Lu's conjecture for minimal surfaces

TL;DR

This paper proves Lu's conjecture for minimal 2-spheres and certain minimal surfaces, unifying gap theorems for submanifolds in spheres and extending previous results.

Contribution

It confirms Lu's conjecture for specific minimal surfaces, broadening the understanding of scalar curvature gaps in submanifold geometry.

Findings

Proved Lu's conjecture for minimal 2-spheres.

Established results for minimal surfaces under certain curvature conditions.

Unified gap theorems for high-codimensional submanifolds.

Abstract

After Chern's conjecture on the discreteness of the constant scalar curvatures of compact minimal submanifolds $M^n$ in unit spheres $\mathbb{S}^{n+q}$, Z. Q. Lu proposed a conjecture regarding the second gap, based on his ingenious refinement of the known first gap theorem. This refinement unifies Simons' first gap theorem for hypersurfaces with the corresponding theorems for high-codimensional submanifolds established by Yau, Shen, Li and Li, among others. In this paper, for arbitrary codimension, we prove Lu's conjecture for minimal 2-spheres, and for any minimal surfaces under some slight inequality conditions about the normal scalar curvature.