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).
In this article, we thoroughly investigate the stability inequality for Ricci-flat cones. Perhaps most importantly, we prove that the Ricci-flat cone over CP^2 is stable, showing that the first stable non-flat Ricci-flat cone occurs in the smallest possible dimension. On the other hand, we prove that many other examples of Ricci-flat cones over 4-manifolds are unstable, and that Ricci-flat cones over products of Einstein manifolds and over Kahler-Einstein manifolds with h^(1,1)>1 are unstable in dimension less than 10. As results of independent interest, our computations indicate that the Page…
We investigate the possibility of desingularizing a positively curved metric cone by an expanding gradient Ricci soliton with positive curvature operator. This amounts to study the deformation of such geometric structures. As a consequence, we prove that the moduli space of conical positively curved gradient Ricci expanders is connected.
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.
We prove that a shrinking gradient Ricci soliton which is asymptotic to a Kähler cone along some end is itself Kähler on some neighborhood of infinity of that end. When the shrinker is complete, it is globally Kähler.
It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume growth in the above result is essential. More precisely, we construct a complete Ricci-flat manifold of dimension 4 with non-Euclidean volume growth who has at least two distinct tangent cones at infinity and one of them has a smooth cross section.
We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.
We use the momentum construction of Calabi to study the conical Kähler-Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov-Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts the cone divisor to a single point and Gromov-Hausdorff converges to a two dimensional projective orbifold. This gives the first example of the conical Kähler-Ricci flow contracting the cone divisor to a single point. At the end, we introduce a conjectural picture of the geometry of finite time non-collapsing sing…
Let $D$ be a smooth divisor in a compact complex manifold $X$ and let $β\in (0,1)$. We show that in any positive co-homology class on $X$ there is a Kähler metric with cone angle $2πβ$ along $D$ which has bounded Ricci curvature. We use this result together with the Aubin-Yau continuity method to give an alternative proof of a well-known existence theorem for Kahler-Einstein metrics with cone singularities.
A lower-bound estimate of injectivity radius for complete Riemannian manifolds is discussed in a pure geometric viewpoint and is applied to study tangent cones at infinity of certain gradient Ricci solitons. We also study the asymptotic volume ratio of gradient Ricci solitons.
We show that if $X$ is a limit of $n$-dimensional Riemannian manifolds with Ricci curvature bounded below and $γ$ is a limit geodesic in $X$ then along the interior of $γ$ same scale measure metric tangent cones $T_{γ(t)}X$ are Hölder continuous with respect to measured Gromov-Hausdorff topology and have the same dimension in the sense of Colding-Naber.
In this paper, we prove a pseudolocality-type theorem for $\mathcal L$-complete noncompact Ricci flow which may not have bounded sectional curvature; with the help of it we study the uniqueness of the Ricci flow on noncompact manifolds. In particular, we prove the strong uniqueness theorem for the $\mathcal L$-complete Ricci flow on the Euclidean space. This partially answers a question proposed by B-L.~Chen.
A Ricci soliton $(M,g,v,λ)$ on a Riemannian manifold $(M,g)$ is said to have concurrent potential field if its potential field $v$ is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in \cite{CD2}. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.
In this paper, we prove that if an asymptotically Euclidean manifold $(M^n,g)$ under the condition that $R \ge 0$ has long time existence of Ricci flow, the mass of $(M^n,g)$ is nonnegative. In addition, we give an independent proof of positive mass theorem in dimension $3$.
This is largely an exposition article that expands the author's talk at the Xiamen International Conference on Geometric Analysis in June 2021. We first survey the author's joint work with Jiyuan Han on the Yau-Tian-Donaldson (YTD) conjecture for $g$-weighted Kähler-Ricci solitons ($g$-solitons). We then review recent works of Apostolov-Canderbank-Jubert-Lahdili which establish a connection between a particular $g$-soliton equation with Ricci-flat Kähler cone metrics (or equivalently Sasaki-Einstein metrics). The main interest in this connection is the transformation of a possibly irregular Sa…
This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points. First we describe the known examples of preserved curvature con-ditions and how they have been used to derive geometric results, in particular sphere theorems. We then describe some recent results which give restrictions on general preserved conditions. The paper ends with some open questions on these matters. The Ricci flow is the following evolution equation:
We prove a rigidity result for non-negative scalar curvature perturbations of the Euclidean metric on $\mathbb{R^n}$ , which may be regarded as a weak version of the rigidity statement of the positive mass theorem. We prove our result by analyzing long time solutions of Ricci DeTurck flow. As a byproduct in doing so, we extend known $L^p$ bounds and decay rates for Ricci DeTurck flow and prove regularity of the flow at the initial data.
