$L^2$ 至7维标量曲率有界的Ricci流的曲率界

中文摘要 AI 辅助

本文研究标量曲率一致有界的Ricci流，在这类Ricci流上建立了7维及以下的一致$L^2$黎曼曲率界与$L^4$Ricci曲率界。这些估计改进了Bamler建立的$L^{2-\varepsilon}$曲率界，以及Bamler和Zhang建立的4维情形下的$L^2$曲率界。更一般地，我们对满足积分$L^p$Ricci曲率界（对应$p> n/2$）、一致体积下界和强$\varepsilon$正则性假设的$n$流形，建立了一致$L^2$曲率界。该结果推广了Jiang和Naber著名的$L^2$曲率估计，后者是在Ricci曲率的逐点双侧界下建立的。为克服逐点控制的缺失，我们发展了颈分解定理的$L^p$适配版本，并对颈区域的曲率建立了超凸性估计。此外，我们证明了本文的结构假设在标量曲率有界的Ricci流上是动态自然的。作为应用，将本文的估计与Bamler建立的Ricci流先验$L^{4-\varepsilon}$曲率半径界相结合，我们得到了$n \le 7$维标量曲率有界的Ricci流上的一致$L^2$曲率界。

英文摘要

In this paper, we study Ricci flows with uniformly bounded scalar curvature. On such Ricci flows we establish uniform $L^2$ Riemannian curvature bounds and $L^4$ Ricci curvature bounds up to dimension 7. These estimates improves $L^{2-\varepsilon}$ curvature bounds established by Bamler and $L^2$ curvature bounds in dimension 4 established by Bamler and Zhang. More generally, we establish uniform $L^2$ curvature bounds on $n$-manifolds satisfying an integral $L^p$ Ricci curvature bound for $p> n/2$, a uniform volume lower bound, and a strong $\varepsilon$-regularity assumption. This result extends the celebrated $L^2$ curvature estimates of Jiang and Naber, which were established under pointwise two-sided Ricci curvature bounds. To overcome the lack of pointwise control, we develop an $L^p$ adaptation of the neck decomposition theorem and establish a superconvexity estimate for the curvature on neck regions. Furthermore, we demonstrate that our structural assumptions are dynamically natural on Ricci flows with bounded scalar curvature. As an application, by combining our estimate with the a priori $L^{4-\varepsilon}$ curvature radius bound of Ricci flow established by Bamler, we obtain uniform $L^2$ curvature bounds on Ricci flows with bounded scalar curvature of dimension $n \le 7$.

发表机构

• School of mathematical Sciences, Zhejiang University（浙江大学数学科学学院）

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