Embedded constant $m^{\text{th}}$ mean curvature hypersurfaces on spheres

TL;DR

This paper investigates hypersurfaces with constant $m^{ ext{th}}$ mean curvature in spheres, establishing existence results for certain curvature ranges and generalizing previous findings for specific cases.

Contribution

It provides new existence theorems for compact hypersurfaces with constant $H_m$ in spheres, extending known results for particular $m$ values.

Findings

Existence of hypersurfaces for $H_m$ in specified ranges.

Generalization of previous results for $m=1,2,4$.

Conditions linking $H_m$ to geometric parameters.

Abstract

In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature $H_m$ in a unit sphere $S^{n+1}(1)$ and prove that if the $m^{\text{th}}$ mean curvature $H_m$ takes value between $\dfrac{1}{(\tan \frac{\pi}{k})^m}$ and $\frac{k^2-2}{n}(\frac{k^2+m-2}{n-m})^{\frac{m-2}{2}}$ for $1\leq m\leq n-1$ and any integer $k\geq 2$, then there exists at least one $n$-dimensional compact nontrivial embedded hypersurface with constant $H_m>0$ in $S^{n+1}(1)$. When $m=1$, our results reduce to the results of Perdomo \cite{[P]}; when $m=2$ and $m=4$, our results reduce to the results of Cheng-Li-Wei \cite{[WCL]}.