发表机构
Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究实空间形式中超曲面的非线性平均曲率谱方程，证明谱参数满足特定条件时的极小性及定量局部刚性，涵盖Chen猜想与双调和极小性情形。
AI 中文摘要
我们研究了欧氏空间、球面空间和双曲空间中超曲面的非线性平均曲率谱方程。对于允许的解析响应$F$和环境截面曲率$c$，我们证明了当谱参数满足$\sigma\ge nc$时的极小性，以及当$\sigma<nc$时在梯度点处的严格平均曲率界。该类别包含所有幂$F(t)=t^{p-1}$（$p>1$）以及满足显式宽度条件的正混合。特别地，该结果证明了欧氏余维一中的Chen猜想和常负曲率中的双调和极小性，无需完备性或完整性假设。一个共同的实解析延拓和焦点迹线论证处理了所有三种几何。显式的脐模型表明无条件谱极小性范围是精确的。独立地，我们证明了在三种几何中紧致非极小测地球面以及球面中适当等半径Clifford超曲面的定量局部刚性（模环境等距），对于在模型振幅附近满足$F,F'>0$的光滑法则。
英文摘要
We study nonlinear mean-curvature spectral equations for hypersurfaces in Euclidean, spherical and hyperbolic space. For an admissible analytic response $F$ and ambient sectional curvature $c$, we prove minimality when the spectral parameter satisfies $σ\ge nc$, and strict mean-curvature bounds at gradient points when $σ<nc$. The class includes every power $F(t)=t^{p-1}$, $p>1$, and positive mixtures satisfying an explicit width condition. In particular, the result proves Chen's conjecture in Euclidean codimension one and biharmonic minimality in constant negative curvature, without completeness or holonomicity. A common real-analytic continuation and focal-trace argument treats all three geometries. Explicit umbilical models show that the unconditional spectral minimality range is sharp. Independently, we prove quantitative local rigidity, modulo ambient isometries, of compact nonminimal geodesic spheres in all three geometries and proper equal-radius Clifford hypersurfaces in spheres, for smooth laws with $F,F'>0$ near the model amplitude.
