Berger球面中曲面的第二Jacobi特征值与谱刚性
Second Jacobi eigenvalues and spectral rigidity for surfaces in Berger spheres
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中文总结 AI 辅助
本文为Berger球面中闭曲面第二Jacobi特征值建立上界，结合多种几何工具，并应用于非正欧拉示性数情形，在临界参数处刻画等式条件。
中文摘要 AI 辅助
本文建立了浸入收缩Berger球面中的任意闭双侧曲面的标量Jacobi算子第二特征值的上界。该估计涉及全均曲率平方、欧拉示性数以及由曲面法向与Hopf方向夹角决定的显式非正项。证明结合了将Berger球面实现为复射影平面的测地超曲面、后者到欧氏球面的第一标准嵌入、加权Hersch-Li-Yau平衡论证以及Willmore泛函的共形不变性。不施加极小性或常均曲率假设。作为应用，若$1/3\leq\alpha\leq1$且曲面具有非正欧拉示性数，则其第二Jacobi特征值为非正；当$\alpha>1/3$时严格为负。在临界值$\alpha=1/3$处，等式迫使浸入像与极小Clifford环面全等；在嵌入范畴中，这给出了等式情形的完整刻画。
英文摘要
In this paper, we establish an upper bound for the second eigenvalue of the scalar Jacobi operator of an arbitrary closed two-sided surface immersed in a contracted Berger sphere. The estimate involves the total squared mean curvature, the Euler characteristic, and an explicit nonpositive term determined by the angle between the surface normal and the Hopf direction. The proof combines the realization of a Berger sphere as a geodesic hypersurface of a complex projective plane, the first standard embedding of the latter into a Euclidean sphere, a weighted Hersch--Li--Yau balancing argument, and the conformal invariance of the Willmore functional. No minimality or constant mean curvature assumption is imposed. As an application, if $1/3\leqα\leq1$ and the surface has nonpositive Euler characteristic, then its second Jacobi eigenvalue is nonpositive; it is strictly negative for $α>1/3$. At the critical value $α=1/3$, equality forces the immersed image to be congruent to the minimal Clifford torus; in the embedded category, this yields a complete characterization of the equality case.
发表机构
- Universidade Federal de Alagoas（阿拉戈阿斯联邦大学）
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