Locally Maximizing Metric of Width on Manifolds with Boundary

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].