Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.
problem Kähler-Einstein metrics on singular projective varieties.
method Approximation with constant scalar curvature metrics, RCD space analysis.
result Metric completion of smooth part is non-collapsed RCD space and homeomorphic to original variety.
Study two types of singular Kähler-Einstein metrics on complex varieties.
problem Understanding different types of singular Kähler-Einstein metrics.
method Analyzing Ricci flat Kähler cone metrics and applying to more general spaces.
result A weaker notion of singular Kähler-Einstein metrics is equivalent to a stronger one under certain conditions.
Establishes Hermite-Einstein metrics on complex spaces with singularities.
problem Existence of Hermite-Einstein metrics on complex spaces with singularities.
method Established existence of estimable Hermite-Einstein metrics for stable reflexive coherent sheaves on compact normal Kähler spaces with klt singularities.
result Obtained precise results for varieties with klt singularities.
The paper characterizes Einstein metrics using CPE metrics and vacuum static spaces.
problem Understanding the conditions for Einstein metrics using CPE metrics and vacuum static spaces.
method Analyzing CPE metrics and their relationship with Einstein metrics and vacuum static spaces.
result A necessary and sufficient condition for a CPE metric to be Einstein in terms of σ_2-singular spaces is provided.
The study shows how certain singular Kähler metrics can define Kähler currents and RCD spaces.
problem Understanding singular Kähler-Einstein metrics and their properties.
method Analyzing the properties of singular Kähler-Einstein metrics and their approximations.
result Singular Kähler-Einstein metrics can define Kähler currents and RCD spaces under certain conditions.
Tian's theorem applies to Moishezon spaces with singular metrics.
problem Distribution of currents on Moishezon spaces with singular metrics.
method Proving asymptotic distribution of Fubini-Study currents.
result Curvature currents of metrics on singular Hermitian line bundles.
Extended Vaisman theorem to compact spaces with singularities.
problem Generalizing Vaisman's theorem to spaces with singularities.
method Extended Vaisman's theorem to compact complex spaces with singularities.
result Vaisman's theorem extended to compact spaces with singularities.
The study examines conditions for scalar curvature and Dirac operators on singular spaces.
problem Existence of scalar curvature measures and Dirac operators on singular spaces.
method Investigation of smooth manifolds with singular Riemannian metrics.
result Sufficient conditions for the existence of scalar curvature measures and Dirac operators.
Study wall singularities in spaces with upper curvature bounds.
problem Understanding singularities in spaces with curvature constraints.
method Geometric structure theorem and geometric characterization for codimension one and two.
result Necessary and sufficient conditions for singular sets to be of codimension at least two.
Study on metrics with positive scalar curvature on manifolds with singularities.
problem Understanding metrics with positive scalar curvature on manifolds with singularities.
method Proving homotopy invariance of the space of metrics with positive scalar curvature on manifolds with fibred singularities.
result Proved that the space of metrics with positive scalar curvature is homotopy invariant under certain surgeries.
Gauduchon's theorem extended to singular spaces with smoothing.
problem Extending Gauduchon's theorem to singular spaces.
method Using smoothing techniques for singular spaces.
result Existence of conformally equivalent metrics on singular spaces.
Study 2D spaces with curvature, finding a graph structure.
problem Understanding the geometry of 2D spaces with curvature constraints.
method Analyzing spaces as unions of disks, identifying singular points.
result Obtained a graph structure of topological singular points.
Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.
problem Understanding geometric structures of singular Kahler spaces.
method Proving RCD spaces homeomorphic to 3D projective varieties with bounded Nash entropy and Ricci curvature.
result Compact RCD spaces are equivalent to underlying projective varieties.
New findings on Kähler-Einstein currents dominating Kähler forms on compact spaces.
problem Understanding Kähler-Einstein currents on singular spaces.
method Analyzing currents' properties and applying them to compact spaces with log terminal singularities.
result Kähler-Einstein currents dominate Kähler forms on compact spaces with log terminal singularities.
