Lu's conjecture for minimal surfaces

TL;DR

The paper resolves Lu's conjecture for minimal submanifolds in spheres by combining spectral analysis of the fundamental matrix with Simons-type identities and normal curvature bounds. It proves the conjecture for minimal $2$-spheres, showing that $\lambda_1=\lambda_2=S/2$ and that the gap above the Calabi model is sharp, with the optimal second gap $5/2$ realized only in the Calabi case $q=4$, $K=1/6$. For general minimal surfaces under mild $\rho^{\perp}$-bounds, it provides explicit gap inequalities for $\max_p(S+\lambda_2)$, together with integral identities and rigidity results including the Clifford torus under certain pinching, and broader inequalities linking $S$, $\lambda_2$ and the normal curvature. Together, these results extend classical gap theorems (Simons, Chern, Peng-Terng) to high codimension and connect Lu's conjecture with Simon's conjecture in dimension two, enriching the landscape of rigidity phenomena for minimal submanifolds in spheres.

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.

Theorems & Definitions (30)

• Conjecture 1.1: Lu's Conjecture Lu
• Conjecture 1.2: Simon's Conjecture Simon2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Lemma 3.1