Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics
Xiaoxiang Jiao, Wenduo Zou
TL;DR
The paper addresses how many $c$-CMC hypersurfaces can appear on a closed $(n+1)$-manifold when perturbing the ambient metric, within the framework of $c$-CMC min-max theory. It introduces the $\mathcal{A}^{c}$-min-max value $W^{c}(M,g)$ and proves its continuous dependence on the metric; it then proves that for any $g$ and any $c>0$, if the number of $c$-CMC hypersurfaces is finite, one can find a nearby metric $h$ with strictly more such hypersurfaces, with an explicit bound on $\|g-h\|_{L^{\frac{n+1}{2}}}$ in terms of $W^{c}(M,g)$, $c$, and counts. The results include corollaries under positive Ricci curvature and the existence of sequences of metrics that yield arbitrarily many $c$-CMC hypersurfaces. Overall, the work advances multiplicity theory for CMC hypersurfaces by linking metric perturbations to the growth of $c$-CMC families and giving quantitative control on the perturbation.
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 $(M,g)$.