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.
The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.
problem Proving Strichartz and spectral projection theorems on curved surfaces.
method Using large negative curvature neighborhoods, the study proves theorems on asymptotically conic and Euclidean ends surfaces.
result The study proves theorems without loss of interval on specific types of curved surfaces.
We study the asymptotics of the determinant of Laplacian on a translation surface (a compact Riemann surface equipped with a conformal flat conical metric with trivial holonomy) of genus g with 2g-2 conical points of angle 4πas two conical points collide.
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.
Constructs geodesic lines in curved surfaces using min-max methods.
problem Constructing geodesic lines in curved surfaces where minimization schemes fail.
method Employing min-max methods to construct uncountably many geometrically distinct geodesic lines.
result Proves the existence of uncountably many properly embedded geodesic lines with Morse index ≤ 1.
As part of the general investigation of Ricci flow on complete surfaces with finite total curvature, we study this flow for surfaces with asymptotically conical (which includes as a special case asymptotically Euclidean) geometries. After establishing long-time existence, and in particular the fact that the flow preser…
Study estimates self-shrinker index with conical ends, proving index bound.
problem Estimating the index of self-shrinkers with asymptotically conical ends.
method Constructing Gaussian Harmonic forms and extending index estimates.
result Proves Morse index of self-shrinkers is at least (2g+r-1)/3.
We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface M=Γ\H2 associated with a Fuchsian group of the 1st kind Γ containing parabolic elements. M is t…
The paper proves smoothness of mean curvature flow for generic initial data in 3D and 4D.
problem Smoothness of mean curvature flow for generic initial data.
method Long-time existence and uniqueness result for ancient mean curvature flows.
result Smooth mean curvature flow until disappearance in a round point for low-entropy hypersurfaces in 4D.
In this paper, we establish a framework for the analysis of linear parabolic equations on conical surfaces and use them to study the conical Ricci flow. In particular, we prove the long time existence of the conical Ricci flow for general cone angle and show that this solution has the optimal regularity, namely, the ti…
Vortices on conical surfaces embedded in hyperbolic space.
problem Embedding Abelian vortices on conical surfaces in hyperbolic space.
method Analyzing elliptic sinh-Gordon and Tzitzeica equations, asymptotic analysis of Painleve III ODE.
result Radial solutions can be globally embedded in hyperbolic space.
Krein's formula for conic Laplacians on compact Riemann surfaces
problem Establishing Krein's formula for self-adjoint extensions of conic Laplacians on compact Riemann surfaces
method Using finite-dimensional symplectic space of critical asymptotic boundary data
result Deriving a trace identity for the resolvent difference and proving a comparison formula for the positive-spectrum zeta determinants
Existence proof of noncompact self-shrinkers with arbitrary genus.
problem Existence of noncompact self-shrinkers with arbitrary genus.
method Employing min-max techniques to rigorously prove existence.
result Confirmation of one asymptotically conical end for large genus self-shrinkers.
Regularized zeta function for polyhedra calculated from Riemann surface invariants.
problem Calculating a spectral invariant for polyhedra using zeta function regularization.
method Holomorphic invariants and conical points of the metric, sewing two polyhedra, self-adjoint extensions.
result Explicit expression for spectral invariant through Riemann surface invariants.
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.
New self-expander found between two given asymptotic ones.
problem Finding new self-expanders between given asymptotic ones.
method Developed a min-max theory for asymptotically conical self-expanders of mean curvature flow.
result Existence of a new asymptotically conical self-expander trapped between two given ones.
Study X-ray transform on conic spaces, proving injectivity under certain conditions.
problem Injectivity of geodesic X-ray transform on conic metrics.
method Injectivity under non-trapping and no conjugate point assumptions.
result Injectivity of geodesic X-ray transform for asymptotically conic metrics.
Study on potential behavior in special geometric spaces.
problem Understanding potential behavior in specific geometric spaces.
method Analyzing asymptotic behavior of p-capacitary potentials and weak Inverse Mean Curvature Flow. result Characterized the behavior of potentials in Asymptotically Conical manifolds.
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.
Integer degree found for conical self-expanders.
problem Existence of integer degree for conical self-expanders.
method Proper projection map from parameterizations to asymptotic cones.
result Existence of integer degree for conical self-expanders.
Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete m-dimensional manifolds with singularities on sets of codimension at least 2. With certain restrictions on the metric, we establish that the spectrum is discrete …
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.
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.
Study instantons on asymptotically conical Spin(7)-manifolds, identifying deformation spaces.
problem Deformation theory of instantons on specific Spin(7)-manifolds.
method Relating deformation complex to spinors, identifying kernel of twisted negative Dirac operator.
result Virtual dimension of moduli space calculated using index theorem and Dirac operator spectrum.
