Shouhei Honda
In this short note we provide several conjectures on the regularity of measured Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature bounded below, from the point of view of the synthetic treatment of lower bounds on Ricci curvature for metric measure spaces.
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We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties $\quad$- $\mathrm{tr}(\mathrm{Hess}f)=Δf$ on $U\subset\mathsf{X}$ for every $f$ sufficiently regular, $\quad$- $\mathfrak{m}=c\mathscr{H}^n$ on $U\subset\mathsf{X}$ for some $c>0$, where $U\subset \mathsf{X}$ is open and $\mathsf{X}$ is a - possibly collapsed - RCD space of es…
We show characterizations of non-collapsed compact $RCD(K, N)$ spaces, which in particular confirm a conjecture of De Philippis-Gigli on the implication from the weakly non-collapsed condition to the non-collapsed one in the compact case. The key idea is to give the explicit formula of the Laplacian associated to the pull-back Riemannian metric by embedding in $L^2$ via the heat kernel. This seems the first application of geometric flow to the study of RCD spaces.
Integral type Gauss-Green formula on non-collapsed RCD spaces and its applications
利用Laplacian的强局部性和特征函数逼近方法，在非塌缩RCD空间上证明了积分型Gauss-Green公式，并推广了Colding的单调性公式，建立了超曲面在给定点的平均曲率与以该点为中心的小球体积之间的渐近关系。
A characterization of non-collapsed $RCD(K, N)$ spaces via Einstein tensors
本文通过热核嵌入诱导度量展开式中第二主项的无散度性质，刻画了紧致非坍缩 $RCD(K, N)$ 空间，并证明该结果无法推广至非紧情形。
Topology of non-collapsed three-dimensional RCD spaces
本文研究了非坍缩三维RCD空间的拓扑结构，证明其具有局部有限的拓扑奇点，并展示了非定向性在Gromov-Hausdorff收敛下的稳定性。
The volume entropy of a compact metric measure space is known to be the exponential growth rate of the measure lifted to its universal cover at infinity. For a compact Riemannian $n$-manifold with a negative lower Ricci curvature bound and a upper diameter bound, it was known that it admits an almost maximal volume entropy if and only if it is diffeomorphic and Gromov-Hausdorff close to a hyperbolic space form. We prove the quantitative rigidity of almost maximal volume entropy for $\operatorname{RCD}$-spaces with a negative lower Ricci curvature bound and Riemannian manifolds with a negative …
Cohomogeneity one RCD-spaces
本研究针对上齐性一RCD空间展开，证明非塌缩情形下的切片定理，推导完整拓扑结构与正则轨道表示定理，给出构造方法与完整刻画，并完成本质维数≤4的非塌缩上齐性一RCD空间的分类。
We provide a general theory for parallel transport on non-collapsed ${\sf RCD}$ spaces obtaining both existence and uniqueness results. Our theory covers the case of geodesics and, more generally, of curves obtained via the flow of sufficiently regular time dependent vector fields: the price that we pay for this generality is that we cannot study parallel transport along a single such curve, but only along almost all of these (in a sense related to the notions of Sobolev vector calculus and Regular Lagrangian Flow in the nonsmooth setting). The class of ${\sf ncRCD}$ spaces contains finite dim…
Gromov-Hausdorff and intrinsic flat convergence of RCD(K,N) and Kato spaces
本文研究了RCD(K,N)和Kato空间在Gromov-Hausdorff和内在平坦收敛下的定向性稳定性及极限一致性。
We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov-Hausdorff convergence.
Given a metric measure space $(X,d,\mathfrak{m})$ that satisfies the Riemannian Curvature Dimension condition, $RCD^*(K,N),$ and a compact subgroup of isometries $G \leq Iso(X)$ we prove that there exists a $G-$invariant measure, $\mathfrak{m}_G,$ equivalent to $\mathfrak{m}$ such that $(X,d,\mathfrak{m}_G)$ is still a $RCD^*(K,N)$ space. We also obtain some applications to Lie group actions on $RCD^*(K,N)$ spaces. We look at homogeneous spaces, symmetric spaces and obtain dimensional gaps for closed subgroups of isometries.
We prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric immersion in a Euclidean space via an eigenmap, then the eigenmap is a locally bi-Lipschitz embedding map to a sphere, which generalizes a fundamental theorem of Takahashi in submanifold theory to a non-smooth setting. Applications of these results include a topological sphere theorem and topological finiteness theorems, which are new even for closed Riemannian m…
Monotonicity formula and stratification of the singular set of perimeter minimizers in RCD spaces
本文在RCD(0,N)度量测度锥中建立周长极小化集合的单调性公式及刚性陈述，并由此得到非塌缩RCD空间中奇异分层的尖锐Hausdorff维数估计，以及非负Ricci曲率流形中全局周长极小化子blow-down锥的存在性。
On the non-collapsed RCD spaces with local bounded covering geometry
本文研究具有局部有界覆盖几何的非坍缩RCD空间，证明了殆平坦流形定理的对应结论及正则纤维化定理，验证了RCD+CBA框架下的相关猜想。
Fine representation of Hessian of convex functions and Ricci tensor on RCD spaces
本文在 RCD 空间上为分布 Ricci 张量建立极分解，并推广至凸函数 Hessian 与截面曲率算子，解决了若干开放问题。
Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions. The goal of the paper is to prove the corresponding stability properties for synthetic Ricci curvature lower bounds. Specifically, we show that such stability holds for quotients of $\mathsf{RCD}^{…
Singular Kähler-Einstein metrics and RCD spaces
该研究聚焦奇异射影簇上的Kähler-Einstein度量，证明在常数量曲率度量的逼近条件下，其光滑部分的度量完备化为非坍塌RCD空间且与原簇同胚，拓展了奇异几何空间的研究范畴。
Improvement of the dimension bound for unweighted RCD spaces
该研究针对无权重RCD空间的维数界问题，证明有限维无权重RCD空间的维数界可自动改进为其Hausdorff维数，且在额外假设n-可求长性时，该结论对N=∞的情形也成立，推进了RCD空间理论的相关研究。
RCD(0,N)-spaces with small linear diameter growth
本文研究了RCD(0,N)空间修正基本群的结构性质，推广了Sormani的相关工作，证明了在假设小线性直径增长条件下该基本群是有限生成的。
Inradius collapsed manifolds with a lower Ricci curvature bound
本文研究了内径坍缩流形的几何性质，证明其极限空间具有特定的对称性和曲率约束，且边界成分数有限。
Margulis Lemma on $\text{RCD}(K,N)$ spaces
本文将 Margulis 引理推广到 $\ ext{RCD}(K,N)$ 空间，并通过改进正则 Lagrangian 流的正则性估计作为关键工具。
Contractibility of moduli spaces of RCD(0,2)-structures
本文研究RCD(0,2)-空间，分类了所有容许该结构的紧拓扑空间，描述了对应模空间并证明其均具有可缩性，为非光滑度量测度空间的模空间研究提供了新结果。
We generalize to the ${\rm RCD}(0,N)$ setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with non-negative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in [AFM] we also introduce the notion of electrostatic potential in ${\rm RCD}$ spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in ${\rm RCD}(K,N)$ spaces and on a new functional v…
In this paper, we will establish an elliptic local Li-Yau gradient estimate for weak solutions of the heat equation on metric measure spaces with generalized Ricci curvature bounded from below. One of its main applications is a sharp gradient estimate for the logarithm of heat kernels. These results seem new even for smooth Riemannian manifolds.
We prove an upper bound on the rank of the abelianised revised fundamental group (called "revised first Betti number") of a compact $RCD^{*}(K,N)$ space, in the same spirit of the celebrated Gromov-Gallot upper bound on the first Betti number for a smooth compact Riemannian manifold with Ricci curvature bounded below. When the synthetic lower Ricci bound is close enough to (negative) zero and the aforementioned upper bound on the revised first Betti number is saturated (i.e. equal to the integer part of $N$, denoted by $\lfloor N \rfloor$), then we establish a torus stability result stating th…
We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed manifolds with (sectional curvature) local bounded covering geometry, to manifolds with (global) Ricci boundedcovering geometry.
Examples of tangent cones of non-collapsed Ricci limit spaces
本研究拓展了Colding-Naber关于非坍缩Ricci极限空间切锥同胚不唯一的结果，构造了5维及以上极限空间，使任意容许核心度量的有限流形族都可作为同一点切锥的截面。
Removability of non-isolated singularities for Einstein metrics and RCD spaces
本文建立了爱因斯坦度量及里奇曲率有下界度量的非孤立奇点可去性结果，证明特定奇异集外有界的度量可延拓，还可用于证明Schoen的标量曲率奇点猜想。
Ricci entropy, RCD structures and Kahler spaces
本文研究具有klt奇点的紧致正规Kahler空间上的奇异Kahler度量，证明多重位势理论中Ricci电流下界与RCD理论中合成Ricci下界等价，从而每个Kahler类具有RCD结构，并推出Calabi-Yau空间基本群几乎阿贝尔及紧致性结果。
We prove existence and uniqueness of optimal maps on $RCD^*(K,N)$ spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation.
