozio-Montezuma's results.

Abstract

In this paper we use min-max theory to study the existence free boundary minimal hypersurfaces (FBMHs) in compact manifolds with boundary $(M^{n+1}, \partial M, g)$, where $2\leq n\leq 6$. Under the assumption that $g$ is a local maximizer of the width of $M$ in its comformal class, and all embedded FBMHs in $M$ are properly embedded, we show the existence of a sequence of properly embedded equidistributed FBMHs. This work extends the result of Ambrozio-Montezuma [2].