On Ricci flows with closed and smooth tangent flows

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-16 DOI:10.1007/s00526-024-02778-6

Pak-Yeung Chan, Zilu Ma, Yongjia Zhang

引用次数: 0

Abstract

In this paper, we consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler (Structure theory of non-collapsed limits of Ricci flows, 2020. arXiv:2009.03243). The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. Among other things, we prove: (1) that in these cases the tangent flow must be unique, (2) that if a Ricci flow with finite-time singularity has a closed singularity model, then the singularity is of Type I and the singularity model is the tangent flow at the singular point; this answers a question proposed in Chow et al. (The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects. Mathematical surveys and monographs, vol 163. AMS, Providence, 2010), (3) a dichotomy theorem that characterizes ancient Ricci flows admitting a closed and smooth backward sequential limit.

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关于具有封闭平稳切线流的利玛窦流

在本文中，我们考虑的是巴姆勒（《利玛窦流非塌缩极限的结构理论》，2020 年，arXiv:2009.03243）意义上的利玛窦流（Ricci flows admitting closed and smooth tangent flows）。对于古老的利玛窦流来说，切向流可以是无穷远处的切向流；对于发展出有限时间奇点的利玛窦流来说，切向流可以是奇点处的切向流。除其他外，我们还证明了：（1）在这些情况下，切向流必须是唯一的；（2）如果具有有限时间奇点的利玛窦流有一个封闭的奇点模型，那么奇点属于 I 型，奇点模型就是奇点处的切向流；这回答了 Chow 等人（《利玛窦流：技术与应用》（The Ricci flow: techniques and applications.第三部分。几何分析方面。Mathematical surveys and monographs, vol 163.AMS, Providence, 2010），(3) 一个二分法定理，描述了古代利玛窦流的特征，它允许一个封闭和光滑的后向序列极限。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.