乘积复射影空间上Poincaré常数与对数Sobolev上界的显式计算与尺度优化

Explicit Computation and Scale Optimization of the Poincaré Constant and Logarithmic Sobolev Upper Bounds on Products of Complex Projective Spaces

摘要: 本文研究缩放乘积复射影空间$\left(M,g_a\right)$ 上的Poincaré常数与对数Sobolev常数。以凯勒Bochner-Beckner不等式、复射影空间显式谱及熵张量化为工具，计算Ricci下界和第一非零特征值，证明凯勒Bochner-Poincaré估计给出的常数上界在整个尺度族上均达到等号；构造保留各因子信息的对数Sobolev张量化上界，并证明其严格优于整体Ricci上界。进一步，在固定体积下求得该上界的唯一极小尺度及体积保持扰动的尖点行为，并通过四类模型给出比例与渐近结果。

Abstract: This paper investigates the Poincaré constant and the logarithmic Sobolev constant on the scaled product of complex projective spaces $\left(M,g_a\right)$ . Using the Kähler Bochner-Beckner inequality, the explicit spectrum of complex projective spaces, and entropy tensorization, we compute the Ricci curvature lower bound and the first nonzero eigenvalue, and prove that the upper bound for the Poincaré constant obtained by the Bochner method is attained sharply throughout the entire family of scaled metrics. We further construct a tensorized upper bound for the logarithmic Sobolev constant that preserves the geometric information of each factor and prove that it is strictly sharper than the upper bound derived from the global Ricci curvature lower bound. Moreover, under a fixed-volume constraint, we determine the unique scale minimizing this upper bound and analyze its cusp-type behavior under volume-preserving perturbations. Finally, ratio comparisons and asymptotic results are obtained for four classes of model spaces.