arXiv research

Study conic singular manifolds, proving Lipschitz normal embedding.

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 finds conditions for conformal deformations to constant scalar curvature in conic metrics.

Study on spherical conical metrics and their reducibility on compact Riemann surfaces.

Proves existence of Yamabe metrics on conical manifolds with conical points and links.

The paper classifies Landsberg metrics on a 2D Lie group and proves a conjecture.

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.

Unique circle patterns on spheres found for spherical conical metrics.

Classifies 4D toric Hermitian ALF metrics with conical singularities.

Study describes ALF instantons with conical singularities.

Proves conditions for positive scalar curvature on certain manifolds with conical singularities.

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…

Characterizes conical angles for metrics with dihedral symmetry.

Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.

Constructs scalar-flat Kähler metrics with varying conical singularities.

Survey on metrics with conic singularities on Riemann surfaces.

Analytic torsion studied for fibred boundary metrics, with applications to conic degeneration.

In the category of metrics with conical singularities along a smooth divisor with angle in $(0, 2π)$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the Kähler-Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coord…

We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric. We also construct explicitly some conical metrics whose curvature is not integrable.

The paper explores eigenfunctions of spherical conical metrics using harmonic maps to spheres.

The paper proves the existence of Sasaki-Einstein metrics on specific Sasakian manifolds.

The paper examines conditions for Gromov-Hausdorff convergence of metric quotients and provides examples of conic-flat surfaces.

Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.

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 $\mathcal{T}^{\mathrm{conic}}_{γ,k}$ of conic constant curvature metrics on a surface of genus $γ$ with $k$ …

The paper solves navigation problems on conic Kropina manifolds and establishes curvature relationships.

Proves existence of Yamabe metrics on conical 4-manifolds using min-max method.

The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.

Study examines heart and football-shaped metrics, verifying geometric structure.

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.

Recently it was shown by H. Guenancia and M. Paun that a singular metric satisfying the conical Kahler-Einstein equation with a simple normal crossing divisor is equivalent to a conical metric along that divisor. In this note, we present an alternative proof of their theorem.

In this paper, we study the stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle $2πβ$ along the divisor, then for any $β'$ sufficiently close to $β$, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein me…

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

The paper finds conical higher cscK metrics on minimal ruled surfaces with conical singularities.

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

We prove that the conical Kähler-Ricci flows introduced in \cite{CYW} exist for all time $t\in [0,+\infty)$. These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class $C_{1,β}$ is negative or zero, the corresponding conical Kähler-Ricci flows co…

The paper shows how to stabilize perturbed Kähler-Ricci solitons.

For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrar…

We show that the monodromy of a spherical conical metric is reducible if and only if it has a real-valued eigenfunction with eigenvalue 2 in the holomorphic extension of the associated Laplace--Beltrami operator. Such an eigenfunction produces a meromorphic vector field, which is then related to the developing maps of …

The purpose of this paper is to prove the uniqueness of conical Kähler-Einstein metrics, under the condition that the twisted $Ding$-functional is proper. This is a generalization of the author's previous work, and we shall first investigate the uniqueness of twisted Kähler-Einstein metrics, and then use these smooth p…

This is the continuation of our paper \cite{GS}, to study the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background Kähler metric of conical singularities along a divisor of simple normal crossings. As an application, …

We consider the constant Q-curvature metric problem in the given conformal class on conic 4-manifolds and study related differential equations.

We give criterions for the existence of toric conical Kahler-Einstein and Kahler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry-Emery-Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with c…

We prove that on one Kähler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical Kähler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also establish a curvature estimate for conic metrics along a simple normal crossing divisor which …

In this paper we introduce the "interpolation-degneration" strategy to study Kahler-Einstein metrics on a smooth Fano manifold with cone singularities along a smooth divisor that is proportional to the anti-canonical divisor. By "interpolation" we show the angles in $(0, 2π]$ that admit a conical Kahler-Einstein metric…

In this paper we determine a complete list of rational surface singularities which have metrically conical bilipschitz type of its inner metric. We achieve this by using the thick-thin decomposition of Birbrair, Neumann and Pichon.

Following Donaldson's oppenness theorem on deforming a conical Kähler-Einstein metric, we prove a parabolic Schauder-type estimate with respect to conical metrics. As a corollary, we show that the conical Kähler-Ricci Flow exists for short time. The key is to establish the relevant heat kernel estimates, where we use t…

Polyhomogeneous expansions for Calabi-Yau metrics near singularities.