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.
A maximal surface $\sb$ with isolated singularities in a complete flat Lorentzian 3-manifold N is said to be entire if it lifts to a (periodic) entire multigraph $\tilde{\sb}$ in ł3. In addition, $\sb$ is called of finite type if it has finite topology, finitely many singular points and $\tilde{\sb}$ is finitely …
Study ruled surfaces with finite multiplicity, focusing on their curves and singularities.
problem Understanding ruled surfaces with finite multiplicity.
method Analyzing striction curves and singularities of ruled surfaces.
result Geometric meanings of invariants related to ruled surfaces.
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.
Proves finitely generated graded rings for klt singularities.
problem Understanding the structure of klt singularities.
method Analyzes graded rings associated with minimizers of normalized volume functions.
result Graded rings are finitely generated for klt singularities.
Extends Serre-Swan theorem to all finitely generated modules over smooth functions.
problem Classical Serre-Swan theorem limitations.
method Introduces tepui fibrations and singular vector bundles.
result Realizes all finitely generated modules over smooth functions.
The paper provides criteria and curvatures for singularities of curves in R^N.
problem Tackles the classification and characterization of singularities of curves in R^N.
method Systematic procedure for constructing criteria, explicit criteria for multiplicities 2-4, generalized curvatures.
result Generalized curvatures reinterpret Fukui's theorem for curves of finite multiplicities.
Proves finite step termination of Kähler-Einstein metric singularity formation.
problem Singularity formation of Kähler-Einstein metrics.
method Finite step termination of bubble trees for singularity formation.
result Finite step termination of Kähler-Einstein metric singularity formation proved in non-collapsing situation.
Functorial semi-norms on singular homology give refined "size" information on singular homology classes. A fundamental example is the l^1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the l^1-semi-norm.
Study on singularities of frontal surfaces, classifying under equivalence.
problem Classifying singularities of frontal surfaces.
method Classification under left-right-equivalence, introduction of frontalisation, definition of cuspidal and transverse double point curves.
result Frontal surfaces have finite codimension if and only if the curves are reduced.
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.
Every graph can be represented as a singular set of a special surface.
problem Representing any finite graph as the singular set of a compact 3D surface.
method Constructing a calibrated 3-dimensional homologically area minimizing surface with a special Lagrangian form.
result The singular set of the surface is precisely the given graph.
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.
We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves …
We describe two-dimensional potential Schrodinger and Dirac operators which are finite-gap at one energy level and have singular spectral curves. It appears that the singularities can be rather complicated. Such Dirac operators appear as the spectral curves of tori immersed into the three-space.
We show that the number of entire maximal graphs with finitely many singular points that are conformally equivalent is a universal constant that depends only on the number of singularities, namely 2^$ for graphs with n+1 singularities. We also give an explicit description of the family of entire maximal graphs with a f…
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.
Study bounds singular set of minimal hypersurfaces with index control.
problem Estimating singular set size of minimal hypersurfaces.
method Finite index and null singular set conditions on integral varifolds.
result Local measure bounds on singular set and upper Minkowski content.
Indices of vector fields and 1-forms studied for singular varieties and actions.
problem Understanding indices of vector fields and 1-forms in various contexts.
method Generalization to singular varieties and actions of finite groups.
result New insights into indices of vector fields and 1-forms.
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 paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show tha…
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 calculate finite volumes of moduli spaces of flat surfaces with conical singularities.
problem Calculating volumes of moduli spaces of flat surfaces with prescribed conical singularities.
method Induction on the Euler characteristics of the punctured surface for almost all orders of the singularities.
result Explicit computation of volumes is possible.
Characterizes monodromies of projective structures on finite-type surfaces.
problem Understanding monodromies of projective structures on finite-type surfaces.
method Geometrical/topological study of local conical projective structures.
result Any representation can be represented as the holonomy of a branched projective structure.
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.
We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal crossing singularities only, so that the fundamental group of S is isomorphic to G. …
The paper proves ideal triangulations and disk unfolding for singular flat surfaces.
problem Proving ideal triangulations and disk unfolding for singular flat surfaces.
method Using geodesic triangulation and finite geodesic connections.
result Each singular flat surface has an ideal triangulation and can be unfolded into a flat disk.
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.
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.
The study examines vector fields with integer singularities in 3D balls.
problem Characterizing the strong Lp-closure of vector fields with finitely many integer singularities. method Characterization and decomposition of vector fields with finitely many integer singularities.
result Decomposition theorem for elements in LZ1(B), revealing information about mass-minimizing currents. The paper defines subgroups of camomile type and studies singular braids and links.
problem Understanding subgroups of camomile type in singular braid groups.
method Presentations, defining relations, and group theory.
result The center of SGn is a direct factor in SPn but not in SPn. We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.
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.
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.
An HCMU metric is a conformal metric which has a finite number of singularities on a compact Riemann surface and satisfies the equation of the extremal Kähler metric. In this paper, we give a necessary and sufficient condition for the existence of a kind of HCMU metrics which has both cusp singularities and conical sin…
We study singularities of Lagrangian mean curvature flow in $\C^n$ when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonia…
Curvature criteria for A-simple singularities and their parallel curves identified.
problem Determining singularity types of A-simple singularities and their parallel curves.
method Defined curvature parameters and criteria for A-simple singularities.
result Criteria to determine singularity types of A-simple singularities and their parallel curves.
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. 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.
We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic p≥2. In this cas…
Defines Perelman's functionals on manifolds with non-isolated conical singularities.
problem Defining functionals on manifolds with non-isolated conical singularities.
method Starting from a spectral point of view for the Perelman's λ-functional, defining the spectrum of Schrödinger operator and proving the existence of discrete eigenvalues.
result Proves the existence of the infimum of W-functional and obtains asymptotic behavior of eigenfunctions.
We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the si…
Study describes singularities of height functions on specific singular surfaces.
problem Analyzing singularities of height functions on singular surfaces.
method Using geometric language and blowing-ups, investigate singularities of height functions and dual surfaces.
result Characterized singularities of height functions and dual surfaces on specific singular surfaces.
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…
We determine the global behavior of every C^2-solution to the two-dimensional degenerate Monge-Ampere equation, u_{xx}u_{yy}-u_{xy}^2=0, over the finitely punctured plane. With this, we classify every solution in the once or twice punctured plane. Moreover, when we have more than two singularities, if the solution u is…
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.
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.