For an embedded submanifold $Σ\subset\mathbb{R}^{N}$, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds $Σ$ in the same way. For this purpose, we derive asymptotics for the approximation of the Ricci curvature proposed in [AW19]. Specifically, we prove Proposition 3.2 in [AW19].
Let $X$ be the Gromov-Hausdorff limit of a sequence of pointed complete Kähler manifolds $(M^n_i, p_i)$ satisfying $Ric(M_i)\geq -(n-1)$ and the volume is noncollapsed. We prove that, there exists a Lie group isomorphic to $\mathbb{R}$, acting isometrically, on the tangent cone at each point of $X$. Moreover, the action is locally free on the cross section. This generalizes the metric cone theorem of Cheeger-Colding to the Kähler case. We also discuss some applications to complete Kähler manifolds with nonnegative bisectional curvature.
The main objective of this thesis is the study of the evolution under the Ricci flow of surfaces with singularities of cone type. A second objective, emerged from the techniques we use, is the study of families of Ricci flow solitons in dimension 2 and 3. The Ricci flow is an evolution equation for Riemannian manifolds, introduced by R. Hamilton in 1982. It is from the achievements made by G. Perelman with this technique in 2002 when the Ricci flow has been established in a discipline itself, generating a great interest in the community. This thesis contains four original results. First result…
This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions greater than 4, if a Ricci flow invariant condition is weaker than "Einstein with nonnegative scalar curvature", then this condition has to be "nonnegative scalar curvature". As a corollary, we obtain that a Ricci flow invariant curvature condition which is stronger than "nonnegat…
We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any $n\in\mathbb{N}_{0}$ and for any negative line bundle $L$ over a compact Kähler manifold $D$, the total space of the vector bundle $L^{\oplus (n+1)}$ admits a unique AC expanding gradient Kähler-Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if $c_{1}(K…
Let C be a cone in the space of algebraic curvature tensors. Moreover, let (M,g) be a compact Einstein manifold with the property that the curvature tensor of (M,g) lies in the cone C at each point on M. We show that (M,g) has constant sectional curvature if the cone C satisfies certain structure conditions.
Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean space when it develops a singularity of type I, and proved that its rescaled flow converges to a self-shrinker in the Euclidean space. In this paper, we generalize this result for a Ricci-mean curvature flow moving along a Ricci flow constructed from a gradient shrinking Ricci soliton.
We show that, up to the flow of the soliton vector field, there exists a unique complete steady gradient Kähler-Ricci soliton in every Kähler class of an equivariant crepant resolution of a Calabi-Yau cone converging at a polynomial rate to Cao's steady gradient Kähler-Ricci soliton on the cone.
We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone, whose maximal Euclidean factor has dimension $k$, then $π_1(M)$ is finitely generated. In particular, this confirms the Milnor conjecture for a manifold whose universal cover has Euclidean volume growth and the unique tangent cone at infinity.
We obtain an Euclidean volume growth results for complete Riemannian manifolds satisfying a Euclidean Sobolev inequality and a spectral type condition on the Ricci curvature. We also obtain eigenvalue estimates, heat kernel estimates, Betti number estimates for closed manifolds whose Ricci curvature is controlled in the Kato class.
We prove that a crepant resolution of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat Kähler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup of SL(n,C). We consider the case in which X is toric. A result of A. Futaki, H. Ono, and G. Wang guarantees the existence of a Ricci-flat Kähler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions.
In this paper we show that all conformal metrics to a pseudo-euclidean space invariant under the translation group, and all the conformal metrics product manifold also invariant by translation where F m it is Ricci flat semi-Riemannian manifold, are gradient Ricci almost soliton. We also proved that all conformal metrics to a euclidean space and invariant by rotation are gradient Ricci almost soliton.
