Nicola Gigli
We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting to define Hessian, covariant/exterior derivatives and Ricci curvature.
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Comparison estimates on nonsmooth spaces with integrable Ricci lower bounds via localization
本文通过局部化方法，在具有可积 Ricci 下界的度量测度空间上推广了 Bishop-Gromov 比较、Myers 直径估计和 Cheng 比较原理，获得了几何与函数不等式。
There are two primary goals to this paper. In the first part of the paper we study smooth metric measure spaces (M^n,g,e^{-f}dv_g) and give several ways of characterizing bounds -Kg\leq \Ric+\nabla^2f\leq Kg on the Ricci curvature of the manifold. In particular, we see how bounded Ricci curvature on M controls the analysis of path space P(M) in a manner analogous to how lower Ricci curvature controls the analysis on M. In the second part of the paper we develop the analytic tools needed to in order to use these new characterizations to give a definition of bounded Ricci curvature on general me…
Collapse to nonsmoothable manifolds under lower sectional and positive Ricci curvature bounds
构造截面曲率有下界、正里奇曲率下界的坍缩序列，极限为不可光滑流形，并研究其拓扑与几何性质。
We prove that `volume cone implies metric cone' in the setting of RCD spaces, thus generalising to this class of spaces a well known result of Cheeger-Colding valid in Ricci-limit spaces.
We consider Ricci flow on a closed surface with cone points. The main result is: given a (nonsmooth) cone metric g_0 over a closed surface there is a smooth Ricci flow g(t) defined for (0,T], with curvature unbounded above, such that g(t) tends to g_0 as t tends to 0. This result means that Ricci flow provides a way for instantaneously smoothening cone points. We follow an argument of P. Topping modifying his reasoning for cusps of negative curvature; in that sense we can consider cusps as a limiting zero-angle cone, and we generalize to any angle between 0 and 2π.
In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of t…
Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary
本文在具有低里奇曲率下界的非光滑度量测度空间中证明了平均凸边界子集内切半径的锐上界，推广了Kasue和Li的光滑情形结果，并给出稳定性及RCD情形下的等号刻画。
Isoperimetry on manifolds with Ricci bounded below: overview of recent results and methods
本文综述了Ricci下界流形上等周问题的最新进展，重点介绍等周轮廓二阶微分不等式及其证明方法，并展示了其在等周不等式和等周集存在性等方面的应用，同时指出了若干开放问题。
On the uniqueness of Ricci flows from certain Alexandrov spaces
本研究证明从奇异Reifenberg空间出发、满足尺度不变曲率界等条件的Ricci流唯一性，并据此在极限空间上规范定义光滑结构，推广了Perelman稳定性定理。
By using optimal mass transport theory we prove a sharp isoperimetric inequality in ${\sf CD} (0,N)$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for normed spaces and Riemannian manifolds to a nonsmooth framework. In the case of $n$-dimensional Riemannian manifolds with nonnegative Ricci curvature, we outline an alternative proof of the rigidity result of S. Brendle (2021). As applications of the isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp const…
Causal convergence conditions through variable timelike Ricci curvature bounds
该研究提出非光滑整体双曲正则长度度量时空概念，引入带可变Ricci曲率界的熵类时曲率维数条件，统一广义相对论能量条件，推广相关几何估计与奇点定理，给出稳定性结果与类时几何不等式。
An overview of the stability of Sobolev inequalities on Riemannian manifolds with Ricci lower bounds
本文综述了具有Ricci曲率下界的黎曼流形上Sobolev不等式稳定性的最新进展，重点介绍了光滑与非光滑几何方法的结合，并针对非负Ricci曲率和欧氏体积增长情形给出了自包含的稳定性证明概述，最后讨论了相关结果和开放问题。
Displacement convexity of Boltzmann's entropy characterizes the strong energy condition from general relativity
本文将黎曼流形中里奇曲率下界与熵的测地凸性之间的对应关系推广到洛伦兹时空，证明 Hawking-Penrose 强能量条件等价于 Boltzmann-Shannon 熵在洛伦兹距离分数次幂诱导的概率测度空间中的测地凸性，为在非光滑时空中研究强能量条件奠定基础。
This is the first of a series of papers, where we introduce a new class of estimates for the Ricci flow, and use them both to characterize solutions of the Ricci flow and to provide a notion of weak solutions to the Ricci flow in the nonsmooth setting. In this first paper, we prove various new estimates for the Ricci flow, and show that they in fact characterize solutions of the Ricci flow. Namely, given a family $(M,g_t)_{t\in I}$ of Riemannian manifolds, we consider the path space $P\mathcal{M}$ of its space time $\mathcal{M}=M\times I$. Our first characterization says that $(M,g_t)_{t\in I}…
Trading linearity for ellipticity: a nonsmooth approach to Einstein's theory of gravity and the Lorentzian splitting theorems
本文介绍在洛伦兹号差下发展非光滑度量几何的最新进展，提出通过牺牲达朗贝尔算子线性、利用负齐次 p-达朗贝尔算子恢复椭圆性，从而得到低正则性分裂定理，并简化了 Yau 猜想相关证明。
Fibrations, the First Betti Number, and Almost Nonnegative Ricci Curvature
本文在几乎非负Ricci曲率和额外正则性假设下，证明了流形纤维化为b1-环面的定理，去除了Yamaguchi结果中所需的截面曲率上界，并推广到非光滑RCD空间。
On the existence of nonconstant solutions of system of Lane-Emden equations on $\mathrm{RCD}^*(-K,N)$ spaces
本文在RCD*(-K,N)空间上建立Lane-Emden方程组正解的椭圆梯度估计，证明刘维尔型定理，推广光滑黎曼空间的对应结果，研究该类空间上非线性椭圆方程组的解。
This paper builds on the theory of generalised functions begun in [1]. The Colombeau theory of generalised scalar fields on manifolds is extended to a nonlinear theory of generalised tensor fields which is diffeomorphism invariant and has the sheaf property. The generalised Lie derivative for generalised tensor fields is introduced and it is shown that this commutes with the embedding of distributional tensor fields. It is also shown that the covariant derivative of generalised tensor fields commutes with the embedding at the level of association. The concept of generalised metric is introduce…
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