Constructs Kahler-Einstein metrics near isolated log canonical singularities.
problem Constructing metrics near singularities in complex geometry.
method Constructs Kahler-Einstein metrics with negative scalar curvature near isolated log canonical singularities.
result Metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable.
Paper develops techniques for singular metrics on vector bundles.
problem Developing techniques for singular metrics on vector bundles.
method Introducing non-pluripolar products and defining I-good singularities. result Derives a Chern--Weil type formula for Hermitian vector bundles with I-good singularities. The paper proves compactness of metrics with isolated singularities on a sphere.
problem The moduli space of metrics with constant Q-curvature and positive scalar curvature on a sphere with punctures.
method Defined asymptotic necksize and radial Pohozaev invariant, proved sequential compactness.
result Any bounded set in the moduli space is sequentially compact.
Investigates singular Finsler foliations on (α,β)-spaces and their relation to Riemannian foliations.
problem Understanding conditions for singular Finsler foliations to be singular Riemannian foliations.
method Analyzes (α,β)-spaces and verifies conditions for SFFs to be SRFs, extending Molino's conjecture. result Equifocality of regular leaves for SFFs under certain conditions.
In this article we introduce a generalization of locally conformally Kaehler metrics from complex manifolds to complex analytic spaces with singularities and study which properties of locally conformally Kaehler manifolds still hold in this new setting. We prove that if a complex analytic space has only quotient singul…
Uniformly Euclidean metrics with isolated singularities on certain manifolds are Ricci flat and have nonnegative synthetic Ricci curvature.
problem Proving the existence of Ricci flat metrics with isolated singularities on specific manifolds.
method Demonstrating nonnegative synthetic Ricci curvature using the RCD(0, n) condition.
result Uniformly Euclidean metrics with isolated singularities on Mn=Tn#M0 are Ricci flat and extend smoothly over the singularity. Study precise asymptotic behavior of functions in singular metric spaces.
problem Singular metric spaces with incomplete geometry.
method Expansions of quasi-harmonic and eigenfunctions.
result More precise description of asymptotic behavior at infinity.
New Calabi-Yau metrics found on complex symmetric spaces.
problem Finding Calabi-Yau metrics on complex symmetric spaces.
method Complete Calabi-Yau metrics with prescribed horospherical singular tangent cone.
result First examples of Calabi-Yau smoothings of singular tangent cones.
Paper introduces a complete metric topology for low energy spaces.
problem Defining a topology for low energy spaces with prescribed singularity.
method Introduces a completely metrizable topology stronger than capacity convergence.
result Low energy spaces have a natural completely metrizable topology.
New minimal surfaces found using a modified metric connection.
problem Finding minimal surfaces in Euclidean 3-space.
method Used a special semi-symmetric metric connection instead of the Levi-Civita connection.
result Found non-trivial minimal surfaces other than planes.
Estimates for complex Monge-Ampère equations lead to insights on moduli spaces and singular metrics.
problem Uniform estimates for complex Monge-Ampère equations on Kähler manifolds.
method Refined techniques to control degenerate equations and analyze families of singular Kähler-Einstein metrics.
result Uniform integrability properties and insights into moduli spaces of stable varieties.
New Einstein RCD spaces found with cone singularities.
problem Existence of Einstein RCD spaces with cone singularities.
method Characterization of RCD spaces and cone singularity analysis.
result Existence of smooth non-compact 4-manifolds with ALE Ricci-flat RCD(0,4) metrics.
The paper classifies metrics with constant negative Q-curvature in Euclidean spaces.
problem Classifying metrics with constant negative Q-curvature in Euclidean spaces.
method Variational techniques and finite volume conditions.
result Existence and classification of singular and nonsingular metrics with constant negative Q-curvature.
We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities.
The abstract discusses how Kähler-Einstein metrics relate to algebraic geometry.
problem Understanding the connection between Kähler-Einstein metrics and algebraic geometry.
method Exploring metric limits and rescalings of Kähler-Einstein metrics in relation to moduli spaces and singularities.
result Proposes tentative conjectural pictures connecting Kähler-Einstein metrics and algebraic geometry.
