Study conic singular manifolds, proving Lipschitz normal embedding.
problem Understanding metric properties of conic singular manifolds.
method Analyzing interplay between conic and asymptotically conic behavior.
result Proves Lipschitz normal embedding for conic singular sub-manifolds.
Proves conditions for positive scalar curvature on certain manifolds with conical singularities.
problem Conditions for positive scalar curvature on manifolds with isolated conical singularities.
method Analyzes isolated conical singularities and uses Geroch type results.
result No metric with positive scalar curvature on X#Tn with isolated conical singularity. 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.
Proves positive mass theorem for AF spin manifolds with conical singularities.
problem Proving the positive mass theorem for singular metrics on AF manifolds.
method Analyzes AF spin manifolds with isolated conical singularities, allowing topological singularities.
result Proves the positive mass theorem for AF spin manifolds with conical singularities.
Study describes ALF instantons with conical singularities.
problem Understanding ALF instantons with conical singularities.
method Applied techniques from previous work.
result Only 4D subfamilies can be smoothly compactified with conical singularities.
Study Schauder estimates for conical singularities using Kähler metrics.
problem Derive interior Schauder estimates for equations with conical singularities.
method Use linear elliptic and parabolic equations with a Kähler metric of conical singularities.
result Prove short-time existence of conical Kähler-Ricci flow.
Survey on Ricci flow on spaces with conical singularities.
problem Analyzing Ricci flow on spaces with isolated conical singularities.
method Ricci de Turck flow preserving conical singularities, stability of Ricci flat metrics, preservation of positive scalar curvature under certain conditions.
result Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is preserved under the flow.
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.
Desingularizes conically singular Cayley submanifolds.
problem Constructing fibrations of compact Spin(7) manifolds by Cayley submanifolds.
method Desingularization through gluing rescaled asymptotically conical submanifolds.
result Conically singular Cayley submanifolds can be desingularized.
Constructs surfaces with conical singularities using variational methods.
problem Creating Hamiltonian Stationary Surfaces with specific singularities.
method Variational methods and convergence process similar to Ginzburg-Landau analysis.
result Obtained surfaces with prescribed conical singularities related to optimal Wente constants.
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.
Developed theory of Perelman's W-functional on manifolds with conical singularities.
problem Analyzing manifolds with conical singularities using Perelman's W-functional.
method Theory development and mathematical analysis on manifolds with isolated conical singularities.
result Existence and asymptotic order of minimizers for the W-functional on manifolds with conical singularities.
Study curve shortening flow on Riemann surfaces with conic singularities.
problem Analyzing curve shortening flow on surfaces with conic singularities.
method Generalized Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. Reproofed Gage-Hamilton-Grayson theorem. Proved CSF can't touch conic singularities with cone angles ≤ π.
result CSF can't touch conic singularities with cone angles ≤ π for embedded simple closed curves.
Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.
problem Understanding the Lipschitz geometry of conic singular sub-manifolds.
method Analyzing the metric properties of conic singular sub-manifolds in compact non-Euclidean manifolds.
result Connected conic singular sub-manifolds are Lipschitz Normally Embedded.
Proves immediate transversality for conic singularities.
problem Transversality issues in Morse complexes with conic singularities.
method Proves immediate transversality for conic singularities in Morse complexes.
result Immediate transversality holds for conic singularities.
The paper analyzes Perelman's entropies on manifolds with conical singularities.
problem Analyzing manifolds with conical singularities using Perelman's entropies.
method Employing singular Ricci de Turck flow and Perelman's entropies to study manifolds with conical singularities.
result Entropy is monotone along the singular Ricci de Turck flow and helps in proving properties of Ricci solitons.
The paper proves stability of Ricci de Turck flow near conical singularities.
problem Stability of Ricci de Turck flow near conical singularities.
method Constructing a Ricci de Turck flow starting close to a Ricci-flat metric with isolated conical singularities.
result The flow converges to a singular Ricci-flat metric under certain conditions.
