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 finds conditions for conformal deformations to constant scalar curvature in conic metrics.
problem Finding conditions for conformal deformations to constant scalar curvature in conic metrics.
method Analyzes conformal deformations within a class of incomplete Riemannian metrics that generalize conic orbifold singularities.
result Determines sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1.
Based on C. Li and Y. Rubinstein's upper bisectional curvature bound estimate for the conic Kähler metric, we can construct a smoothing sequence for the conic metric with uniformly upper bisectional curvature bound. For the conic metric along a simple normal crossing divisor with triple or higher multiple points we may…
Study on spherical conical metrics and their reducibility on compact Riemann surfaces.
problem Existence and geometric structure of reducible spherical conical metrics.
method Analysis of monodromy groups and geometric cutting of surfaces.
result Existence of reducible spherical conical metrics with saddle points on the same geodesic.
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.
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 paper classifies Landsberg metrics on a 2D Lie group and proves a conjecture.
problem Classifying Landsberg metrics on a 2D Lie group.
method Analyzing left invariant conic Finsler metrics on a 2D non-Abelian Lie group.
result Any left invariant conic Landsberg metric on G must be Berwald. Unique conical Kähler-Einstein metric found on Fano manifold.
problem Existence of conical Kähler-Einstein metrics on Fano manifolds.
method Proved existence of a unique conical Kähler-Einstein metric with admissible cone angles.
result Existence and uniqueness of conical Kähler-Einstein metric.
Study constant Q-curvature metrics on conic 4-manifolds.
problem Find metrics with constant Q-curvature on conic 4-manifolds.
method Analyze related differential equations in the given conformal class.
result Solve constant Q-curvature problem on conic 4-manifolds.
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.
The purpose of this paper is to generalize the convexity of Mabuchi's functional to the conic setting. We first established a frame to study conic cscK metrics, and then the conic Mabuchi functional was introduced in such a way that conic cscK metrics are its critical points. Finally we proved the convexity of the coni…
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.
Unique circle patterns on spheres found for spherical conical metrics.
problem Non-uniqueness in circle packing for spherical metrics.
method Prescribed geodesic total curvature instead of cone angles.
result Unique existence of circle patterns for spherical conical metrics.
Proves conic version of YTD conjecture on log Fano manifolds.
problem Existence of conic Kahler-Einstein metrics on log Fano manifolds.
method Proof of the YTD conjecture for conic Kahler-Einstein metrics.
result Proven existence of conic Kahler-Einstein metrics on log Fano manifolds.
New connection between complex analysis and PDE for spherical conical metrics.
problem Understanding spectral properties of spherical conical metrics.
method Analyzing monodromy, eigenfunctions, and developing maps.
result Monodromy reducibility implies real-valued eigenfunction with eigenvalue 2.
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.
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.
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. 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.
We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff limit. This is a gene…
Researchers find necessary conditions for conic 4-spheres with constant σ₂ metrics.
problem Finding constant σ₂ metrics on conic 4-spheres.
method Analyzing earlier works and using necessary conditions to find existence.
result Uniqueness of conic 4-spheres with constant σ₂ metrics under certain conditions.
Characterizes conical angles for metrics with dihedral symmetry.
problem Understanding metrics with specific symmetry properties.
method Using recent results on local invariants of quadratic differentials.
result Complete characterization of conical angles for dihedral spherical metrics.
Extends Cheeger-Colding Theory to conic Kahler-Einstein metrics.
problem Proving Yau-Tian-Donaldson conjecture for Fano varieties with specific singularities.
method Extension of Cheeger-Colding Theory.
result Technical tool for proving Y-T-D conjecture.
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.
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.
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.
The abstract discusses finding metrics and connections on conics in CP^2.
problem Finding metrics and connections on conics in CP^2.
method Application of the method of [4]
result Metric and connection found on the space of conics in CP^2.
Compactifies Riemann surface metrics with conical singularities.
problem Constant curvature metrics on Riemann surfaces with conical singularities.
method Compactified spaces and study local deformation of metrics.
result Sharp regularity theorem for coalescing conic points.
Local rigidity shown for certain spherical conical metrics.
problem Deformation of spherical conical metrics with cone angles > 2π.
method Synthetic geometry approach.
result Local rigidity in the choice of cone positions for a specific family of metrics.
Investigates uniqueness of conic cscK metrics with cone angles less than π.
problem Uniqueness of conic constant scalar curvature Kähler metrics with cone singularities.
method Introduced a new Hölder space $\cC^{4,\a,\b}$ to study regularities, proved regularity of conic cscK metrics, and established reductivity through careful study of the conic Lichnerowicz operator.
result Any $\cC^{2,\a,\b}$ conic cscK metric is indeed of class $\cC^{4,\a,\b}$.
