arXiv research

This is a continuation of the previous articles on Kahler cone metrics. In this article, we introduce weighted function spaces and provide a self-contained treatment on cone angles in the whole interval $(0,1]$. We first construct geodesics in the space of Kahler cone metrics (cone geodesics). We next determine the ver…

Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.

New Calabi-Yau metrics with conical singularities are created near complex lines.

An extra large metric is a spherical cone metric with all cone angles greater than 2 pi and every closed geodesic longer than 2pi. We show that every two-dimensional extra large metric can be triangulated with vertices at cone points only. The argument implies the same result for Euclidean and hyperbolic cone metrics, …

The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.

The Lee-Gauduchon cone is a convex cone of cohomology classes for complex manifolds.

Study shows expanding Ricci solitons from specific metric cones.

The Kähler cone of a compact manifold carries a natural Riemannian metric, given by the intersection product of its cohomology ring. We write down the curvature tensor of this metric by embedding the Kähler cone in the space of hermitian metrics on the underlying manifold. After discussing weak functorality and complet…

No semistability found for Calabi-Yau metrics near cones.

We prove that `volume cone implies metric cone' in the setting of RCD spaces, thus generalising to this class of spaces a well known result of Cheeger-Colding valid in Ricci-limit spaces.

New Calabi-Yau metrics found on complex symmetric spaces.

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.

Unique metric found for discrete curvature on spherical cone-metrics.

In this paper, we prove Matsushima's theorem for Kähler-Einstein metrics on a Fano manifold with cone singularities along a smooth divisor that is not necessarily proportional to the anti-canonical class. We then give an alternative proof of uniqueness of Kähler-Einstein cone metrics by the continuity method. Moreover,…

Anti-de Sitter spacetimes embed cone-metrics as bent Cauchy surfaces.

The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.

The study finds PK cone metrics on complex manifolds near hyperplane arrangements.

Promotes Poisson deformations to hyperkähler structures.

Study on cones over metric spaces with curvature bounds.

We show that the existence of constant scalar curvature Kähler (cscK) metrics with cone singularities is equivalent to the properness of log $K$-energy. We also prove their equivalence to the geodesic stability. They are extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjectu…

We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ 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 develop some foundations for the study of Kahler-Einstein metrics with cone singularities transverse to a divisor. The main goal is a treatment of the deformation of the cone angle.

Study left-invariant pseudo-Riemannian metrics on Lie groups focusing on null cone Lie algebras.

A unique hyperbolic metric is found for each spherical cone-metric on the boundary of a hyperbolizable 3-manifold.

New Calabi-Yau metrics constructed with detailed geometry at infinity.

Extremal metrics lead to scalar-flat Kähler cones.

Tian initiated the study of incomplete Kähler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle $2π(1-α)$ for $α\in (0, 1)$. In this paper we study how the existence of such Kähler-Einstein metrics depends on $α$. We show that in the negative s…

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As…

Study on Kähler-Einstein metrics with polynomial convergence rates.

The study shows how nonnegative Ricci curvature and metric cones imply the existence of abelian subgroups in the fundamental group of open manifolds.

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…

Flexible metrics found on a genus 2 surface.

Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.

Kähler cones over Sasakian manifolds are flat if projectively induced.

Study examines causal properties of Finsler spacetimes with cone Killing vectors.

Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the 'large-scale structure' of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the complet…

This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle is less than 2π. We show that these are in a natrual way projective algebraic var…

Recently, Atiyah and LeBrun proved versions of the Gauss-Bonnet and Hirzebruch signature Theorems for metrics with edge-cone singularities in dimension four, which they applied to obtain an inequality of Hitchin-Thorpe type for Einstein edge-cone metrics. Interestingly, many natural examples of edge-cone metrics in dim…

The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.

We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an applicat…

Study spherical metrics on flat torus with cone singularities.

New examples of manifolds in tangent cones of non-collapsed Ricci limit spaces.

The paper proves Schauder estimates on cone products and characterizes harmonic functions.

Study finds existence and non-uniqueness of cone spherical metrics on compact Riemann surfaces.

Study convex hyperbolic cone-metrics on 3-manifold boundaries, proving unique bent realizations.

Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.

Characterizes rigid and flexible hyperbolic cone metrics and billiards.

We consider the Calabi-Yau metrics on $\mathbf{C}^n$ constructed recently by Yang Li, Conlon-Rochon, and the author, that have tangent cone $\mathbf{C}\times A_1$ at infinity for the $(n-1)$-dimensional Stenzel cone $A_1$. We show that up to scaling and isometry this Calabi-Yau metric on $\mathbf{C}^n$ is unique. We al…