AI 中文总结
该研究在正曲率4维流形中，证明满足特定曲率条件的稳定极小超曲面必为全测地，还构造了反例，相关刚性证明结合了谱分裂理论等方法。
AI 中文摘要
设$M^3\to X^4$是一个完备、连通、双侧稳定极小浸入。我们证明，若环境截面曲率非负且环境标量曲率有正的一致下界，则$M$是全测地的，且其法里奇曲率为零。该证明未施加任何弱有界几何假设或上曲率界。我们还在$\mathbb{R}^4$上构造了一个具有严格正截面曲率的完备度量，该度量容许一个完备、嵌入、单端、非抛物、双侧稳定极小超曲面，其微分同胚于$\mathbb{R}^3$且非全测地。刚性证明结合了谱分裂理论、扭曲$\mu$-泡构造和调和函数水平集论证。该例子通过对文献[CLS]中的例子进行紧支集变形得到。
英文摘要
Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption and no upper curvature bound are imposed. We also construct a complete metric of strictly positive sectional curvature on $\mathbb{R}^4$ admitting a complete, embedded, one-ended, nonparabolic, two-sided stable minimal hypersurface diffeomorphic to $\mathbb{R}^3$ which is not totally geodesic. The rigidity proof combines spectral splitting theory, a warped $μ$-bubble construction, and a harmonic function level set argument. The example is obtained by a compactly supported deformation of an example in \cite{CLS}.
Commentspages 19
