arXiv research

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.

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.

The article explores Helfrich flow with spontaneous curvature, finding singularities and convergence behaviors.

Estimates for VPMCF show ancient MCF solutions and finite-time behavior.

Study Kähler-Ricci flow on manifolds with singularities.

We prove that the scalar curvature of a homogeneous Ricci flow solution blows up at a forward or backward finite-time singularity.

Model predicts crashes in rational expectation bubbles using percolation theory.

Study shows scalar curvature rate for conical Kähler-Ricci flow near singularity.

Study Hermitian curvature flow on complex surfaces, finding singularities and limits.

Study shows how solutions to Yamabe flow can develop Type II singularities.

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

In this paper, we first derive a pinching estimate on the traceless Ricci curvature in term of scalar curvature and Weyl tensor under the Ricci flow. Then we apply this estimate to study finite-time singularity behavior. We show that if the scalar curvature is uniformly bounded, then the Weyl tensor has to blow up, as …

We establish fundamental results for a parabolic flow of Riemannian metrics introduced by Bahuaud-Helliwell in arXiv:1010:4287v1 which is based on the Fefferman-Graham ambient obstruction tensor. First, we obtain local $L^2$ smoothing estimates for the curvature tensor and use them to prove pointwise smoothing estimate…

No finite-time singularities in Yang-Mills flow in 4D.

This short note studies the collapsing behavior of the Kähler-Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion X -> B where B is a Kähler manifold of lower dimension than X. We give cohomological and curvature conditions under which the fibers collapse at the optimal rate ~(T-t)^{1/2}

Study finite singularities in G2 structure flows using Shi-type estimates.

We show precompactness results for solutions to parabolic fourth order geometric evolution equations. As part of the proof we obtain smoothing estimates for these flows in the presence of a curvature bound, an improvement on prior results which also require a Sobolev constant bound. As consequences of these results we …

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

Mean curvature flow's singularity is shown to occur at a finite time.

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

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

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

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…

Enhances understanding of Kähler-Ricci flow singularities.

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

Constructs finite-time singularities in Lagrangian mean curvature flow with precise dynamics.

We present a novel analysis extending the recent work of Mizuno et al. [2002] on the hyperinflations of Germany (1920/1/1-1923/11/1), Hungary (1945/4/30-1946/7/15), Brazil (1969-1994), Israel (1969-1985), Nicaragua (1969-1991), Peru (1969-1990) and Bolivia (1969-1985). On the basis of a generalization of Cagan's model …

Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.

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

Researchers found a new type of singularity in surface evolution equations.

Chen's flow leads to finite-time singularities for closed submanifolds.

We show that if on a compact Kahler threefold there is a solution of the Kahler-Ricci flow which encounters a finite time collapsing singularity, then the manifold admits a Fano fibration. Furthermore, if there is finite time extinction then the manifold is Fano and the initial class is a positive multiple of the first…

Continuity method on Fano fibrations converges to singular metrics.

Curve shortening flow converges to a point with entropy bound.

New heat flow for harmonic maps avoids singularities but not bubbles.

Strict type-II blowup in harmonic map flow is proven to have Hölder continuous body map.

New examples of Laplacian solitons found on solvable Lie groups.

Study shows how embryo wounds heal through a mathematical model.

Flow approach solves Toda system equations.

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

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

Paper studies smoothness of bi-conformal heat flow on 4-manifolds.

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 …

Paper studies heat flow for maps on manifolds, avoiding singularities.

Paper shows perturbed Taub-Bolt metric becomes singularity under Ricci flow.

Paper finds only one unique regular shrinker with 2 closed regions.

Ancient solutions to Ricci flow in 3D are mostly cylinders or solitons.

Study on singularities of Chern-Ricci flow on complex manifolds.