arXiv research

Ricci flows with bounded scalar curvature cannot develop Type I singular points.

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

Study on axially symmetric surfaces' flow, showing all singularities are of type I.

3D Ricci flows have bounded diameter before Type I singularities.

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

Local singularity analysis for Ricci flows with applications to bounded scalar curvature.

We define Type I singularities for the mean curvature flow associated to a density $ψ$ ($ψ$MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the $ψ$-curvature due to the density. We describe a family of curves whose e…

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

We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber. As a by-product we conclude that the volume of a finite-volume singular s…

Integral estimates for Ricci flows up to singular time.

Study of Lagrangian mean curvature flow with equivariant symmetry.

We provide a condition for spatial curves which rules out the development of a type I singularity. The condition is that after the last time for which an inflection point develops, if the torsion is ever everywhere non-negative, the curve cannot develop a type I singularity.

The purpose of this article is to examine the possible shapes of type I singularities that form in the mean curvature flow of submanifolds of arbitrary codimension, assuming that the initial submanifold satisfies a particular curvature pinching condition.

The paper confirms Ilmanen's conjecture about mean curvature flows.

Study classifies bubbles of Type I singularities in Kähler-Ricci flow on compact surfaces.

We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-Müller-Topping that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result …

New metrics found on orbifold resolutions with specific curvature.

We prove that the Ricci flow on CP^n blown-up at one point starting with any rotationally symmetric Kahler metric must develop Type I singularities. In particular, if the total volume does not go to zero at the singular time, the parabolic blow-up limit of the Type I Ricci flow along the exceptional divisor is a comple…

Study on triaxial Bianchi IX metrics and Ricci flow singularity.

Paper solves the minimal generating set problem for singular Reidemeister moves.

In a singular Type I Ricci flow, we consider a stratification of the set where there is curvature blow-up, according to the number of the Euclidean factors split by the tangent flows. We then show that the strata are characterized roughly in terms of the decay rate of their volume, which in our context plays the role o…

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

New distance comparison principle for curve shortening flow in higher dimensions.

Study neckpinch singularities in Ricci flow with cylindrical symmetry.

Study of splitting maps in Type I Ricci flows for understanding singular set structure.

Let $(M,g,φ)$ be a solution to the Ricci flow coupled with the heat equation for a scalar field $φ$. We show that a complete, $κ$-noncollapsed solution $(M,g,φ)$ to this coupled Ricci flow with a Type I singularity at time $T<\infty$ will converge to a non-trivial Ricci soliton after parabolic rescaling, if the base po…

Ancient solutions of Ricci flow with Type I growth are classified.

The Kähler-Ricci flow's singularities are analyzed with bounds and convergence results.

Sharp Li-Yau equality proven for shrinking Ricci solitons without curvature assumptions.

Two new proofs show Ricci flow breathers are special solutions.

We show that a rescale limit at any degenerate singularity of Ricci flow in dimension 3 is a steady gradient soliton. In particular, we give a geometric description of type I and type II singularities.

Proves analogous result for harmonic Ricci flow, agreeing with gradient Ricci solitons.

Solves Ricci flow singularities by healing pinched discs.

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based …

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

In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of $\mathcal{F}$-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabil…

Study Ricci flow on CP1-bundles over Kähler-Einstein manifolds.

Curve Shortening Flow preserves circularity for convex projections.

Study four-dimensional Ricci flow using branching curves.

Numerical simulations show stability of Type-II singularities in noncompact hypersurfaces.

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 consider the Ricci flow on $\mathbb{CP}^n$ blown-up at one point starting with any $U(n)$-invariant Kähler metric. It is known that the Kähler-Ricci flow must develop Type I singularities. We show that if the total volume does not go to zero at the singular time, then any Type I parabolic blow-up limit of the Ricci …

Study curve shortening flow in high dimensions with boundary constraints.

The paper shows how stable minimal spheres emerge in certain 3D spaces under Ricci flow.

Study Ricci flow on $\mathbb{R}^{n+1}$, focusing on asymptotic behavior and singularities.

Study shows curvature behavior for Kähler-Ricci flow with finite singularities.

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

In this paper we consider the class $\mathcal{A}$ of those solutions $u(x,t)$ to the conjugate heat equation $\frac{d}{dt}u = -Δu + Ru$ on compact Kähler manifolds $M$ with $c_1 > 0$ (where $g(t)$ changes by the unnormalized Kähler Ricci flow, blowing up at $T < \infty$), which satisfy Perelman's differential Harnack i…