We prove that a crepant resolution of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat Kähler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup…
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Kähler cones over Sasakian manifolds are flat if projectively induced.
problem Characterizing Kähler cones over Sasakian manifolds.
method Relating Kähler potentials and using Ricci-flatness.
result Kähler cones over regular Sasakian manifolds are flat if projectively induced.
Asymptotically conical Ricci flat Kähler metrics on with cone singularities along a complex curvemath.DG
We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.
We extend Calabi ansatz over Kähler-Einstein manifolds to Sasaki-Einstein manifolds. As an application we prove the existence of a complete scalar-flat Kähler metric on Kähler cone manifolds over Sasaki-Einstein manifolds. In particular there exists a complete scalar-flat Kähler metric on the toric Kähler cone manifold…
Extremal metrics lead to scalar-flat Kähler cones.
problem Proving scalar-flat Kähler cone metrics from extremal Kähler metrics.
method Analyzing complex cones and Sasaki joins of extremal manifolds.
result Scalar-flat Kähler cone metrics exist for large cones.
In this article, we thoroughly investigate the stability inequality for Ricci-flat cones. Perhaps most importantly, we prove that the Ricci-flat cone over CP^2 is stable, showing that the first stable non-flat Ricci-flat cone occurs in the smallest possible dimension. On the other hand, we prove that many other example…
Using methods of A. Grigor'yan and L. Saloff-Coste we prove that on a manifold with a conical end the heat kernel has a Gaussian bound. This result is applied to asymptotically conical Kähler manifolds. It is a result of the author and R. Goto that a crepant resolution of a Ricci-flat Kähler cone admits a Ricci-flat Kä…
Study shortest geodesics on flat cone spheres with conical singularities.
problem Understanding the distribution of shortest geodesics on flat cone spheres.
method Proved a recurrent relation on the distribution of the length of shortest geodesics with respect to Thurston's volume form.
result Proved a recurrent relation on the distribution of the length of shortest geodesics.
The paper provides uniform length estimates for trajectories on flat cone surfaces.
problem Estimating the length of trajectories on flat cone surfaces.
method Using self-intersection numbers and constants depending only on the flat metric, the paper focuses on convex flat cone spheres with a positive curvature gap and a fixed number of singularities.
result Uniform two-sided estimates for trajectory lengths on convex flat cone spheres are obtained.
The study proves an expansion theorem for scalar-flat asymptotically conical Kähler metrics.
problem Proving an expansion theorem for scalar-flat asymptotically conical Kähler metrics.
method Using weak decay conditions and the standard Kähler metric of the metric cone, the expansion theorem is proven.
result Each scalar-flat AC Kähler metric admits an expansion with a main term given by the standard Kähler metric of the metric cone and a leading error term of O(r^{2-2n}).
The author has proved that a crepant resolution Y of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,\R). These manifolds are generalizations of the Ricci-flat ALE Kähler spaces known by the work of P. Kronheimer, D. Joyce and others. …
We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Ka…
Study spherical metrics on flat torus with cone singularities.
problem Existence and uniqueness of spherical metrics with specific cone angles.
method Analysis of Green function critical points and torus geometry.
result Existence of unique spherical metrics determined by torus geometry.
Continuous metrics on ample bundles lie in infinite-dimensional cones.
problem Understanding the structure of positive metrics on ample line bundles.
method Analyzing bounded graded filtrations and embedding into Mabuchi-flat cones.
result Continuous metrics embed isometrically into the space of positive metrics.
Exposes new connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
problem Exploring connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
method Surveying and reviewing recent works, transforming complex Monge-Ampère equations, and proving the YTD conjecture.
result Established a transformation from irregular Sasaki-Einstein metrics to -solitons on quasi-regular quotients.
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 paper extends Siegel-Veech formula to convex flat cone spheres.
problem No formula exists for flat surfaces with irrational cone angles.
method Defined a generalized Siegel-Veech transform and Siegel-Veech measure.
result The Siegel-Veech measure is absolutely continuous and piecewise real analytic.
Paper proves flat metrics from holomorphic quadratic differentials can be identified by length spectrum.
problem Identifying flat metrics from holomorphic quadratic differentials.
method Proved using length spectrum on closed oriented surfaces.
result Flat metrics from holomorphic quadratic differentials can be distinguished by their length spectrum.
The Liouville theorem and -estimate for Calabi-Yau cones establish uniqueness and asymptotic behavior of metrics.
problem Establishing uniqueness and asymptotic behavior of metrics on Calabi-Yau cones.
method Developed a Liouville theorem and -estimate for Ricci-flat, conical Kähler manifolds.
result Uniformly bounded Kähler metrics on a ball around the apex are asymptotic to the Ricci-flat cone metric with polynomial decay.
Unique tangent cones found for Kahler-Einstein metrics on singular varieties.
problem Finding unique tangent cones for Kahler-Einstein metrics on singular varieties.
method Analyzing unique Ricci flat currents with local bounded potential.
result Local tangent cones of the unique Kahler-Einstein metric are unique.
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.
In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked length spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a…
The study computes indicial roots and metric convergence orders for Ricci-flat conifolds.
problem Understanding the convergence behavior of Ricci-flat conifolds.
method Analyzing the Lichnerowicz Laplacian and tensor fields on cones, computing indicial roots and metric convergence orders.
result Lower bounds for metric convergence orders on Ricci-flat conifolds.
