Embedded hypersurfaces with constant $m^{\text{th}}$ mean curvature in a unit sphere

TL;DR

This paper constructs new examples of compact embedded hypersurfaces with constant $m^{th}$ mean curvature in a unit sphere, expanding the known class of such geometric objects for various curvature values.

Contribution

It provides explicit constructions of nontrivial embedded hypersurfaces with constant $m^{th}$ mean curvature in spheres, especially for the case of $H_4$ within specific bounds.

Findings

Existence of compact embedded hypersurfaces with constant $H_4$ for certain curvature ranges.

Construction methods for hypersurfaces with prescribed constant $m^{th}$ mean curvature.

Extension of known examples to higher dimensions and specific curvature intervals.

Abstract

In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature in a unit sphere $S^{n+1}(1)$ and construct many compact nontrivial embedded hypersurfaces with constant $m^{\text{th}}$ mean curvature $H_m>0$ in $S^{n+1}(1)$, for $1\leq m\leq n-1$. In particular, if the $4^{\text{th}}$ mean curvature $H_4$ takes value between $\dfrac{1}{(\tan \frac{\pi}{k})^4}$ and $\dfrac{k^4-4}{n(n-4)}$ for any integer $k\geq3$, then there exists an $n$-dimensional ($n\geq 5$) compact nontrivial embedded hypersurface with constant $H_4$ in $S^{n+1}(1)$.