Chern conjecture on minimal hypersurfaces

TL;DR

This paper investigates conditions under which complete minimal hypersurfaces in a sphere are classified as either totally geodesic or Clifford tori, based on scalar curvature and second fundamental form bounds.

Contribution

It establishes a new curvature bound that classifies minimal hypersurfaces with constant scalar curvature in spheres as either totally geodesic or Clifford tori.

Findings

Hypersurfaces with $S \,\leq\, 1.8252 n - 0.712898$ are classified.

The classification applies to hypersurfaces with constant scalar curvature and constant $f_3$.

The result advances understanding of the Chern conjecture for minimal hypersurfaces.

Abstract

In this paper, we study $n$-dimensional complete minimal hypersurfaces in a unit sphere. We prove that an $n$-dimensional complete minimal hypersurface with constant scalar curvature in a unit sphere with $f_3$ constant is isometric to the totally geodesic sphere or the Clifford torus if $S\leq 1.8252 n-0.712898$, where $S$ denotes the squared norm of the second fundamental form of this hypersurface.