The paper extends Perelman's λ-functional to manifolds with conical singularities.
problem Analyzing the spectrum of Schrödinger operators on manifolds with conical singularities.
method Proving spectral properties and extending Perelman's λ-functional.
result The spectrum of the Schrödinger operator on compact manifolds with conical singularities is discrete and asymptotic behavior of eigenfunctions near the singularity is obtained.
Survey on metrics with conic singularities on Riemann surfaces.
problem Conformal metrics with conic singularities on Riemann surfaces.
method Study of metrics with constant positive curvature and conic singularities.
result Simplified proofs for interesting cases, including those by C. L. Chai, C. S Lin, and C. L. Wang.
Study heat trace expansion on manifolds with conic points.
problem Analyzing heat diffusion on manifolds with sharp corners.
method Using the Singular Asymptotics Lemma to derive an expansion.
result Detailed asymptotic expansion reveals geometric insights.
The abstract proves spherical surface decompositions with conical singularities.
problem Decomposing surfaces with spherical metrics and conical singularities.
method Geometric triangulations and irreducible components of standard shapes.
result Spherical polygons, including half-spherical concave polygons, can be arbitrarily complicated.
We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work on the unobstructed case. We interpret the analytic obstructions geometrically via the obstruction theory for defor…
Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.
problem Existence of conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundaries.
method Variational argument to derive existence results for surfaces with at least two boundary components.
result First result in this setting for surfaces with conical singularities of both positive and negative orders.
Constructs scalar-flat Kähler metrics with varying conical singularities.
problem Creating scalar-flat Kähler metrics with specific singularities.
method Using LeBrun's ansatz, constructs metrics with varying conical singularities.
result Constructs complete scalar-flat Kähler metrics with prescribed conical singularities.
Proves mass theorem for AF manifolds with conical singularities.
problem Proving the positive mass theorem for specific types of manifolds.
method Conformal blow up technique applied to AF manifolds with isolated conical singularities.
result Positive mass theorem proven for the specified manifolds.
Proves curvature comparison theorem for manifolds with conical singularities.
problem Comparing scalar mean curvature of manifolds with conical singularities.
method Uses Dirac operator and index theory to prove curvature comparison theorem.
result Proves curvature comparison theorem without knowing the index of the twisted Dirac operator.
New approach to prescribing Gaussian curvature on spheres with conical singularities.
problem Prescribing Gaussian curvature on the 2-sphere with conical singularities.
method Variational methods not relying on Moser-Trudinger inequality, plus precompactness theorem.
result Sufficient conditions for a positive function to be the Gaussian curvature of a conformal conical metric.
The Yamabe flow preserves conical singularities under certain conditions.
problem Preserving conical singularities under the Yamabe flow.
method Using maximal Lq-regularity theory for conically degenerate operators. result Asymptotic expansion of the evolving metric near conical tips.
Resolves singular cscK varieties to smooth ones.
problem Resolving conically singular cscK varieties to smooth cscK manifolds.
method Glueing result for crepant resolutions of cscK varieties with discrete automorphism groups.
result Shows a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.
Classifies 4D toric Hermitian ALF metrics with conical singularities.
problem Classifying specific types of 4D Riemannian metrics.
method Explicit formulas provided for classification.
result Examples of metrics with conical singularities have infinitely many distinct topologies.
Gradient flow solves curvature problem on surfaces with conical points.
problem Solving curvature problems on surfaces with conical points.
method Introduced and analyzed a gradient flow on closed Riemann surfaces with conical singularities.
result Gradient flow proves long-time existence and convergence under certain assumptions.
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.
New special Lagrangian submanifolds with multiple conical singularities found.
problem Existence of special Lagrangian submanifolds with multiple conical singularities.
method Extending Caffarelli-Hardt-Simon perturbation argument to special Lagrangian setting and proving a bridge principle.
result Existence of conically singular special Lagrangian submanifolds with prescribed regular tangent cones.
