arXiv research

Study finite time singularities in Ricci flow with bounded scalar curvature.

Finite singular times for symmetric network curvature flow.

The study shows stability of neckpinch singularities in mean curvature flows.

Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.

A mean curvature flow starting from a closed embedded hypersurface in $R^{n+1}$ must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact embedded $(n-1)$-dimensional Lipschitz submanifolds plus a set of dimension at most …

Mean curvature flow shows a surface fattening at its first singular point.

Classifies invariant hypersurfaces with singularities.

Gromov-Hausdorff convergence of time-slices of singular Ricci flows

Mean curvature flow shows singularities on smooth surfaces.

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…

Parabolic geometric flows are smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of [CM6] and here is that, by bringing in the dynamical properties of the flow, we obtain also smoothing for large time for generic initi…

Study resolves flow through cylindrical singularities, proving nonfattening.

Proves heat expansion for Laplacian on a singularity.

Parabolic geometric flows have the property of smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of this paper is that, by bringing in the dynamical properties of the flow, we obtain also smoothing for long time for g…

The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.

The paper studies singularities in a complex flow related to mean curvature.

Ricci flow singularities on compact Kähler surfaces are of Type I.

Alternative approach to regularize time-dependent singular Lagrangian systems.

The paper constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.

On a compact Kähler manifold with semi-ample canonical line bundle and Kodaira dimension one, we observe a relation between the infinite-time singularity type of the Kähler-Ricci flow and the characteristic indexes of singular fibers of the semi-ample fibration.

We establish several quantitative results about singular Ricci flows, including estimates on the curvature and volume, and the set of singular times.

In an earlier paper of the authors it was shown that the sheaf theoretically based recently developed abstract differential geometry of the first author can in an easy and natural manner incorporate singularities on arbitrary closed nowhere dense sets in Euclidean spaces, singularities which therefore can have arbitrar…

Study curve shortening flow on Riemann surfaces with conical singularities.

Enhances understanding of Kähler-Ricci flow singularities.

In the present paper we study a type of generic singularity of mean curvature flow modelled on the bubble-sheet $\mathbb S^1\times\mathbb R^3$ , and we derive an asymptotic profile for a neighborhood of singularity.

Two new methods improve forecasting of functional time series data.

Proves smoothness of conical singularities in mean curvature flow.

Consider the Kahler-Ricci flow with finite time singularities over any closed Kahler manifold. We prove the existence of the flow limit in the sense of current towards the time of singularity. This answers affirmatively a problem raised by Tian on the uniqueness of the weak limit from sequential convergence constructio…

Study of singularities in mean curvature flow with focus on $\mathbb{S}^3 imes\mathbb{R}$.

Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In …

Study verifies Joyce's conjectures for circle-invariant Lagrangian surfaces.

For the Kähler-Ricci flow on a compact Kähler manifold with semi-ample canonical line bundle, we prove the singularity type at infinity does not depend on the choice of the initial metric. We also provide new simple proofs for some existing classification results on infinite-time singularity type of the Kähler-Ricci fl…

The paper extends Ricci flow theory with Type-I scalar curvature bounds, proving entropy convergence and characterizing singular sets.

Constructs ancient solutions to mean curvature flow with prescribed singular sets.

Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.

The paper analyzes finite-time singularities in Spin(7)-structure flows using Shi-type estimates.

In this paper we investigate the singularities of Lagrangian mean curvature flows in $\mathbf{C}^m$ by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asympto…

We study "warped Berger" solutions $\big(\mc S^1\times\mc S^3,G(t)\big)$ of Ricci flow: generalized warped products with the metric induced on each fiber $\{s\}\times\mathrm{SU}(2)$ a left-invariant Berger metric. We prove that this structure is preserved by the flow, that these solutions develop finite-time neckpinch …

Study calculates ring structure in instanton homology for a surface with points.

Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.

In this note we study finite-time singularities in the Chern-Ricci flow. We show that finite-time singularities are characterized by the blow-up of the scalar curvature of the Chern connection.

Paper proves strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot ν(p,t)$, for $t\in [0,T)$. We prove that the mean curvature blows up at the first singular time $T$ if all singu…

New study confirms some mean curvature flow solutions have bounded mean curvature.

We study mean curvature flow of smooth, axially symmetric surfaces in $\mathbb{R}^3$ with Neumann boundary data. We show that all singularities at the first singular time must be of type I.

In this short paper, we show that Kähler-Ricci flows over closed manifolds would have scalar curvature blown-up for finite time singularity. Certain control of the blowing-up is achieved with some mild assumption.

We consider one of the generic regimes of formation of singularities. We obtain a detailed description of a possibly small, but fixed, neighborhood of the blowup point, up to (and including) the blowup time, and find that it is mean convex. This confirms a conjecture by Ilmanen. And we find that the singularity is isol…

Given any embedded Lagrangian on a four dimensional compact Calabi-Yau, we find another Lagrangian in the same Hamiltonian isotopy class which develops a finite time singularity under mean curvature flow. This contradicts a weaker version of the Thomas-Yau conjecture regarding long time existence and convergence of Lag…