曲率挤压条件下共形 Killing 向量场的 Bochner-Yano 型定理
Bochner-Yano Type Theorems for Conformal Killing Vector Fields under Curvature Pinching Conditions
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中文总结 AI 辅助
研究曲率挤压条件下闭黎曼流形上共形 Killing 向量场，通过建立新 Bochner 型恒等式和 Moser 迭代得梯度估计，证明 Ricci 曲率有界时非平凡共形 Killing 向量场处处非零，推广了相关经典定理。
中文摘要 AI 辅助
本文研究曲率挤压条件下闭黎曼流形上的共形 Killing 向量场。通过为共形 Killing 向量场的 1 - 形式对偶建立新的 Bochner 型恒等式，经 Moser 迭代程序得到精确梯度估计。基于此估计，证明在 Ricci 曲率合适上界下，非平凡共形 Killing 向量场处处非零。进而得出，在非零 Euler 特征的偶数维流形上，共形 Killing 向量场恒为零，这意味着此类流形的共形变换群有限。结果将 Yano 和 Bochner 的经典刚性定理从非正 Ricci 曲率情形推广到小正 Ricci 曲率情形，且将 Chen 和 Han 的近期结果从 Killing 向量场推广到共形 Killing 向量场。
英文摘要
In this article, we investigate conformal Killing vector fields on closed Riemannian manifolds under a curvature pinching condition. By establishing a new Bochner-type identity for the $1$-form dual to a conformal Killing vector field, we derive a sharp gradient estimate via a Moser iteration procedure. Based on this estimate, we prove that, under a suitable upper bound on the Ricci curvature, every nontrivial conformal Killing vector field must be nowhere vanishing. Consequently, on even-dimensional manifolds with non-zero Euler characteristic, every conformal Killing vector field vanishes identically, which in turn implies that the conformal transformation group of such a manifold is finite. Our results extend the classical rigidity theorems of Yano and Bochner from the setting of non-positive Ricci curvature to that of small positive Ricci curvature, and generalize the recent results of Chen and Han from Killing vector fields to conformal Killing vector fields.
