Volume gap between the minimal submanifold and the unit sphere

TL;DR

This paper improves the volume gap between minimal submanifolds and the unit sphere by refining coefficients and applying eigenvalue estimates, advancing understanding in geometric analysis.

Contribution

It introduces modifications to existing volume gap estimates and employs Cheng-Yang eigenvalue bounds to enlarge the gap.

Findings

Enhanced volume gap estimates for minimal submanifolds.

Application of Cheng-Yang eigenvalue bounds to geometric analysis.

Refined coefficients lead to sharper geometric inequalities.

Abstract

In this paper, following the method of Cheng-Li-Yau, we first modify the coefficients in the constant $B_n$ to improve the volume gap. Further, we also enlarge our gap by applying an estimate of Cheng-Yang for eigenvalues of Laplacian.