Study the spectral geometry of surfaces with curved conic singularities.
problem Understanding the spectral properties of surfaces with conic singularities.
method Using the heat trace expansion, express spectral geometry terms through the geometry and curvature of the singularities.
result The first few terms in the heat trace expansion are expressed through the geometry and curvature of the singularities.
Paper proves estimates for metrics with conic singularities.
problem Proving estimates for metrics with conic singularities.
method Reformulated Alexandrov's maximum principle for zero order estimates and Chen-Cheng's framework for higher order estimates.
result A priori estimates for conic singularities metrics with prescribed scalar curvature.
Study on prescribing positive curvature with conical singularities on a sphere.
problem Prescribing positive curvature with conical singularities on a sphere.
method Fine analysis of bubble trees and an area identity in the convergence process.
result Criterion for nonexistence in an open region of the prescribing data.
Article provides conditions for spherical metrics with conical singularities.
problem Existence of metrics with conical singularities on a 2-sphere.
method Criterion based on linear inequalities in prescribed angles.
result Necessary and sufficient condition for metric existence.
New Calabi-Yau metrics with conical singularities are created near complex lines.
problem Creating Calabi-Yau metrics with conical singularities near complex lines.
method Using branched covering arguments to construct metrics with conical singularities.
result Calabi-Yau metrics with unstable conical singularities are successfully created.
Proves existence of Yamabe metrics on conical manifolds with conical points and links.
problem Existence of Yamabe metrics on singular manifolds with conical points and links.
method Derives a counterpart of Aubin's result, uses conical links and Fourier analysis, adds lower-order correction to standard bubbles.
result Derives asymptotic expansions on the Yamabe quotient for generic type metrics.
Anomaly term vanishes for smooth conical spaces, non-trivial for cones over tori.
problem Analyzing anomaly terms in spaces with conical singularities.
method Proof of vanishing anomaly term for smooth conical spaces; demonstration of non-triviality for cones over tori.
result Anomaly term vanishes for smooth conical spaces, non-trivial for cones over tori.
Deformations of singular Cayley submanifolds studied.
problem Constructing fibrations of compact Spin(7) manifolds.
method Deformation theory of conically singular and asymptotically conical Cayley submanifolds.
result Detailed description of the deformation theory.
Backwards uniqueness proved for flows with asymptotically conical singularities.
problem Proving uniqueness of mean curvature flows with specific singularities.
method Developed new global tools to handle singularities, asymptotic structure, and smooth parts of flows.
result Backwards uniqueness for mean curvature flows with asymptotically conical singularities proved.
Study shows scalar curvature rate for conical Kähler-Ricci flow near singularity.
problem Behavior of scalar curvature near finite time singularities in conical Kähler-Ricci flow.
method Investigated normalized conical Kähler-Ricci flow with finite maximal existence time, proving scalar curvature bounded by C/(T−t)2. result Scalar curvature of ωt is bounded by C/(T−t)2 under a contraction associated with limiting cohomology class [ωT]. Study finds existence of Q-curvature metrics on even-dimensional manifolds with conical singularities.
problem Existence of Q-curvature metrics on manifolds with conical singularities. method Blow-up analysis of a 2mth-order PDE and variational min-max argument. result First existence result for supercritical conic manifolds (except spheres).
The study creates Kähler-Einstein metrics with conical singularities.
problem Producing Kähler-Einstein metrics with specific singularities.
method Using the Calabi ansatz for metrics with conical singularities.
result Created regular Calabi-Yau cones and Kähler-Einstein metrics with negative Ricci.
Paper proves rigidity of de-Sitter tori with conical singularities.
problem Global rigidity of de-Sitter tori with singularities.
method Introduced constant curvature Lorentzian surfaces with conical singularities and proved rigidity via topological dynamics.
result De-Sitter tori with a single singularity are determined by their lightlike bi-foliation.
Paper proves unique conical tangent flows in all dimensions.
problem Proving uniqueness of conical singularities in mean curvature flows.
method Based on Chodosh-Schulze's work in hypersurfaces, extended to all codimensions.
result Uniqueness of asymptotically conical tangent flows in all codimensions.
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.