In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under a…
Uniform bounds prove connection between Kähler metrics and RCD spaces.
problem Bounding Nash entropy and Calabi energy for Kähler metrics.
method Proving uniform Sobolev bounds for Kähler manifolds.
result Establishes connection to RCD spaces and provides examples.
Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space X that admits Poincaré inequalities for a continuum of mutually singular measures.
Newly discovered Eguchi-Hanson metric arises from edge metrics.
problem Understanding limits of compact singular Einstein spaces.
method Constructing Kahler-Einstein edge metrics on Calabi-Hirzebruch manifolds.
result Eguchi-Hanson metric emerges as a Gromov-Hausdorff limit.
Solves Plateau-Douglas problem for singular configurations in general metric spaces.
problem Existence of minimal surfaces for singular configurations.
method Generalized approach via minimal sequences in metric spaces.
result Existence of minimal surfaces for singular configurations in general metric spaces.
Study of scattering on singular Yamabe spaces using asymptotically hyperbolic manifolds.
problem Understanding conformal geometry of compact manifolds with boundary.
method Application of scattering theory to singular Yamabe metrics.
result Definition of extrinsic GJMS operators and Q-curvatures on boundary.
The paper constructs stable Higgs bundles for hyperbolic metrics with singularities.
problem Existence of conformal hyperbolic metrics with prescribed singularities.
method Stable parabolic Higgs bundles of rank two.
result Alternative proof of Heins' theorem and extension of Hitchin's work.
We investigate cohomogeneity-one metrics whose principal orbit is an Aloff-Wallach space SU(3)/U(1). In particular, we are interested in metrics whose holonomy is contained in Spin(7). Complete metrics of this kind which are not product metrics have exactly one singular orbit. We prove classification results for metric…
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 paper proves prevalent existence and partially determines moduli space of area-minimizing surfaces with fractal singular sets.
problem Existence and moduli space of area-minimizing surfaces with fractal singular sets.
method Proof of prevalent existence, determination of moduli space, refinement of strata.
result Sharp results on moduli space and refinement of strata, showing fractal singularities do not completely dissolve under generic perturbations.
Uniform Sobolev inequality for Kähler metrics with entropy bound.
problem Establishing Sobolev inequalities for Kähler metrics with entropy bound.
method Uniform Sobolev inequality for Kähler metrics with entropy bound and no lower Ricci curvature bound.
result Derive various geometric estimates for Kähler-Einstein currents.
Study asymptotics of hyperkähler geometry on singular fibers of Hitchin moduli space.
problem Asymptotic hyperkähler geometry of SL2(C)-Hitchin moduli space over singular fibers. method Extension of exponential convergence results to locally fiducial Higgs bundles and subintegrable systems.
result Hyperkähler metric converges exponentially to semi-flat metric on subintegrable systems.
Study on metrics with singularities on spheres, showing moduli space structure.
problem Constant Q-curvature metrics on spheres with singular points.
method Analysis of moduli space, Gromov-Hausdorff topology, symplectic structure construction.
result Moduli space structure is a real analytic variety with formal dimension equal to the number of punctures.
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.
We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of smooth compact manifolds with smooth metrics, is tangen- tial to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of smooth manifolds with rough metri…
By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with n cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension n−3. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a n…
Study weak super Ricci flow through neckpinch in metric measure spaces.
problem Understanding Ricci flow behavior at singularities.
method Introduce weak super Ricci flow and show conditions for continuation.
result Weak super Ricci flow properties at singularities.
The article studies Ricci-flat metrics on complex projective space.
problem Understanding Ricci-flat metrics on complex projective space.
method Description and explicit computation of curvature.
result All compact, minimal submanifolds are contained in the zero section.
Uniform volume estimate for Kähler metrics in big cohomology classes.
problem Estimating volume for singular Kähler metrics in big cohomology classes.
method Generalized mixed energy estimate for functions in complex Sobolev space to big cohomology classes.
result Uniform non-collapsing volume estimate for local Kähler metrics.