The sharp diameter bound of stable minimal surfaces

TL;DR

This work proves a sharp diameter bound for complete stable minimal surfaces in complete 3-manifolds with scalar curvature $R\ge1$, establishing $\mathrm{Diam}(\Sigma)<\frac{2\sqrt{6}\pi}{3}$. The main strategy reduces the problem to a general 2D eigenvalue-diameter estimate for surfaces with a curvature potential: if $\lambda_1(-\Delta_{\Sigma}+βK_{\Sigma})\ge\lambda$ with $β>\tfrac14$, then $\mathrm{Diam}(\Sigma)<\frac{2β\pi}{\sqrt{\lambda(4β-1)}}$, proven via a positive eigenfunction and a 1D variational method along a minimizing path, with explicit ODE-based model surfaces achieving sharpness. This 2D result is then lifted to 3D by observing that stable minimal surfaces satisfy $\lambda_1(-\Delta_{\Sigma}+K_{\Sigma})\ge\tfrac12$ when $R\ge1$, yielding the main bound, and the authors construct a sequence of metrics on $S^2\times S^1$ that almost attain equality. The findings partly connect to filling radius and width invariants, situating the result within the broader landscape of curvature-driven diameter and area bounds.

Abstract

For three dimensional complete Riemannian manifolds with scalar curvature no less than one, we obtain the sharp upper bound of complete stable minimal surfaces' diameter.

Figures (1)

• Figure 1: Figure of $\Sigma_k$ as $k\to\infty$.

Theorems & Definitions (10)

• Theorem 1.1
• Conjecture 1.2
• Definition 2.1
• Lemma 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Corollary 2.6
• Lemma 3.1
• Theorem 3.2