Spherical Spectral Synthesis and Two-Radius Theorems on Damek-Ricci Spaces

Damek-Ricci空间上的微分共形刚性

Infinitesimal Conformal Rigidity on Damek-Ricci Spaces

本文研究了Damek-Ricci空间上的共形向量场，证明其必为Killing向量场，采用解析方法给出了构造性证明。

Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces

In this article we consider solvable hypersurfaces of the form $N \exp(\R H)$ with induced metrics in the symmetric space $M = SL(3,\C)/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\C) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $α\in [0,π/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $α= 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for th…

Damek-Ricci空间中的球状等参超曲面

Sphere-like isoparametric hypersurfaces in Damek-Ricci spaces

本文在Damek-Ricci空间中通过过度膨胀超球面构造新的球状等参超曲面族，连接测地球面与已知等参超曲面，并研究其齐次性和焦簇的全测地条件。

Damek-Ricci空间上某些有理等参函数的分类

Classification of Certain Rational Isoparametric Functions on Damek-Ricci Spaces

本文分类了Damek-Ricci空间上以标准坐标多项式除以t形式的有理等参函数，涵盖已知例子并产生新例，确定焦点簇与平均曲率，并处理了C H^k中R H^k管状邻域的例外情形。

关于超几何型调和流形的体积熵的一个注记

A note on the volume entropy of harmonic manifolds of hypergeometric type

本文研究超几何型调和流形在 Ricci 曲率归一化条件下的体积熵不等式，证明上界仅由截面曲率为 -1 的实双曲空间达到，并指出 Damek-Ricci 空间达到下界的例子仅有四种。

Codimension one Ricci soliton subgroups of solvable Iwasawa groups

Recently, Jablonski proved that, to a large extent, a simply connected solvable Lie group endowed with a left-invariant Ricci soliton metric can be isometrically embedded into the solvable Iwasawa group of a non-compact symmetric space. Motivated by this result, we classify codimension one subgroups of the solvable Iwasawa groups of irreducible symmetric spaces of non-compact type whose induced metrics are Ricci solitons. We also obtain the classifications of codimension one Ricci soliton subgroups of Damek-Ricci spaces and generalized Heisenberg groups.

Ricci Bounds for Euclidean and Spherical Cones

We prove generalized lower Ricci bounds for Euclidean and spherical cones over compact Riemannian manifolds. These cones are regarded as complete metric measure spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0,n+1) and that the spherical cone over the same manifold fulfills the curvature-dimension condition CD(n,n+1).

Rigidity of spherical product Ricci solitons

We show that $S^2\times S^2$ is isolated as a shrinking Ricci soliton in the space of metrics, up to scaling and diffeomorphism. We also prove the same rigidity for $S^2\times N$, where $N$ belongs to a certain class of closed Einstein manifolds. These results are the Ricci flow analogues of our results for Clifford-type shrinking solitons for the mean curvature flow.

Kähler-Ricci 流在 $\mathbf G$-球面 Fano 流形上的收敛性

Kähler-Ricci flow on $\mathbf G$-spherical Fano manifolds

本文证明 $\mathbf G$-球面 Fano 流形上 Kähler-Ricci 流的极限为具有奇异 Kähler-Ricci 孤子的 $\mathbf G$-球面 $\mathbb Q$-Fano 簇，其结构由 Cartan 环面元素诱导的退化给出。

具有旋转对称性的球面里茨蒂( Ricci tori )

Spherical Ricci tori with rotational symmetry

本文研究了具有旋转对称性的球面里茨蒂度量，通过构造两参数族度量并分析其在三维球面中的诱导度量，证明了存在嵌入的紧致球面里茨表面。

紧致子流形在欧几里得和球面空间形式中受限曲率的刚性结果

Rigidity Results for Compact Submanifolds with Pinched Ricci Curvature in Euclidean and Spherical Space Forms

本文研究了在欧几里得和球面空间形式中具有受限曲率的紧致子流形，证明其要么是爱因斯坦克利福德 тор斯，要么是拓扑球体，或具有部分同调群消失，扩展了多项经典刚性结果。

A note on blowup limits in 3d Ricci flow

We prove that Perelman's ancient ovals occur as blowup limit in 3d Ricci flow through singularities if and only if there is an accumulation of spherical singularities.

