Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics

TL;DR

This paper demonstrates that for a given closed Riemannian manifold, one can find a metric deformation that increases the number of constant mean curvature hypersurfaces, with bounds depending on the original metric and hypersurface count.

Contribution

It proves the existence of metric modifications that increase the number of constant mean curvature hypersurfaces while providing explicit bounds on the metric change.

Findings

Existence of a metric h with more c-CMC hypersurfaces than g.

Explicit upper bounds for the L^{(n+1)/2} norm of (g-h).

The number of c-CMC hypersurfaces can be increased via metric deformation.

Abstract

Given a closed Riemannian manifold $(M^{n+1},g)$,$3\leq n+1\leq7$.In this paper,we will prove that for any $c>0$,suppose the number of closed $c-CMC$ hypersurfaces is finite,then there exists a metric $h$ on $M$ such that the $c-CMC$ hypersurfaces in $(M,g)$ are also $c-CMC$ hypersurfaces in $(M,h)$ and the number of $c-CMC$ hypersurfaces in $(M,h)$ is strictly greater than the number of $c-CMC$ hypersurfaces in $(M,g)$.Moreover,we will give a precise upper bound for the $L^{\frac{n+1}{2}}$ norm of $(g-h)$,which depends on the metric $g$ and the number of $c-CMC$ hypersurfaces in $(M,g)$.