Finite Diffeomorphism Types of Four Dimensional Ricci Flow with Bounded Scalar Curvature

Wen Shuai JIANG

数学学报(英文) ›› 2021, Vol. 37 ›› Issue (11) : 1751-1767.

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数学学报(英文) ›› 2021, Vol. 37 ›› Issue (11) : 1751-1767. DOI: 10.1007/s10114-021-0149-4

Finite Diffeomorphism Types of Four Dimensional Ricci Flow with Bounded Scalar Curvature

    Wen Shuai JIANG
作者信息 +

Finite Diffeomorphism Types of Four Dimensional Ricci Flow with Bounded Scalar Curvature

    Wen Shuai JIANG
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摘要

In this paper, we consider Ricci flow on four dimensional closed manifold with bounded scalar curvature, noncollasping volume and bounded diameter. Under such conditions, we can show that the manifold has finitely many diffeomorphism types, which generalizes Cheeger-Naber's result to bounded scalar curvature along Ricci flow. In particular, this implies the manifold has uniform L2 Riemann curvature bound. As an application, we point out that four dimensional Ricci flow would not have uniform scalar curvature upper bound if the initial metric only satisfying lower Ricci curvature bound, lower volume bound and upper diameter bound.

Abstract

In this paper, we consider Ricci flow on four dimensional closed manifold with bounded scalar curvature, noncollasping volume and bounded diameter. Under such conditions, we can show that the manifold has finitely many diffeomorphism types, which generalizes Cheeger-Naber's result to bounded scalar curvature along Ricci flow. In particular, this implies the manifold has uniform L2 Riemann curvature bound. As an application, we point out that four dimensional Ricci flow would not have uniform scalar curvature upper bound if the initial metric only satisfying lower Ricci curvature bound, lower volume bound and upper diameter bound.

关键词

Ricci flow / scalar curvature

Key words

Ricci flow / scalar curvature

引用本文

导出引用
Wen Shuai JIANG. Finite Diffeomorphism Types of Four Dimensional Ricci Flow with Bounded Scalar Curvature. 数学学报(英文版), 2021, 37(11): 1751-1767 https://doi.org/10.1007/s10114-021-0149-4
Wen Shuai JIANG. Finite Diffeomorphism Types of Four Dimensional Ricci Flow with Bounded Scalar Curvature. Acta Mathematica Sinica, 2021, 37(11): 1751-1767 https://doi.org/10.1007/s10114-021-0149-4

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基金

Supported by NSFC (Grant No. 11701507) and the Fundamental Research Funds for the Central Universities 2019QNA3001 and DECRA 190101471

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