Mean curvature flow with a conical end becomes stable.
problem Stability of mean curvature flow with a conical end.
method Analysis of asymptotic behavior and entropy.
result Expanding solution is asymptotically stable.
Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.
problem Degeneration of asymptotically conical Ricci-flat Kähler metrics.
method Analysis of Kähler class degeneration and convergence of metrics.
result Construction of singular Calabi-Yau metrics and their metric geometry.
The paper shows how to stabilize perturbed Kähler-Ricci solitons.
problem Stabilizing perturbed Kähler-Ricci solitons.
method Normalized Kähler-Ricci flow starting from perturbed metrics.
result The flow converges to an asymptotically conical gradient expanding Kähler-Ricci soliton.
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.
Study sharp asymptotic estimates for conical ends using frequency functions.
problem Proving sharp asymptotic estimates for almost eigenfunctions of drift Laplacians on conical ends.
method Used a weighted variant of Almgren's frequency functions.
result Obtained a purely elliptic proof of uniqueness of self-shrinkers and self-expanders of the mean curvature flow.
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.
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.
The study shows that certain nearly G2 and nearly Kähler conifolds cannot be resolved by gluing asymptotically conical G2 and Calabi-Yau manifolds.
problem The inability to resolve nearly G2 and nearly Kähler conifolds by gluing asymptotically conical G2 and Calabi-Yau manifolds.
method Topological analysis of asymptotically conical G2 and Calabi-Yau manifolds to show conditions under which resolutions are impossible.
result For certain rates of the metric, the G2 4-form and Kähler form cannot be simultaneously exact, leading to non-existence of resolutions.
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.
Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.
problem Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
method Established optimal upper bounds for cone angles of Kähler-Einstein metrics with conical singularities.
result Optimal upper bounds for conical Kähler-Einstein metrics on K-unstable del Pezzo surfaces.
Study of mean curvature flows with conical singularities using mathematical techniques.
problem Understanding the dynamics of mean curvature flows near conical singularities.
method Feynman-Kac formula and invariant cone method for noncompact settings.
result Generic initial perturbations avoid conical singularities in mean curvature flows.
Compactness proven for specific types of self-expanders in mean curvature flow.
problem Proving compactness for asymptotically conical self-expanders of mean curvature flow.
method Analyzing families of self-expanders and showing compactness in locally smooth topology.
result Properness of the projection map for specified classes of self-expanders.
Generalized Gauss-Bonnet formula for conical metrics on compact Riemann surfaces.
problem Proving a generalized Gauss-Bonnet formula for conical metrics.
method Analyzing Gaussian curvature and Lebesgue integrability.
result Proved a generalized Gauss-Bonnet formula for conical metrics.
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.
Strict convexity proven for certain self-expanders in high dimensions.
problem Convexity of self-expanders in mean curvature flow.
method Investigation of convexity properties for asymptotically conical self-expanders.
result Strict convexity proven for n-dimensional self-expanders. Study shows smooth manifold of self-expanders in mean curvature flow.
problem Understanding the structure of self-expanders in mean curvature flow.
method Analyzes asymptotically conical self-expanders as a smooth Banach manifold.
result Non-degenerate self-expanders are generic.
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.
Backward propagation rules for warped products under Ricci flow.
problem Understanding how warped product structures behave under Ricci flow.
method Establishing sufficient conditions for backward propagation of warped product structures.
result Asymptotically conical shrinkers are multiply-warped products over Einstein manifolds.
The study bounds and characterizes surfaces containing smooth conics and twistor fibers in a flag threefold.
problem Bounding and characterizing surfaces containing smooth conics and twistor fibers in a flag threefold.
method Analyzing the family of smooth conics and using algebraic properties to construct surfaces.
result The only smooth cases of surfaces containing infinitely many twistor fibers are of bidegree (1,1).
In this paper we define a new convergence called "asymptotically conic convergence" in which a smooth family of Riemannian metrics on a fixed compact manifold degenerate to a metric with isolated conic singularity. Our results are: convergence of the spectrum of the geometric Laplacians and uniform convergence of the c…
We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.
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 linear differential operators on special manifolds.
problem Analyzing elliptic differential operators on specific types of manifolds.
method Examining a linear elliptic differential operator of the form Δ + V - λ on quasi-asymptotically conical manifolds.
result Established an isomorphism theorem for these operators.
Proves unknottedness of certain 3D shapes with multiple ends.
problem Determining the structure of complex 3D shapes.
method Used mean curvature flow to analyze shapes with multiple ends.
result Proves unknottedness of shapes with multiple asymptotically conical ends.