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.
Any expanding homogeneous Ricci soliton (in particular any homogeneous Einstein manifold of negative scalar curvature) can be obtained, up to isometry, from a Lie subgroup of a nilpotent Iwasawa group $N$ whose induced metric is a Ricci soliton. By nilpotent Iwasawa group we mean the nilpotent Lie group $N$ of the Iwasawa decomposition associated with a symmetric space of non-compact type. Motivated by this fact, in this paper we classify codimension one Lie subgroups of any nilpotent Iwasawa group $N$ whose induced metric is a Ricci soliton.
We construct explicit proper p-harmonic functions on rank-one Lie groups of Iwasawa type. This class of Lie groups includes many classical Riemannian manifolds such as the rank-one symmetric spaces of non-compact type, Damek-Ricci spaces, rank-one Einstein solvmanifolds and Carnot spaces.
We prove a splitting theorem for complete gradient Ricci soliton with nonnegative curvature and establish a rigidity theorem for codimension one complete shrinking gradient Ricci soliton in $\mathbb R^{n+1}$ with nonnegative Ricci curvature.
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…
We study solutions to generalized Ricci flow on four-manifolds with a nilpotent, codimension $1$ symmetry. We show that all such flows are immortal, and satisfy type III curvature and diameter estimates. Using a new kind of monotone energy adapted to this setting, we show that blowdown limits lie in a canonical finite-dimensional family of solutions. The results are new for Ricci flow.
It is natural to ask whether solvsolitons are global maxima for the Ricci pinching functional F:=scal^2/|Ric|^2 on the set of all left-invariant metrics on a given solvable Lie group S, as it is to ask whether they are the only global maxima. A positive answer to both questions was given in a recent paper by the same authors when the Lie algebra s of S is either unimodular or has a codimension-one abelian ideal. In the present paper, we prove that this also holds in the following two more general cases: 1) s has a nilradical of codimension-one; 2) the nilradical n of s is abelian and the funct…
In this paper, we prove that if the initial submanifold $M_0$ of dimension $n(\ge6)$ satisfies an optimal pinching condition, then the mean curvature flow of arbitrary codimension in hyperbolic spaces converges to a round point in finite time. In particular, we obtain the optimal differentiable sphere theorem for submanifolds in hyperbolic spaces. It should be emphasized that our pinching condition implies that the Ricci curvature of the initial submanifold is positive, but does not imply positivity of the sectional curvature of $M_0$.
We consider Finsler submanifolds $M^n$ of nonnegative Ricci curvature in a Minkowski space $\mathbb{M}^{n+p}$ which contain a line or whose relative nullity index is positive. For hypersurfaces, submanifolds of codimension two or of dimension two, we prove that the submanifold is a cylinder, under a certain condition on the inertia of the pencil of the second fundamental forms. We give an example of a surface of positive flag curvature in a three-dimensional Minkowski space which is not locally convex.
We introduce a new technique to the study and identification of submanifolds of simply-connected symmetric spaces of compact type based upon an approach computing $k$-positive Ricci curvature of the ambient manifolds and using this information in order to determine how highly connected the embeddings are. This provides codimension ranges in which the Cartan type of submanifolds satisfying certain conditions which generalize being totally geodesic necessarily equals the one of the ambient manifold. Using results by Guijarro--Wilhelm our approach partly generalizes recent work by Berndt--Olmos o…
In this paper, we extend the theory of Ricci flows satisfying a Type-I scalar curvature condition at a finite-time singularity. In [Bam16], Bamler showed that a Type-I rescaling procedure will produce a singular shrinking gradient Ricci soliton with singularities of codimension 4. We prove that the entropy of a conjugate heat kernel based at the singular time converges to the soliton entropy of the singular soliton, and use this to characterize the singular set of the Ricci flow solution in terms of a heat kernel density function. This generalizes results previously only known with the stronge…
In this paper we prove a compactness result for Ricci flows with bounded scalar curvature and entropy. It states that given any sequence of such Ricci flows, we can pass to a subsequence that converges to a metric space which is smooth away from a set of codimension $\geq 4$. The result has two main consequences: First, it implies that singularities in Ricci flows with bounded scalar curvature have codimension $\geq 4$ and, second, it establishes a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. In the course of the proof, we will also establish the fol…
In this paper, we show that the uniform L^4-bound of the transverse Ricci curvature along the Sasaki-Ricci flow on a compact quasi-regular transverse Fano Sasakian (2n+1)-manifold M. When M is dimension up to seven and the space of leaves of the characteristic foliation is well-formed, we first show that any solution of the Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold Sasaki-Ricci soliton on the limit space which is a S^1-orbibundle over the unique singular Kaehler-Ricci soliton on a normal projective variety with codimension two orbifold singularitie…
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…
We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear stability, we turn our attention to left-invariant soliton metrics on (non-compact) simply connected solvable Lie groups and prove linear stability of many such metrics. These include an open set of two-step solvsolitons, all two-step nilsolitons, two infinite families of three-step solvable Einstein metrics, all nilsolitons of dimensions six or less, and all…
We establish codimension 4 regularity of noncollapsed sequences of metrics with bounds on natural generalizations of the Ricci tensor. We obtain a priori L2 curvature estimates on such spaces, with diffeomorphism finiteness results and rigidity theorems as corollaries.
