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.
Study on Kähler-Ricci flow's infinite-time singularities.
problem Understanding singularities in Kähler-Ricci flow.
method Relates flow's singularity type to fibration's indexes.
result Observation of singularity type's relation to fibration indexes.
Study on estimating curvature and volume in singular Ricci flows.
problem Quantitative analysis of singular Ricci flows.
method Estimates on curvature and volume, set of singular times.
result Established several quantitative results about singular Ricci flows.
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.
The paper shows singularities in geometric flows can wander off.
problem Understanding the dynamics and behavior of singularities in geometric flows.
method Combining dynamical properties with smoothing effects for long time behavior.
result Singularities in certain geometric flows are shown to wander off, not returning to a dilated or translated copy of themselves.
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.
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.
This paper tackles the dynamics of singularities in geometric flows.
problem Understanding the behavior of singularities in geometric flows over long time.
method By incorporating dynamical properties, the paper shows smoothing for long time for generic initial conditions.
result The singularities are shown to be the simplest possible in an important special case.
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 …
Mean curvature flow shows a surface fattening at its first singular point.
problem Understanding the behavior of surfaces under mean curvature flow.
method Proving existence of a genus-g surface with specific properties under mean curvature flow. result The genus-g surface fattens at the first singular time, and as g increases, the shrinker converges to a multiplicity 2 plane. Classifies invariant hypersurfaces with singularities.
problem Classifying invariant hypersurfaces with singularities.
method Analyzing O(p)imesO(q)-invariant constant mean curvature hypersurfaces. result Solved Wu-yi Hsiang's conjecture.
Gromov-Hausdorff convergence of time-slices of singular Ricci flows
problem Gromov-Hausdorff convergence of time-slices of singular Ricci flows
method Completion of singular Ricci flow with respect to a natural spacetime distance
result Gromov-Hausdorff convergence at the first singular time
Uniqueness of conical flows helps understand singularities in surface flows.
problem Understanding singularities in surface flows.
method Analyzing asymptotically conical tangent flows.
result Uniqueness of multiplicity-one asymptotically conical tangent flows.
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…
Mean curvature flow shows singularities on smooth surfaces.
problem Understanding singularities in mean curvature flow.
method Analyzing spherical or nondegenerate neck pinches.
result First singular time has isolated singularities.
Study of bubble-sheet singularity in mean curvature flow.
problem Analyzing a specific type of singularity in mean curvature flow.
method Deriving an asymptotic profile for a neighborhood of the singularity.
result An asymptotic profile for a neighborhood of the singularity is derived.
Study resolves flow through cylindrical singularities, proving nonfattening.
problem Analyzing free boundary flow through cylindrical singularities.
method Foundational results for free boundary Brakke flows and classification of ancient flows.
result Proves all cylindrical singularities have a mean-convex neighborhood, leading to well-posed flow.
The paper classifies solutions to Kapustin-Witten equations with specific singularities.
problem Classifying solutions to Kapustin-Witten equations with Nahm pole singularities.
method Analytical classification of solutions with detailed singularity analysis.
result Classification of solutions with Nahm pole and generalized Nahm pole singularities.
Proves heat expansion for Laplacian on a singularity.
problem Analytic hypersurface with isolated singularity and Laplacian heat expansion.
method Local parametrization, Newton scheme, quasihomogeneous tangent cone, local models with irregular singularities.
result Existence of small time heat expansion for Laplace operator.
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.
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.
Alternative approach to regularize time-dependent singular Lagrangian systems.
problem Regularizing time-dependent singular Lagrangian systems.
method Employing the coisotropic embedding theorem and the Tulczyjew isomorphism.
result Uniqueness of the Lagrangian regularization to first order.
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 constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.
problem Infinite-time singularities in Lagrangian mean curvature flow.
method Constructing solutions by gluing special Lagrangian 'Lawlor necks' and analyzing dynamics of neck size.
result The flow decomposes initial data into a union of special Lagrangians intersecting at one point.
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.
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.
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.
Two new methods improve forecasting of functional time series data.
problem Forecasting of functional time-dependent data.
method Functional Singular Spectrum Analysis (FSFA) based forecasting methods.
result Our methods outperform existing algorithms for periodic stochastic processes.
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…
Proves smoothness of conical singularities in mean curvature flow.
problem Resolving singularities in mean curvature flow.
method Analyzes smooth hypersurfaces with isolated conical singularities.
result Smoothness of level set flow through asymptotically conical singularities.
Study confirms mean convexity of singularity neighborhood in mean curvature flow.
problem Understanding the structure of singularities formed by mean curvature flow.
method Detailed analysis of a small neighborhood around the blowup point.
result The neighborhood is mean convex, confirming a conjecture.
Study of singularities in mean curvature flow with focus on S3imesR.
problem Understanding singularities in mean curvature flow.
method Detailed analysis of singularities modeled on S3imesR, using normal form transformations and rescaled MCF. result Proves mean convexity and singularity isolation in a small neighborhood, conjectures singularity formation in entire neighborhood.
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.
Study on axially symmetric surfaces' flow, showing all singularities are of type I.
problem Understanding singularity formation in axially symmetric mean curvature flow.
method Analysis of Neumann boundary conditions and type of singularities.
result All singularities at first time are of type I.
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.
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…
Constructs ancient solutions to mean curvature flow with prescribed singular sets.
problem Creating mean-convex ancient solutions with specific singular sets.
method Constructs solutions with a prescribed singular set Kimes{0} using mean curvature flow in a Riemannian metric. result Constructs ancient solutions with a first-time singular set exactly Kimes{0}. Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.
problem Understanding the nature and behavior of singularities in level set flow.
method Analytical approach using Lojasiewicz inequality and curvature blow-up rates.
result The arrival time is C2 near a critical point if and only if it satisfies a Lojasiewicz inequality. 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 paper we investigate the singularities of Lagrangian mean curvature flows in Cm 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}×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 shows how solutions to Yamabe flow can develop Type II singularities.
problem Existence and detailed analysis of Type II singularities in Yamabe flow.
method Detailed asymptotic analysis and blow-up rate calculation.
result Yamabe flow solutions can converge to a steady soliton after blow-up.
Study calculates ring structure in instanton homology for a surface with points.
problem Calculating the ring structure in instanton homology for a surface with points.
method Used singular instanton Floer homology and excision formula.
result Proved an excision formula for instanton Floer homology when n=1.
Study shows no C0 extensions across null boundaries in expanding singularities.
problem Existence of C0-extensions across null boundaries in expanding singularities. method Analysis of globally hyperbolic space-times with expanding singularities.
result No C0-extensions across compact boundaries exist, while the boundary must be null. Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.
problem Understanding neck pinches in Lagrangian flows.
method Introduced nondegenerate neck pinch and teardrop singularities, proving stability and answering questions.
result Nondegenerate neck pinches are stable and can be perturbed to nondegenerate 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.
problem Proving strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
method Established Lojasiewicz inequality for pointed W-entropy under cylindrical geometry assumption. result Strong uniqueness of cylindrical tangent flows at first singular time of Ricci flow proved.
Consider a family of smooth immersions F(⋅,t):Mn→Rn+1 of closed hypersurfaces in Rn+1 moving by the mean curvature flow ∂t∂F(p,t)=−H(p,t)⋅ν(p,t), for t∈[0,T). We prove that the mean curvature blows up at the first singular time T if all singu…