We describe a framework for constructing the general Ricci-flat metric on the anticanonical cone over the del Pezzo surface of rank one.
Calabi-Yau metrics with cone singularities along intersecting complex lines: the unstable casemath.DG
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.
By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension . The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a n…
The paper studies Kähler compactifications of C^n and their applications in geometry.
problem Compactifying complex manifolds with Kähler structures.
method Analyzes Kähler orbifold compactifications of C^n and uses symplectic homology.
result Proves the uniqueness of asymptotically conical Ricci-flat Kähler metrics on C^3.
We shall construct a natural Higgs bundle structure on the complexified Kähler cone of a compact Kähler manifold, which can be seen as an analogy of the classical Higgs bundle structure associated to a variation of Hodge structure. In the proof of the flat-ness of our Higgs bundle, we find a commutator identity that ca…
By a classical theorem of Gallot (1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is dec…
We construct Ricci flat Kahler metrics with cone singularities along a complex hypersurface. This construction is inspired in part by R. Mazzeo's program in the case of negative Einstein constant, and uses the linear theory developed recently by S. Donaldson.
We show that a polarized affine variety admits a Ricci flat Kähler cone metric, if and only if it is K-stable. This generalizes Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to Kähler cones, or equivalently, Sasakian manifolds. As an application we show that the five-sphere admits infinitely many f…
We study the behaviour of families of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold when the Kahler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ri…
The study proves a Liouville theorem for certain asymptotically conical Calabi-Yau manifolds.
problem Characterizing complete Calabi-Yau manifolds with specific geometric properties.
method Analyzing Ricci-flat Kähler metrics on cones and their asymptotic conical structures.
result Liouville theorem holds for asymptotically conical Calabi-Yau manifolds.
Given a weighted line arrangement in the projective plane, with weights satisfying natural constraint conditions, we show the existence of a Ricci-flat Kähler metric with cone singularities along the lines asymptotic to a polyhedral Kähler cone at each multiple point. Moreover, we discuss a Chern-Weil formula that expr…
The study finds PK cone metrics on complex manifolds near hyperplane arrangements.
problem Finding metrics on complex manifolds near singularities.
method Analyzing flat torsion-free meromorphic connections on \(\mathbb{C}^n\) with simple poles at hyperplanes.
result Metric completion of certain connections yields PK cone metrics on \(\mathbb{C}^n\).
The study classifies Einstein metrics on a specific total space, revealing various behaviors and transitions.
problem Classifying conformally Kähler, -invariant, Einstein metrics.
method Drawing on Derdziński's results from the 80s, the study classifies metrics on for all .
result The study discovers infinitely many 1-parameter families of Einstein metrics with different behaviors, including asymptotically hyperbolic, ALF, and cone angle limits.
We prove a removal of singularities result for Bach-flat metrics in dimension 4 under the assumption of bounded L^2 norm of curvature, bounded Sobolev constant and a volume growth bound. This result extends the removal of singularities result for special classes of Bach-flat metrics obtained in \cite{TVMOD}. For the pr…
This is a survey of our recent work on degenerations of Ricci-flat Kahler metrics on compact Calabi-Yau manifolds with Kahler classes approaching the boundary of the Kahler cone.
This paper classifies Calabi-Yau manifolds near cones.
problem Classifying Calabi-Yau manifolds near cones.
method Complete classification of smooth complete Calabi-Yau manifolds asymptotic to a given cone.
result Classification of all smooth complete Calabi-Yau manifolds near a given cone.
It is well known that in compact local Lipschitz neighborhood retracts in Euclidean space flat convergence for integer rectifiable currents amounts just to weak convergence. In the present paper we extend this result to integral currents in complete metric spaces admitting a local cone type inequality. These include in…
The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
problem Existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.
method Explicit K-stability condition, degeneration, and asymptotic cone analysis.
result Uniqueness of -invariant Calabi-Yau metrics on affine spherical manifolds.
We pursue the symplectic description of toric Kahler manifolds. There exists a general local classification of metrics on toric Kahler manifolds equipped with Hamiltonian two-forms due to Apostolov, Calderbank and Gauduchon(ACG). We derive the symplectic potential for these metrics. Using a method due to Abreu, we rela…
Criterion for surfaces in Heisenberg group to be graphs using flat cones.
problem Characterizing surfaces in Heisenberg group as graphs.
method Using planar cones to define intrinsic rectifiability.
result Criterion for topological surfaces to be intrinsic Lipschitz graphs.
New conical metrics found on toric varieties with convex cones.
problem Finding conical metrics on toric affine varieties.
method Existence result for inhomogeneous Monge-Ampere equation, transversal a priori estimates.
result Existence of conical Ricci flat Kahler metrics on Q-Gorenstein affine toric varieties.
Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an ap…
We describe a framework for constructing the Ricci-flat metrics on the total space of the canonical bundle over (the del Pezzo surface of rank one). We construct explicitly the first-order deformation of the so-called `orthotoric metric' on this manifold. We also show that th…
We generalise to the genus one case several results of Thurston concerning moduli spaces of flat Euclidean structures with conical singularities on the two dimensional sphere. More precisely, we study the moduli space of flat tori with cone points and a prescribed holonomy . In his paper `Flat Surfaces' Veech ha…
In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin-Thorpe inequality for oriented Ricci-flat -manifolds with curvature decay and controlled holonomy. As a…