Ricci Curvature and Betti Numbers of Hessian Manifolds
INSA de Toulouse Département de Génie Mathématique Université de Toulouse（图卢兹大学 INSA 数学工程系）；ENAC Laboratoire ENAC Université de Toulouse（图卢兹大学 ENAC 实验室）
Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a generalization of Reeb's formula (that the total mean curvature of the leaves is zero) and its companion (that twice total second mean curvature of the leaves equals to the total Ricci curvature in the normal direction). We also extend results by Brito, Langevin and Rosenberg (that to…
In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds $(M^n,g)$ with bounded Ricci curvature, as well as their Gromov-Hausdorff limit spaces $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow} (X,d)$, where $d_j$ denotes the Riemannian distance. Our main result is a solution to the codimension $4$ conjecture, namely that $X$ is smooth away from a closed subset of codimension $4$. We combine this result with the ideas of quantitative stratification to prove a priori $L^q$ estimates on the full curvature $|Rm|$ for all $q<2$. In the case of Einstein manifolds, we im…
In this paper we characterize non-collapsed limits of Ricci flows. We show that such limits are smooth away from a set of codimension $\geq 4$ in the parabolic sense and that the tangent flows at every point are given by gradient shrinking solitons, possibly with a singular set of codimension $\geq 4$. We furthermore obtain a stratification result of the singular set with optimal dimensional bounds, which depend on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds. As an application of our th…
We construct smooth solutions to Ricci flow starting from a class of singular metrics and give asymptotics for the forward evolution. The singular metrics heal with a set of points (of codimension at least three) coming out of the singular point. We conjecture that these metrics arise as final-time limits of Ricci flow encountering a Type-I singularity modeled on $\mathbb{R}^{p+1} \times S^q$. This gives a picture of Ricci flow through a singularity, in which a neighborhood of the manifold changes topology from $D^{p+1} \times S^{q}$ to $S^p \times D^{q+1}$ (through the cone over $S^p \times S…
Let the warped product $M^n=L^m\times_φF^{n-m}$, $n\geq m+3\geq 8$, of Riemannian manifolds be an Einstein manifold with Ricci curvature $ρ$ that admits an isometric immersion into Euclidean space with codimension two. Under the assumption that $L^m$ is also Einstein, but not of constant sectional curvature, it is shown that $ρ=0$ and that the submanifold is locally a cylinder with an Euclidean factor of dimension at least $n-m$. Hence $L^m$ is also Ricci flat. If $M^n$ is complete, then the same conclusion holds globally if the assumption on $L^m$ is replaced by the much weaker condition that…
This paper consists of two parts. First, motivated by classic results, we determine the subsets of a given nilpotent Lie algebra $\mathfrak{g}$ (respectively, of the Grassmannian of two-planes of $\mathfrak{g}$) whose sign of Ricci (respectively, sectional) curvature remains unchanged for an arbitrary choice of a positive definite inner product on $\mathfrak{g}$. In the second part we study the subsets of $\mathfrak{g}$ which are, for some inner product, the eigenvectors of the Ricci operator with the maximal and with the minimal eigenvalue, respectively. We show that the closures of these sub…
We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension one. Using a recent result of Datar and Szekelyhidi, we effectively determine the existence of Kahler-Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kahler-Einstein Fano threefolds, and Fano threefolds admitting a non-trivial Kahler-Ricci soliton.
In this paper we investigate the problem of non-analytic embeddings of Lorentzian manifolds in Ricci-flat semi-Riemannian spaces. In order to do this, we first review some relevant results in the area, and then motivate both the mathematical and physical interest in this problem. We show that any $n$-dimensional compact Lorentzian manifold $(M^{n},g)$, with $g$ in the Sobolev space $H_{s+3}$, $s>\frac{n}{2}$, admits an isometric embedding in an $(2n+2)$-dimensional Ricci-flat semi-Riemannian manifold. The sharpest result available for this type of embeddings, in the general setting, comes as a…
Diameter estimates for Kähler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for $L^\infty$ estimates for the Monge-Ampère equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, diameter bounds are obtained for long-time solutions of the Kähler-Ricci flow and finite-time solutions when the limiting class is big, as well as for special fibrations of Calabi-Yau manifolds.
This paper studies the structure and stability of boundaries in noncollapsed $\text{RCD}(K,N)$ spaces, that is, metric-measure spaces $(X,\mathsf{d},\mathscr{H}^N)$ with lower Ricci curvature bounded below. Our main structural result is that the boundary $\partial X$ is homeomorphic to a manifold away from a set of codimension 2, and is $N-1$ rectifiable. Along the way we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov-Hausdorff limits $(M_i^N,\mathsf{d}_{g_i},p_i) \rightarrow (X,\mathsf{d},p)$ of smooth manifolds with boundary…
