arXiv research

In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger-Gromov convergence of the underlying manifolds and the uniform $L^1$-bound of the solutions. We also prove a local integral estimate of fundamental solutions.

Study of convergence in Lorentzian spacetimes using temporal functions.

We prove Cheeger-Gromov convergence for a subsequence of a given sequence of manifolds-with-boundary of bounded geometry. The method of the proof is to reduce, via height functions, the problem to the setting of Hamilton's compactnes theorem for manifolds without boundary.

For a sequence $\{(M_i, g_i, x_i)\}$ of pointed Riemannian manifolds with boundary, the sequence $\{(M_i,\tilde g_i,x_i)\}$ is its conformal satellite if the metric $\tilde g_i$ is conformal to $g_i$, that is, $\tilde g_i=u^{\frac{4}{n-2}}_ig_i$. Assuming the manifolds $(M_i,g_i,x_i)$ have uniformly bounded geometry, w…

This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a given manifold M - the Cheeger-Gromov theory - and extensions thereof to Ricci curvature in place of full curvature. This theory is then applied to study a collection of different issues in mathematical aapects of General Relativi…

Study geometric structure of Ricci shrinker ends without global curvature assumptions.

Ricci flow converges to Taub-NUT metric under specific conditions.

Study of pluriclosed flow on Oeljeklaus-Toma manifolds, showing convergence to a soliton.

Study of curvature flow on complex Lie groups, leading to soliton convergence.

Study of Bach flow on specific nilmanifolds, converging to a soliton.

The paper shows convergence of Sasaki-Ricci flow on Sasakian 5-manifolds.

In this paper, we prove the existence of a Kahler Ricci soliton on any smooth Fano horospherical manifold by a study of the Kahler-Ricci flow. Indeed, we prove that the renormalized Kahler Ricci flow converges in the sense of Cheeger Gromov and that this limit is a Kahler-Ricci soliton.

In this paper we show that the convergence of complete Kahler-Einstein hypersurfaces in complex torus in the sense of Cheeger-Gromov will canonically degenerate the underlying manifolds into "pair of pants" decomposition. We also construct minimal Lagrangian tori that represent the vanishing cycles of the degeneration.

We show that for any solvable Lie group of real type, any homogeneous Ricci flow solution converges in Cheeger-Gromov topology to a unique non-flat solvsoliton, which is independent of the initial left-invariant metric. As an application, we obtain results on the isometry groups of non-flat solvsoliton metrics and Eins…

Compactness theorem for Riemannian manifolds with volume and curvature bounds.

In this paper, we prove that the $L^4$-norm of Ricci curvature is uniformly bounded along a Kähler-Ricci flow on any minimal algebraic manifold. As an application, we show that on any minimal algebraic manifold $M$ of general type and with dimension $n\le 3$, any solution of the normalized Kähler-Ricci flow converges t…

Proves stability of convex spheres with similar geodesic lengths.

The paper studies Sasaki-Ricci solitons on Sasakian manifolds up to seven dimensions.

The study extends convergence theorems for Ricci-limit spaces with bounded curvature.

We present new lower bounds on the complexity of Dehn surgery manifolds of knots, using our recent result on the Cheeger-Gromov rho invariants and triangulations. As an application, we give explicit examples of closed hyperbolic 3-manifolds with fixed first homology for which the gap between the Gromov norm and the com…

A natural question in mathematical general relativity is how the ADM mass behaves as a functional on the space of asymptotically flat 3-manifolds of nonnegative scalar curvature. In previous results, lower semicontinuity has been established by the first-named author for pointed $C^2$ convergence, and more generally by…

Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed $C^0$ Cheeger--Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. I…

Solves linearity problem for acyclic groups, bounds Cheeger-Gromov ρ-invariants.