Isoperimetric regions in spherical cones and Yamabe constants of $M\times S^1$

Given closed Riemannian manifold $(M^n, g)$ of positive Ricci curvature $Ricci(g) \geq (n-1)g$ we study isoperimetric regions on the spherical cone over $M$. When $g$ is Einstein we use this to compute the Yamabe constant of $(M \times {\bf R}, g + dt^2)$ and so to obtain lower bounds for the Yamabe invariant of $M\times S^1$.

Harmonic Manifolds and the Volume of Tubes about Curves

H. Hotelling proved that in the n-dimensional Euclidean or spherical space, the volume of a tube of small radius about a curve depends only on the length of the curve and the radius. A. Gray and L. Vanhecke extended Hotelling's theorem to rank one symmetric spaces computing the volumes of the tubes explicitly in these spaces. In the present paper, we generalize these results by showing that every harmonic manifold has the above tube property. We compute the volume of tubes in the Damek-Ricci spaces. We show that if a Riemannian manifold has the tube property, then it is a 2-stein D'Atri space.…

Ricci Bounds for Euclidean and Spherical Cones (revised/extended version)

We prove generalized lower Ricci bounds for Euclidean and spherical cones over complete Riemannian manifolds. These cones are regarded as complete metric measure spaces. In general, they will be neither manifolds nor Alexandrov spaces. We show that the Euclidean cone over an n-dimensional Riemannian manifold whose Ricci curvature is bounded from below by n-1 satisfies the curvature-dimension condition CD(0; n+1) and that the spherical cone over the same manifold fulfills the curvature-dimension condition CD(n; n+1). More generally, for each N > 1 we prove that the condition CD(N-1;N) for a wei…

Ricci flow from some spaces with asymptotically cylindrical singularities

We prove the existence of Ricci flow starting from a class of metrics with unbounded curvature, which are doubly-warped products over an interval with a spherical factor pinched off at an end. These provide a forward evolution from some known and conjectured finite-time local singularities of Ricci flow, generalizing previous examples. The class also includes metrics with non-compact singular ends which become instantaneously compact. Furthermore, we prove local stability of the forward evolution, which allows us to glue it to other manifolds and create a forward evolution from spaces which ar…

球面背景几何中的无限组合 Ricci 流

Infinite combinatorial Ricci flow in spherical background geometry

本文研究球面背景几何中带预定总测地曲率的无限组合 Ricci 流，证明流方程全局解的存在性，并在附加条件下证明其长时间收敛。

从 $\mathbb{S}^2$ 到具有正 Ricci 曲率的近球面的收缩传输映射

Contractive transport maps from $\mathbb{S}^2$ to nearly spherical surfaces with positive Ricci curvature

本文证明每个近球面正曲率曲面都可表示为标准球面在收缩保体积映射下的像，方法结合了 Ricci 流、Kim-Milman 构造与多尺度 Bakry-Émery 判据。

On the Curvature ODE associated to the Ricci flow

In the vector space of algebraic curvature operators we study the reaction ODE $$\frac{dR}{dt} = R^2+R^{#}= Q(R)$$ which is associated to the evolution equation of the Riemann curvature oper- ator along the Ricci flow. More precisely, we analyze the stability of a special class of zeros of this ODE up to suitable normalization. In particular, we show that the ODE is unstable near the curvature operators of the Riemannian product spaces $M \times \mathbb{R}^k, \ k \geq 0$ where $M$ is an Einstein (locally) symmetric space of compact type and not a spherical space form when $k = 0.$

Ricci Solitons on Pseudo-Riemannian Hypersurfaces of 4-dimensional Minkowski space

In this article, we get classification theorems for a Ricci soliton on the pseudo-Riemannian hypersurface of the Minkowski space $\mathbb{E}^4_1$ taking the potential vector field as the tangent component of the position vector of the pseudo-Riemannian hypersurface, denoted by $(M,g,{\bf x}^T,λ)$ in both Riemannian and Lorentzian settings. First, we obtain the necessary and sufficient condition that a pseudo-Riemannian hypersurface $(M,g)$ in $\mathbb{E}^4_1$ admits a Ricci soliton $(M,g,{\bf x}^T,λ)$. In each of the form of the shape operator of a pseudo-Riemannian hypersurface, we obtain cha…

