AI 中文总结
研究二维球面上高斯曲率下界为1的光滑黎曼度量的正拉普拉斯特征值，通过建立谱计数比较，得出特征值相关结论，并应用于完备三维流形上多项式增长调和函数空间，得到维数界和刚性结果。
AI 中文摘要
对于二维球面上高斯曲率下界为1的每一个光滑黎曼度量，我们证明每个正拉普拉斯特征值（计重数）不小于单位圆球的相应特征值。有序谱中任何正位置处的等式迫使该度量与单位圆度量等距。我们还为曲率下界为1的亚历山德罗夫二维球建立了一个精确的有限谱计数比较。在单位圆球的每个正谱阈值处，小于或等于该阈值的拉普拉斯特征值数量（计重数）不超过圆球的相应数量。任何此类阈值处的等式迫使亚历山德罗夫球与单位圆球等距。作为应用，我们得到了具有非负截面曲率和正渐近体积比的完备三维流形上多项式增长调和函数空间的精确欧几里得维数界以及等式情形下的刚性。
英文摘要
For every smooth Riemannian metric on the two sphere whose Gaussian curvature is bounded below by one, we prove that each positive Laplace eigenvalue, counted with multiplicity, is no smaller than the corresponding eigenvalue of the unit round sphere. Equality at any positive position in the ordered spectrum forces the metric to be isometric to the unit round metric. We further establish a sharp finite spectral counting comparison for Alexandrov two spheres with curvature bounded below by one. At every positive spectral threshold of the unit round sphere, the number of Laplace eigenvalues below or at that threshold, counted with multiplicity, does not exceed the corresponding number for the round sphere. Equality at any such threshold forces the Alexandrov sphere to be isometric to the unit round sphere. As an application, we obtain the sharp Euclidean dimension bound for spaces of polynomial growth harmonic functions on complete three dimensional manifolds with nonnegative sectional curvature and positive asymptotic volume ratio, together with rigidity in the equality case.
