Study finite time singularities in Ricci flow with bounded scalar curvature.
problem Understanding finite time singularities in Ricci flow with bounded scalar curvature.
method Analyzing blow-up sequences of locally Type I singularities.
result Every blow-up sequence of a locally Type I singularity has a specific property.
Finite singular times for symmetric network curvature flow.
problem Formation of singularities in network curvature flow.
method Curvature flow of networks with symmetric initial data and two triple junctions.
result The set of singular times is finite.
Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.
problem Finite time singularities in Kähler-Ricci flow on Hirzebruch surfaces.
method Analyze tangent flows based at singular points.
result Tangent flows are K-R flows with orbifold singularities.
Ricci flow singularities on compact Kähler surfaces are of Type I.
problem Understanding finite time singularities of Ricci flow on compact Kähler surfaces.
method Analyzing the Type I property of singularities.
result Non-collapsed finite time singularities are of Type I.
The paper analyzes finite-time singularities in Spin(7)-structure flows using Shi-type estimates.
problem Analyzing finite-time singularities in Spin(7)-structure flows.
method Proves Shi-type derivative estimates and shows that Λ(x,t) must blow up at finite-time singularities.
result Establishes a general analytic framework for studying Spin(7)-structure flows.
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.
The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.
problem Characterizing and understanding Ricci flows with closed and smooth tangent flows.
method Analyzing ancient and finite-time singularity Ricci flows to prove uniqueness and characterizations.
result The tangent flow is unique and characterizes ancient and finite-time singularity flows.
Enhances understanding of Kähler-Ricci flow singularities.
problem Understanding singularities in Kähler-Ricci flow.
method Relates to classic Kähler-Ricci flow and degenerate complex Monge-Ampère equation.
result Improves understanding of finite and infinite time singularities.
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.
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…
Paper shows no finite time singularities for smooth conformal heat flow of harmonic maps.
problem Smoothness of conformal heat flow of harmonic maps.
method Combines harmonic map flow with metric evolution in conformal direction.
result No finite time singularity occurs for the flow, and under certain conditions, maps converge to a point.
Constructs finite-time singularities in Lagrangian mean curvature flow with precise dynamics.
problem Finite-time singularities in Lagrangian mean curvature flow.
method Modulation analysis around shrinking cohomogeneity-one special Lagrangian desingularizations.
result Explicit curvature blow-up rate and precise dynamics of singularities.
New heat flow for harmonic maps avoids singularities but not bubbles.
problem Finite time singularities in harmonic maps.
method Introduces a conformal heat flow for harmonic maps defined by an evolution equation.
result Global weak solution exists, smooth except at most finitely many points.
New study confirms some mean curvature flow solutions have bounded mean curvature.
problem Existence of mean curvature flow singularities with bounded mean curvature.
method Construction of specific solutions in RN for N≥8. result A nontrivial subset of solutions has uniformly bounded mean curvature.
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}×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 Kähler-Ricci flow on manifolds with singularities.
problem Behavior of Kähler-Ricci flow on manifolds with finite-time singularities.
method Use of holomorphic vector fields to prove estimates.
result Proves estimates related to previous work on the flow.
Paper shows perturbed Taub-Bolt metric becomes singularity under Ricci flow.
problem Analyzing stability of Taub-Bolt metric under Ricci flow.
method Box argument and construction of Ricci flows on compact manifolds.
result Compact perturbation of Taub-Bolt metric evolves into finite time singularity.
Study verifies Joyce's conjectures for circle-invariant Lagrangian surfaces.
problem Verifying Joyce's conjectures for specific Lagrangian surfaces.
method Continuation of Lagrangian mean curvature flow through finite time neck pinches.
result Flow converges to a chain of special Lagrangians, verifying conjectures.
The Kähler-Ricci flow's singularities are analyzed with bounds and convergence results.
problem Understanding the singularities and behavior of the Kähler-Ricci flow.
method Li-Yau type and Harnack estimates for weighted Ricci potential functions.
result Finite time singularities are shown to sub-converge to ancient solutions on analytic normal varieties.
Researchers found a new type of singularity in surface evolution equations.
problem Finite-time singularity formation in surface evolution equations.
method Constructed first example of finite time blow-up solutions for the heat flow of the H-system.
result Singularity forms as a scaled least energy H-bubble with decoupled linearized operators.
Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.
problem Finite-time singularities of harmonic map flow in critical dimensions.
method Proving a weighted Lojasiewicz inequality.
result Continuity of body map and no-neck property for bubble-tree decompositions.
The study shows stability of neckpinch singularities in mean curvature flows.
problem Stability of neckpinch singularities in mean curvature flows.
method Analysis of mean curvature flow and perturbations.
result Stability of neckpinch singularities in mean curvature flows.
In this paper, we extend Lotay-Wei's Shi-type estimate from Laplacian flow to more general flows of G2 structures including the modified Laplacian co-flow. Then we prove a version of κ-non-collapsing theorem. We will use both of them to study finite time singularities of general flows of G2 structures.
Study on stability of cylindrical singularities in MCF of finite codimensions.
problem Stability of cylindrical singularities in mean curvature flow.
method Construction of stable manifold, explicit solutions, asymptotic analysis.
result Asymptotic stability of cylindrical singularities under generic perturbations.