In this paper, we prove that Kähler-Ricci flow converges to a Kähler-Einstein metric (or a Kähler-Ricci soliton) in the sense of Cheeger-Gromov as long as an initial Kähler metric is very closed to $g_{KE}$ (or $g_{KS}$) if a compact Kähler manifold with $c_1(M)>0$ admits a Kähler Einstein metric $g_{KE}$ (or a Kähler-…

The ADM mass, viewed as a functional on the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature, fails to be continuous for many natural topologies. In this paper we prove that lower semicontinuity holds in natural settings: first, for pointed Cheeger--Gromov convergence (without any symmetr…

A fundamental tool in the analysis of Ricci flow is a compactness result of Hamilton in the spirit of the work of Cheeger, Gromov and others. Roughly speaking it allows one to take a sequence of Ricci flows with uniformly bounded curvature and uniformly controlled injectivity radius, and extract a subsequence that conv…

In this paper, we prove that on a Fano manifold $M$ which admits a Kähler-Ricci soliton $(\om,X)$, if the initial Kähler metric $\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak Kähler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Moreover, if $\vphi_0$ is also $K_X$-invariant, the…

We prove that, if M is a compact oriented manifold of dimension 4k+3, where k>0, such that pi_1(M) is not torsion-free, then there are infinitely many manifolds that are homotopic equivalent to M but not homeomorphic to it. To show the infinite size of the structure set of M, we construct a secondary invariant tau_(2):…

We give a complementary generalization of the extensions of Bonnet-Myers theorem obtained by Calabi and also Cheeger-Gromov-Taylor.

Enhanced bounds on rho-invariants for 3-manifolds.

Study on Kähler-Ricci flow and conformal submersion singularity formation.

The study bounds invariants of PL manifolds and counts complexity of lens spaces.

Generalizes Candel's theorem on curvature of laminated surfaces.

The paper studies a flow on complex Lie groups, showing convergence to solitons.

This paper continues our exploration of homology cobordism of 3-manifolds using our recent results on Cheeger-Gromov rho-invariants associated to amenable representations. We introduce a new type of torsion in 3-manifold groups we call hidden torsion, and an algebraic approximation we call local hidden torsion. We cons…

The macroscopic version of Urysohn width for scalar curvature is disproven in high dimensions.

In this paper, we prove the extensions of Bonnet--Myers' type theorems obtained by Calabi and Cheeger--Gromov--Taylor via Bakry--Emery Ricci curvature, which generalize the results of \cite{FG, Lim1, Wan, Wang, WW, Wu}.

This paper is concerned with Chern-Ricci flow evolution of left-invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger-Gromov) sense to a Chern-Ricci …

In this paper, we study curvature behavior at the first singular time of solution to the Ricci flow on a smooth, compact n-dimensional Riemannian manifold $M$, $\frac{\partial}{\partial t}g_{ij} = -2R_{ij}$ for $t\in [0,T)$. If the flow has uniformly bounded scalar curvature and develops Type I singularities at $T$, us…

Cheeger and Gromov showed that F-structures are related to collapse with a double-sided curvature bound. We define fibered F-structures and extend some of the Cheeger-Gromov results to the setting of collapse with a lower bound on the curvature operator.

Study of curvature flow on specific Lie groups, leading to soliton solutions.

Solves existence of gravitating vortices with positive curvature.

We study complexities of 3-manifolds defined from triangulations, Heegaard splittings, and surgery presentations. We show that these complexities are related by linear inequalities, by presenting explicit geometric constructions. We also show that our linear inequalities are asymptotically optimal. Our results are used…

We obtain new lower bounds of the minimal genus of a locally flat surface representing a 2-dimensional homology class in a topological 4-manifold with boundary, using the von Neumann-Cheeger-Gromov $ρ$-invariant. As an application our results are employed to investigate the slice genus of knots. We illustrate examples …

We study a flow of $G_2$ structures which induce the same Riemannian metric which is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor along the flow. We show that at a finite-time singularity the torsion must blow-up, so the flow exists as long as the torsion remain…

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the loca…

In arXiv:1005.3255 we proved an orbifold Cheeger-Gromov compactness theorem for complete 4d Ricci shrinkers with a lower bound for the entropy, an upper bound for the Euler characterisic, and a lower bound for the gradient of the potential at large distances. In this note, we show that the last two assumptions in fact …

We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation $\partial_tg_{t}=-{\rm Ric}^{1,1} (g_t)$. The solution $g_t$ always exist for all positive times, and $(1 + t)^{-1}g_t$ converge…