AI 中文总结
该研究针对二维空间形式的测地三角形，证明了其第一非零Neumann特征值的依赖直径的尖锐下界，还推导了非锐角球面三角形的热点定理及等腰球面三角形的相关性质。
AI 中文摘要
我们证明了二维空间形式中给定直径的测地三角形的第一非零Neumann特征值的尖锐下界，该下界由一维模型的第一正径向Neumann特征值给出，可通过退化等腰三角形逼近。当曲率K>0且直径D=π/(2√K)时，等式恰好由双直角三角形达到。我们还证明了直径不超过π/2的非锐角球面三角形的热点定理，并建立了直径为π/2的等腰球面三角形的反对称性和特征值单调性。
英文摘要
We prove a sharp lower bound for the first nonzero Neumann eigenvalue of geodesic triangles of given diameter in two-dimensional space forms. The bound is given by the first positive radial Neumann eigenvalue of an one dimensional model; it is approached by degenerating isosceles triangles. When \(K>0\) and \(D=π/(2\sqrt K)\), equality is attained precisely by birectangular triangles. We also prove a hot-spots theorem for non-acute spherical triangles of diameter at most \(π/2\), and establish antisymmetry and eigenvalue monotonicity for isosceles spherical triangles of diameter \(π/2\).