The paper extends Ricci flow theory with Type-I scalar curvature bounds, proving entropy convergence and characterizing singular sets.
problem Extending Ricci flow theory with Type-I scalar curvature bounds.
method Type-I rescaling procedure and entropy analysis of conjugate heat kernels.
result Entropy of Ricci flow solutions converges to soliton entropy, characterizing singular sets.
Curve shortening flow converges to a point with entropy bound.
problem Analyzing the behavior of curves under shortening flow near singularities.
method Analyzes blow-up limits and uses entropy bounds to prove convergence.
result Initial curves with entropy bound converge to a round point in finite time.
Study shows curvature behavior for Kähler-Ricci flow with finite singularities.
problem Analyzing curvature behavior in Kähler-Ricci flow with finite singularities.
method Assumption of holomorphic map and rational cohomology class, proving L4-like estimate and Type I curvature. result Proves L4-like estimate on Ricci curvature and Type I curvature in L2-sense. We construct periodic monopoles (with singularities), i.e. monopoles on R2×S1 possibly singular at a finite collection of points, by gluing methods.
We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial c…
Paper studies heat flow for maps on manifolds, avoiding singularities.
problem Avoiding singularities in heat flow for maps on manifolds.
method Introduces regularized conformal heat flow for n-harmonic maps. result Regularized n-conformal heat flow does not develop finite time singularities. Strict type-II blowup in harmonic map flow is proven to have Hölder continuous body map.
problem Finite-time singularity of harmonic map flow.
method Analysis of outer energy scale and Hölder continuity proof.
result Strictly type-II blowup body map is Hölder continuous.
Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
problem Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
method Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow
result Identifying the lowest eigenfunction with the scaling mode of the special Lagrangian desingularization
Study on singularities of Chern-Ricci flow on complex manifolds.
problem Understanding finite-time singularities of the Chern-Ricci flow.
method Extending Guedj-Lu's approach to establish uniform a priori estimates for degenerate complex Monge-Ampère equations, applied to Chern-Ricci flows on complex log terminal varieties.
result Showed solutions starting from positive currents are smooth outside some analytic subset.
The article explores Helfrich flow with spontaneous curvature, finding singularities and convergence behaviors.
problem Understanding the long-time behavior of Helfrich flow with spontaneous curvature.
method Analyzing the gradient flow of a locally area- and volume-constrained Willmore flow, and applying it to the Helfrich flow.
result For negative spontaneous curvature, the Helfrich flow exhibits finite-time singularities; for positive spontaneous curvature, it converges globally.
Paper studies smoothness of bi-conformal heat flow on 4-manifolds.
problem Smoothness of bi-conformal heat flow on 4-manifolds.
method Introduces bi-conformal heat flow (bi-CHF) and proves global smoothness without finite time singularities.
result Global smoothness and no finite time singularity for bi-conformal heat flow.
Study curve shortening flow on Riemann surfaces with conical singularities.
problem Evolution of curves on Riemann surfaces with singular points.
method Curve shortening flow governed by a degenerate quasilinear parabolic equation.
result Evolving curves stay fixed at singular points and show collapsing and convergence results.
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…
We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is C3-close at some time to a standard neck will develop a neckpinch singularity in finite time, will become asymptotically rotationally symmetric in a space-time neighborhood of its singul…
We show that timelike maximal cylinders in $\RR^{1 + 2}$ always develop singularities in finite time and that, infinitesimally at a generic singularity, their time slices are evolved by a rigid motion or a self-similar motion. We also prove a mild generalization in non-flat backgrounds.
The paper studies stability and singularities of a two-convex level set flow.
problem Stability and singularities of a two-convex level set flow.
method Assumes two-convex initial hypersurface and finitely many singular times, then shows the singular set has finitely many connected components.
result Near each connected component of the singular set, the perturbed flow has the same type of singular set.
In this paper, we study the generalized Lagrangian mean curvature flow in almost Einstein manifold proposed by T. Behrndt. We show that the singularity of this flow is characterized by the second fundamental form. We also show that the rescaled flow at a singularity converges to a finite union of Special Lagrangian con…
We study finite-time collapsing limits of the continuity method. When the continuity method starting from a rational initial Kähler metric on a projective manifold encounters a finite-time volume collapsing, this projective manifold admits a Fano fibration over a lower dimensional base. In this case, we prove the conti…
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 classifies bubbles of Type I singularities in Kähler-Ricci flow on compact surfaces.
problem Classifying bubbles of Type I singularities in Kähler-Ricci flow.
method Analyzes shrinking gradient Kähler-Ricci solitons and their underlying complex manifolds.
result Proves strong form of Feldman-Ilmanen-Knopf conjecture for compact surfaces.
The paper studies singularities in a complex flow related to mean curvature.
problem Investigating singularities in a complex flow related to mean curvature.
method Constructing two distinct examples of singularities using the line bundle mean curvature flow.
result Found a finite time singularity, ruling out long time existence of the flow.
Ricci flow modelled on specific singularities on closed manifolds.
problem Analyzing singularities in Ricci flows.
method Closed manifold Ricci flow with singularity modeled on asymptotically conical shrinkers.
result Ricci flow solution forms a singularity that matches the given asymptotically conical shrinker.
Smooths metrics with nonnegative scalar curvature near singular sets.
problem Approximating metrics with nonnegative scalar curvature near singularities.
method Ricci-DeTurck flow to approximate metrics.
result Approximated metrics converge to the original metric in C∞ away from the singular set. A mean curvature flow starting from a closed embedded hypersurface in Rn+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 …