Glued spaces and lower curvature bounds
本文综述光滑与非光滑语境下受下曲率界约束的粘合构造定理并提出猜想，聚焦合成下Ricci曲率界，涉及多种空间类型，为相关研究提供理论梳理与猜想方向。
In this paper we prove that in the class of metric measure spaces with Alexandrov curvature bounded from below the Riemannian curvature-dimension condition $RCD(K,N)$ with $K\in \mathbb{R}$ and $N\in [1,\infty)$ is preserved under doubling and gluing constructions.
Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear volume growth condition. Another main result in this paper is a splitting theorem at the noncompact end for a RCD(0, N) space with strongly minimal volume growth. These results generalize some theorems on noncompact manifolds with nonnegative Ricci curvature to non-smooth settings.
Topology of two-dimensional collapsed spaces with lower Ricci bounds
证明了Ricci曲率有下界且本质维数为二的塌缩影测度空间是拓扑曲面（可能带边界）。
Assuming a lower bound on the Ricci curvature of a complete Riemannian manifold, for $p< 1/2$ we show the existence of bounds on the local $L^p$ norm of the Ricci curvature that depend only on the dimension and which improve with volume collapse.
In this paper, we will prove the Weyl's law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below.
Spectral convergence of graph Laplacians with Ricci curvature bounds and in non-collapsed Ricci limit spaces
本文在Ricci曲率有下界的闭流形及其非坍缩Ricci极限空间上，建立了图拉普拉斯算子特征值与特征函数的高概率谱收敛界，为带奇异点的加权拉普拉斯提供谱逼近。
Non-collapsed eGH convergence and dimension
本文研究了非塌陷eGH收敛下等度群维度的下界，推广了Mazur-Rong-Wang的结果，并应用于RCD空间的大等度群研究。
In the framework of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by a positive constant in a synthetic sense, we establish a sharp and rigid reverse-Hölder inequality for first eigenfunctions of the Dirichlet Laplacian. This generalises to the positively curved and non-smooth setting the classical "Chiti Comparison Theorem". We also prove a related quantitative stability result which seems to be new even for smooth Riemannian manifolds.
For $n$-dimensional Riemannian manifolds $M$ with Ricci curvature bounded below by $-(n-1)$, the volume entropy is bounded above by $n-1$. If $M$ is compact, it is known that the equality holds if and only if $M$ is hyperbolic. We extend this result to $\mathsf{RCD}^{\ast}(-(N-1),N)$ spaces. While the upper bound is straightforward, the rigidity case is quite involved due to the lack of a smooth structure in $\mathsf{RCD}^{\ast}$ spaces. As an application we obtain an almost rigidity result which partially recovers a result by Cheng-Rong-Xu for Riemannian manifolds.
Compact harmonic RCD$(K, N)$ spaces are harmonic manifolds
该研究针对Ricci曲率有下界的黎曼流形的对应对象调和RCD$(K,N)$空间，证明满足热核仅依赖时间与点距、测地球交集体积仅依赖半径与球心距任一条件的紧RCD$(K,N)$空间等距于光滑闭黎曼流形。
Moduli spaces of compact RCD(0,N)-structures
该研究围绕紧空间上的RCD(0,N)-结构模空间展开，建立了结构收敛与万有覆叠提升等变收敛的关联，构造并证明了Albanese映射与soul映射的连续性，还给出了具有非平凡有理同伦群的模空间实例。
Rigidity and regularity for almost homogeneous spaces with Ricci curvature bounds
本文研究几乎齐次RCD$(K,N)$空间的Gromov-Hausdorff极限，证明其必为Ricci曲率下界为$K$的幂零Lie群，并建立了拓扑刚性定理及光滑情形下Riemannian orbifold的刚性与$ε$-正则性结果。
On the local topology of non-collapsed Ricci bounded limit spaces
本文证明了非塌缩Ricci有界黎曼流形的Gromov-Hausdorff极限中，正则部分的局部第一Betti数消失，并讨论了相关应用与未解决问题。
We prove that for any complete three-manifold with a lower Ricci curvature bound and a lower bound on the volume of balls of radius one, a solution to the Ricci flow exists for short time. Actually our proof also yields a (non-canonical) way to flow and regularize some interior region of a non-complete initial data satisfying the aformentioned bounds.
Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
该文研究N维${\rm RCD}(K,N)$空间的等周比较定理与渐近等周性质，得到几乎正则性定理及函数不等式推论，多数结论在光滑非紧流形经典框架下也为新结果，合成理论起关键作用。
We will study metric measure spaces $(X,d,m)$ beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds BE$_1(κ,\infty)$ $\bullet$ for which we prove the equivalence with sharp gradient estimates, $\bullet$ the class of which will be preserved under time changes with arbitrary $ψ\in{\mathrm Lip}_b(X)$, and $\bullet$ which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets $Y\subset X$. In the latter case, the distribution-valued Ricci bound will be given by the signed measure $κ= k\, m_Y + \ell\,σ_{\pa…
An $ε$-Regularity Theorem for Non-collapsed Ricci Flow
本文证明了非塌缩Ricci流的$\epsilon$-正则性定理，并据此为Fano Kähler-Ricci流奇点模型建立新估计，同时给出了热方程解一致收敛的判据及其在几何量抛物正则化中的应用。
Using techniques of optimal transportation and gradient flows in metric spaces, we extend the notion of Riemannian Curvature Dimension condition $RCD(K,\infty)$ introduced (in case the reference measure is finite) by Giuseppe Savare', the first and the second author, to the case the reference measure is $σ$-finite; in this way the theory includes natural examples as the euclidean $n$-dimensional space endowed with the Lebesgue measure, and noncompact manifolds with bounded geometry endowed with the Riemannian volume measure. Another major goal of the paper is to simplify the axiomatization of …
One-phase Free Boundary Problems on RCD Metric Measure Spaces
研究在RCD度量测度空间上的一相伯努利型自由边界问题，证明解的存在性与局部Lipschitz正则性，并显示自由边界在除去Hausdorff维度≤N-3的相对闭集外为N-1维拓扑流形。
We prove a general existence theorem for collapsed ancient solutions to the Ricci flow on compact homogeneous spaces and we show that they converge in the Gromov-Hausdorff topology, under a suitable rescaling, to an Einstein metric on the base of a torus fibration. This construction generalizes all previous known examples in the literature.
Almost rigidity results of Green functions with non-negative Ricci curvature
本文综述非光滑情形下 Green 函数的刚性与近刚性结果，并考察非负 Ricci 曲率 RCD 空间中的 Cheeger-Yau 不等式及其应用。
We construct a family of non-collapsed, non-Kähler, non-Einstein steady Ricci solitons in even dimensions greater or equal to four. These solitons exist on complex line bundles over Kähler-Einstein manifolds of positive scalar curvature. They include a four-dimensional $U(2)$-invariant, non-collapsed Riemannian steady soliton on each of the line bundles $O(k)$, $k>2$ of $\mathbb{C}P^1$. Finally, we find Taub-Nut like Ricci solitons and demonstrate a new proof for the existence of the Bryant soliton.
The main goal of the paper is to prove the existence of the universal cover for $RCD^*(K,N)$-spaces. This generalizes earlier work of C. Sormani and the second named author on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental group of such spaces. These are the first topological results for $RCD^{*}(K,N)$-spaces without extra structural-topological assumptions (such as semi-local simple connectedness).
Isometric immersions of RCD$(K,N)$ spaces via heat kernels
本文利用热核构造RCD$(K,N)$空间的等距浸入，证明紧致等距热核浸入空间必为无权重闭光滑黎曼流形，并给出相应黎曼流形类的$C^\infty$-紧性定理。
We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as $t\to-\infty$ and we give their precise asymptotic description. This description applies in particular to the solution constructed by G.Perelman
RCD structures on singular Kahler spaces of complex dimension three
本文证明复三维对数终端奇点射影簇上具有有界Nash熵且Ricci曲率下有界的奇异凯勒度量可诱导出与之同胚的紧RCD空间，并由此建立klt奇点的代数、几何与解析结构之间的联系。
Perelman's entropy and heat kernel bounds on RCD spaces
本文研究了有限维RCD空间上的Perelman W-熵泛函，严格论证了其时间导数公式并推导单调性与刚性性质，同时建立了具有独立意义的热方程解的界。
Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I
本文证明紧致Kähler曲面上有限时间坍缩的Kähler-Ricci流奇点均为Type I型，结合已有结果，所有有限时间奇点均为Type I型。
This note is devoted to the study of sets of finite perimeter over RCD$(K,N)$ metric measure spaces. Its aim is to complete the picture about the generalization of De Giorgi's theorem within this framework. Starting from the results of [2] we obtain uniqueness of tangents and rectifiability for the reduced boundary of sets of finite perimeter. As an intermediate tool, of independent interest, we develop a Gauss-Green integration by parts formula tailored to this setting. These results are new and non-trivial even in the setting of Ricci limits.
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