Stratified spaces and synthetic Ricci curvature bounds

We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K $\in$ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2$π$ and its dimension is at most equal to N. This gives a new wide class of geometric examples of metric measure spaces satisfying the RCD(K, N) curvature-dimension condition, including for instance spherical suspensions, orbifolds, K{ä}hler-Einstein manifolds with a divisor, Einstein manifolds with conical singularities along a curv…

非负Ricci曲率、欧氏体积增长与开4-流形的基本群

Nonnegative Ricci Curvature, Euclidean Volume Growth, and the Fundamental Groups of Open $4$-Manifolds

本文研究具有非负Ricci曲率的4维开流形，证明当万有覆盖具有欧氏体积增长时基本群有限生成，从而证实Pan-Rong猜想在n=4时成立，并进一步刻画了基本群的代数结构。

Three-Dimensional Alexandrov spaces with positive or nonnegative Ricci curvature

We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the $\mathsf{CD}^*(K,N)$ sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional $\mathsf{CD}^*(2,3)$-Alexandrov space must be homeomorphic to a spherical space form or to the suspension of $\mathbb{R}P^2$. We then classify closed three-dimensional $\mathsf{CD}^*(0,3)$-Alexandrov spaces.

Spherical doubly warped spacetimes for radiating stars and cosmology

Spherically symmetric spacetimes are ambient spaces for models of stellar collapse and inhomogeneous cosmology. We obtain results for the Weyl tensor and the covariant form of the Ricci tensor on general doubly warped (DW) spacetimes. In a spherically symmetric metric, the Ricci and electric tensors become rank-2, built with a velocity vector field and its acceleration. Their structure dictates the general form of the energy-momentum tensor in the Einstein equations in DW spherical metrics. The anisotropic pressure and the heat current of an imperfect fluid descend from the gradient of the acc…

具有下界里奇曲率的非塌缩空间拓扑正则性与稳定性

Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below

本文研究非塌缩里奇极限空间的拓扑正则性与稳定性，证实了四维 Colding-Naber 猜想及 Cheeger-Colding-Tian 猜想的特殊情形，并建立流形识别定理与锥刚性结果。

The sphere theorems for manifolds with positive scalar curvature

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0>σ_{n}K_{\max}$, where $σ_n\in (\frac{1}{4},1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. This gives a partial answer to Yau's conjecture on pinching theorem. Moreover, we prove that if $M^n(n\geq3)$ is a compact manifold whose $(n-2)$-th Ricci curvature and normalized scalar curvature satisfy the pointwise condi…

Ricci flow and contractibility of spaces of metrics

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group. This proves the Generalized Smale Conjecture. Our argument is independent of Hatcher's theorem in the $S^3$ case and in particular it gives a new proof of the $S^3$ case.

Ricci curvature bounds for warped products

We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning euclidean and spherical cones. On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov spaces with curvature bounded from below. For the proof we combine techniques developed in these papers. Because the Finsler product metric can degenerate we regard a warped product as metric measure space that is in general neither a Finsler manifold nor an Alexandrov space again …

三维柱对称广义Ricci孤立子

Cylindrical generalized Ricci solitons in three dimensions

构造了一个具有SO(2)×R对称性的完整非紧三维光滑稳态梯度广义Ricci孤立子的显式双参数族，提供了Podestà和Raffero最近发现的球对称孤立子的柱对称对应。

Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any $K\in \mathbb{R}$, $N\geq 1$ and upper diameter bounds) hold, i.e. the isoperimetric profile function of $(X,\mathsf{d},\mathfrak{m})$ is bounded from below by the isoperimetric profile of the model space. Moreover, if equal…

Ricci flow and diffeomorphism groups of 3-manifolds

We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabai's theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than $S^3$ and $RP^3$ and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold $X$, the inclusion $\text{Isom} (X,g)\to \text{Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature.

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds

We prove that if $M$ is a closed $n$-dimensional Riemannian manifold, $n \ge 3$, with ${\rm Ric}\ge n-1$ and for which the optimal constant in the critical Sobolev inequality equals the one of the $n$-dimensional sphere $\mathbb{S}^n$, then $M$ is isometric to $\mathbb{S}^n$. An almost-rigidity result is also established, saying that if equality is almost achieved, then $M$ is close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in the ${\rm RCD}$-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bo…