A Curvature Gap for Minimal Submanifolds in Spheres

Abstract

Let $F:M^n\to\Sn^{n+q}(1)$ be a closed connected minimal immersion in the unit sphere with second fundamental form $h$, $n\ge3$, $q\ge2$, and $S=|h|^2$. We prove that if $M$ is not totally geodesic, then \[ \max_M S\ge \frac{2n}{3}+\frac{n-2}{6300(39n+8)} \ge\frac{2n}{3}+\frac1{787500}. \]

Disclosure

“6300(39n + 8) 6300 · 125 787500 □ Declaration of generative AI use. During the preparation of this manuscript, the author used OpenAI’s ChatGPT, including its Codex tools, to assist with English-language polishing, LATEX formatting, verification of algebraic computations and constants, and internal- consistency checks. In particular, AI-assisted symbolic and numerical checks helpe”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-23T05:19:44.813869Z

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