度量孤立子的一个间隙定理及其应用
A gap theorem for metric solitons and its applications
浏览论文内容
中文总结 AI 辅助
本文针对$\text{F}$-极限度量孤立子，就渐近体积比（AVR）证明了间隙定理，该定理可应用于Ricci流，推导得到间隙定理与$\text{ε}$-正则性定理，明确了两类Ricci流的正则性性质。
中文摘要 AI 辅助
本文针对$\boldsymbol{\text{F}}$-极限度量孤立子，就渐近体积比（AVR）证明了一个间隙定理：若度量孤立子的AVR足够接近1，则该度量孤立子是欧氏的；这是Wang-Wang结果的度量孤立子对应情形。我们的结果可应用于Ricci流，以推导一个间隙定理和一个$\boldsymbol{\text{ε}}$-正则性定理：（1）具有I型数量曲率界且AVR足够接近1的古老Ricci流必为静态欧氏空间；（2）具有局部I型数量曲率界且局部体积比足够接近1的Ricci流，局部正则性足够好（即其曲率半径不能过小）。
英文摘要
In this paper, we prove a gap theorem for $\mathbb{F}$-limit metric solitons with respect to the asymptotic volume ratio (AVR): if the AVR of a metric soliton is sufficiently close to 1, then the metric soliton is Euclidean; this is a metric-soliton counterpart of Wang-Wang. Our result can be applied to Ricci flows to derive a gap theorem and an $\varepsilon$-regularity theorem: (1) an ancient Ricci flow with a type-I scalar curvature bound and AVR close enough to 1 must be the static Euclidean space, (2) a Ricci flow with locally type-I scalar curvature bound and local volume ratio close enough to 1 must be regular enough locally (in the sense that its curvature radius cannot be too small).
