正里奇曲率下尖锐梯度估计的刚性与体积挤压
Rigidity and volume pinching for the sharp gradient estimate in positive Ricci curvature
- Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对正里奇曲率流形，证明了其尖锐梯度估计的几乎刚性，建立梯度亏格与体积亏格的定量控制关系，还得到爱因斯坦流形的间隙定理及闭4维爱因斯坦流形的刚性定理。
AI中文摘要：
Colding建立了非负里奇曲率流形上格林函数的尖锐梯度估计，Manea近期将其推广到正里奇曲率情形。我们证明了正里奇曲率下这类梯度估计的几乎刚性：梯度亏格的平均值，即∫(1-|∇b|²)dV，以定量方式控制体积亏格，其中b是利用格林函数定义的类距离函数。这意味着对满足Ric≥(n-1)g的流形成立定量几乎刚性定理，且存在爱因斯坦流形的间隙定理；我们还通过发现新的单调性公式，证明了闭4维爱因斯坦流形的刚性定理。
英文摘要：
Colding established a sharp gradient estimate for the Green function on manifolds with nonnegative Ricci curvature, which was extended to positive Ricci curvature by Manea recently. We prove almost rigidity of such gradient estimates for positive Ricci curvature. We show that the average of the gradient deficit, namely $\fint (1-|\nabla b|^2)\,dV$, controls the volume deficit in a quantitative manner; here $b$ is the distance-like function defined using the Green function. This implies a quantitative almost rigidity theorem for manifolds with $\mathrm{Ric}\ge(n-1)g$ and a gap theorem for Einstein manifolds. We also prove a rigidity theorem for closed 4-dimensional Einstein manifolds by finding a new monotonicity formula.