The paper proves existence and uniqueness of parabolic metrics on Riemann surfaces with conical singularities.
problem Existence and uniqueness of parabolic metrics on Riemann surfaces with conical singularities.
method Complex analysis
result Proves existence and uniqueness of parabolic metrics on Riemann surfaces with conical singularities.
Analytic torsion studied for fibred boundary metrics, with applications to conic degeneration.
problem Analytic torsion of fibred boundary metrics and conic degeneration.
method Established invariance and gluing formula for renormalized analytic torsion under deformations of metrics.
result Recovery of a result by Sher and Guillarmou about analytic torsion under conic degeneration.
Study conical Kaehler-Einstein metrics' curvature, proving regularity and geometric conditions.
problem Curvature of conical Kaehler-Einstein metrics on Kaehler manifolds.
method Proved regularity for Monge-Ampère equations on Kaehler manifolds with smooth divisors, applied to conical metrics.
result Geometric condition for boundedness of conical singularities.
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.
In the category of metrics with conical singularities along a smooth divisor with angle in (0,2π), we show that locally defined weak solutions (C1,1−solutions) to the Kähler-Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coord…
The paper proves stability of Kähler-Ricci flows on Fano manifolds.
problem Stability of conical Kähler-Ricci flows on Fano manifolds.
method Using the boundedness of Log Mabuchi energy, the paper proves stability of conical Kähler-Einstein metrics.
result For any β' close to β, the conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric.
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 explores eigenfunctions of spherical conical metrics using harmonic maps to spheres.
problem Existence and properties of eigenfunctions for spherical conical metrics.
method Application of multivalued harmonic maps to spheres and algebraic constructions.
result New criteria and examples of metrics with many 2-eigenfunctions.
The paper proves the existence of Sasaki-Einstein metrics on specific Sasakian manifolds.
problem Proving the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five.
method Deriving uniform L^{4}-bounds and analyzing the conic Sasaki-Ricci flow.
result Existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five.
The paper examines conditions for Gromov-Hausdorff convergence of metric quotients and provides examples of conic-flat surfaces.
problem Conditions for Gromov-Hausdorff convergence of metric quotients.
method Analyzes sufficient conditions for Gromov-Hausdorff convergence of metric quotients of a metric space.
result Concrete examples of sequences of two-dimensional conic-flat spheres converging to spheres with singularities.
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.
The study proves convergence of conic 4-spheres' geometry to boundary cases.
problem Convergence of conic 4-spheres' geometry to boundary cases.
method Proved a convergence theorem on the moduli space of constant σ₂ metrics for conic 4-spheres.
result When a numerical condition converges to the boundary case, conic 4-spheres' geometry converges to the boundary case while preserving capacity.
The paper solves navigation problems on conic Kropina manifolds and establishes curvature relationships.
problem Navigation problems on conic Kropina manifolds.
method Analyzes the solution of navigation problems and establishes curvature relationships.
result The solution to navigation problems on conic Kropina manifolds must be either a Randers metric or a Kropina metric.
In this paper we develop a systematic deformation theory for conic constant curvature metrics on a closed surface when all cone angles are less than 2π; in particular, we define and study the Teichmüller space Tγ,kconic of conic constant curvature metrics on a surface of genus γ with k …
The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.
problem Analyzing conical Kähler-Einstein metrics on rank one horosymmetric Fano manifolds.
method Investigates the convergence of these metrics as the cone angle approaches zero, focusing on the behavior on the complement of a codimension one orbit.
result The metrics converge to the Kähler-Einstein metric on the basis and Stenzel's Ricci flat Kähler metrics on the fibers.
Proves existence of Yamabe metrics on conical 4-manifolds using min-max method.
problem Existence of Yamabe metrics on conical 4-manifolds with singular points.
method Min-max scheme adapted to singular setting, leveraging recent positive mass theorems.
result Existence of Yamabe metrics on conical 4-manifolds with finitely-many singular points.
Study examines heart and football-shaped metrics, verifying geometric structure.
problem Analyzing reducible spherical conical metrics and their geometric properties.
method Examined 1-parameter heart shape and 3-parameter football shape families, verified structure theorem, used explicit metric and geodesic calculations.
result Naturally arise from Abelian differentials of the third kind, offer new evidence for spherical geometry and complex analytic structure interaction.
We apply Tian's method in Kahler-Einstein problem to prove that a conic K\''ahler metric with lower Ricci curvature bound can be approximated by smooth K\''ahler metrics with the same lower Ricci curvature bound. Furthermore, conic singularities here can be along a simple normal crossing divisor.