We show that if a shrinking soliton is asymptotic to a cone along an end then the isometry group of the cross-section of the cone embeds in the isometry group of the end of the shrinker. We also provide sufficient conditions for the isometries of the end to extend to the entire shrinker.
We study the Ricci flow for initial metrics which are C^0 small perturbations of the Euclidean metric on R^n. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In proving this stability result, we introduce a monotone integral quantity which measures the deviation of the evolving metric from the Euclidean metric. We also investigate the convergence of the diffeomorphisms relating Ricci harmonic map heat flow to Ricci flow.
In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower diameter bound for compact manifolds with shrinking Ricci solitons. Such a bound can be obtained from an eigenvalue estimate for a twisted Laplacian, called the Witten-Laplacian. In the second part we discuss self-similar solutions to the mean curvature flow on cone manifolds. Many results have been obtained for solutions in $\bfR^n$ or $\bfC^n$. We see tha…
In this paper, we are interested in conical structures of manifolds with respect to the Ricci flow and, in particular, we study them from the point of view of Perelman's functionals. In a first part, we study Perelman's $λ$ and $ν$ functionals of cones and characterize their finiteness in terms of the $λ$-functional of the link. As an application, we characterize manifolds with conical singularities on which a $λ$-functional can be defined and get upper bounds on the $ν$-functional of asymptotically conical manifolds. We then present an adaptation of the proof of Perelman's pseudolocality theo…
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…
We compute the indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones and give a detailed description of the corresponding radially homogeneous tensor fields in its kernel. For a Ricci-flat conifold $(M,g)$ which may have asymptotically conical as well as conically singular ends, we compute at each end a lower bound for the order with which the metric converges to the tangent cone. As a special subcase of our result, we show that any Ricci-flat ALE manifold $(M^n,g)$ is of order $n$ and thereby close a small gap in a paper by Cheeger and Tian.
We show that a simply-connected closed four-dimensional Ricci flow whose Ricci curvature is uniformly bounded below and whose volume does not approach zero must converge to a $C^{0}$ orbifold at any finite-time singularity, so has an extension through the singularity via orbifold Ricci flow. Moreover, a Type-I blowup of the flow based at any orbifold point converges to a flat cone in the Gromov-Hausdorff sense, without passing to a subsequence. In addition, we prove $L^{p}$ bounds for the curvature tensor on time-slices for any $p<2$. In higher dimensions, we show that every singular point of …
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 $M\times S^1$.
We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kahler metrics on the total spaces of (i) holomorphic C^2/Z_p orbifold fibrations over V, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP^1, with generic fibres being the canonical complex cone over V, and (iii) the canonical orbifold line bundle over a fam…
We consider gradient Ricci solitons conformal to a $n$-dimensional pseudo-Euclidean space and we completely describe the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations. As a consequence, the gradient Ricci solitons that arise from the reduced system are invariant under the action of either an $(n-1)$-dimensional translation group or the pseudo-orthogonal group acting on the corresponding $n$-dimensional pseudo-Euclidean space.
We show that the only metric measure space with the structure of an $N$-cone and with two-sided synthetic Ricci bounds is the Euclidean space ${\mathbb R}^{N+1}$ for $N$ integer. This is based on a novel notion of Ricci curvature upper bounds for metric measure spaces given in terms of the short time asymtotic of the heat kernel in the $L^2$-transport distance. Moreover, we establish a beautiful rigidity results of independent interest which characterize the $N$-dimensional standard sphere ${\mathbb S}^N$ as the unique minimizer of $$\int_X\int_X \cos d (x,y)\, m(d y)\,m(d x)$$ among all metri…
In this paper we obtain a simple upper bound for the infimum of the Ricci curvatures of a complete Riemannian manifold with nonzero injectivity radius i(M) depending only on of the i(M). In case of rigidity the Riemannian manifold must be an Euclidean sphere(Euclidean space) conform the injectivity radius be finite(infinite). Furthermore with the additional assumption that the second derivative of the Ricci tensor is null we prove that the same upper bound for the infimum of the Ricci curvatures holds for the supremum of the Ricci curvatures and $M^n$ has, in fact, parallel Ricci tensor.
