arXiv research

Generically, topological insulators have conical points leading to Dirac-like currents.

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

A limit point p of a discrete group of Mobius transformations acting on S^n is called a concentration point if for any sufficiently small connected open neighborhood U of p, the set of translates of U contains a local basis for the topology of S^n at p. For the case of Fuchsian groups (n = 1), every concentration point…

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…

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

The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the image of a non-closed geodesic has 0 distance from the set of conical points. Dynamical properties for the space of geodesics are also proved.

We study the asymptotics of the determinant of Laplacian on a translation surface (a compact Riemann surface equipped with a conformal flat conical metric with trivial holonomy) of genus g with 2g-2 conical points of angle 4πas two conical points collide.

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

We show that for given four points on the sphere and prescribed angles at these points, which are not multiples of $2π$, the number of metrics of curvature 1 having conic singularities with these angles at these points is finite.

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…

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

We compute the fundamental groups of the complements of the family of real conic-line arrangements with up to two conics which are tangent to each other at two points, with an arbitrary number of tangent lines to both conics. All the resulting groups turn out to be big.

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 $k$ …

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 curve shortening flow on Riemann surfaces with conical singularities.

In this paper, we study the Hausdorff dimension of the Floyd and Bowditch boundaries of a relatively hyperbolic group, and show that for the Floyd metric and shortcut metrics respectively, they are are both equal to a constant times the growth rate of the group. In the proof, we study a special class of conical points …

Study Ricci flow on spaces with conical singularities, proving existence and curvature estimates.

Given a hyperbolic subgroup $H$ of a hyperbolic group $G$ for which a Cannon-Thurston map $\hat i:\partial H \ra \partial G$ exists, we study the limit set $Λ_H$ of $H$ with respect to its action on $\partial G$. We prove that the set of conical limit points is exactly the subset of $Λ_H$ consisting of the points to wh…

In this article we address a number of features of the moduli space of spherical metrics on connected, compact, orientable surfaces with conical singularities of assigned angles, such as its non-emptiness and connectedness. We also consider some features of the forgetful map from the above moduli space of spherical sur…

We discuss the constant $σ_{2}$ problem for conic 4-spheres. Based on earlier works of Chang-Han-Yang and Han-Li-Teixeira, we are able to find a necessary condition for the existence problem. In particular, when the condition is sharp, we have the uniqueness result similar to that of Troyanov in dimension 2. It indicat…

Proves smoothness of conical singularities in mean curvature flow.

Characterizes conical angles for metrics with dihedral symmetry.

The paper studies the topology of spherical tori with one conical point.

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

In this paper, we establish a framework for the analysis of linear parabolic equations on conical surfaces and use them to study the conical Ricci flow. In particular, we prove the long time existence of the conical Ricci flow for general cone angle and show that this solution has the optimal regularity, namely, the ti…

The article completes the research of two-point G$^2$ Hermite interpolation problem with spirals by inversion of conics. A simple algorithm is proposed to construct a family of 4th degree rational spirals, matching given G$^2$ Hermite data. A possibility to reduce the degree to cubic is discussed.

Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singul…

We use the momentum construction of Calabi to study the conical Kähler-Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov-Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts the cone divisor to a single point …

We list all the possible fundamental groups of the complements of real conic-line arrangements with two conics which are tangent to each other at two points, with up to two additional lines. For the computations we use the topological local braid monodromies and the techniques of Moishezon-Teicher and van-Kampen. We al…

Paper proposes a method to find approximate SOSP for nonconvex conic optimization problems.

The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.

The Kähler-Ricci flow near conical singularities is described with a $C/t$ curvature bound.

We show that spheres of positive constant curvature with $n$ ($n\geq3$) conic points converge to a sphere of positive constant curvature with two conic points (or called an (American) football) in Gromov-Hausdorff topology when the corresponding singular divisors converge to a critical divisor in the sense of Troyanov.…

Paper proposes a method to find approximate SOSP for nonconvex conic optimization problems.

Study conic Laplacian on \(\mb P^1\) with explicit model and boundary data.

The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.

We investigate the limit the R torsion of a conical frustum as one of the basis is shrunk to a point. We show that, if we take suitable regularization, such a limit gives the intersection torsion of the resulting cone.

In this paper two Poisson structures on the moduli space of hyperbolic surfaces with conical points are compared: the Weil-Petersson one and the ηcoming from the representation variety. We show that they are multiple of each other, if the angles do not exceed 2π. Moreover, we exhibit an explicit formula for ηin terms o…

Derives formulas for determinant of Laplacian on curved surfaces.

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4π$.

Formula derived for spectral determinant of sphere with conical singularities.

In this note we use the Calabi ansatz, in the context of metrics with conical singularities along a divisor, to produce regular Calabi-Yau cones and Kähler-Einstein metrics of negative Ricci with a cuspidal point. As an application, we describe singularities and cuspidal ends of the completions of the complex hyperboli…

We prove the existence and uniqueness of harmonic maps in degree one homotopy classes of closed, orientable surfaces of positive genus, when the target has conic points with cone angles less than $2π$. For a cone point $p$ of cone angle less than or equal $π$ we show that one can minimize, uniquely, in the relative hom…

In this article, we study the properties of the geodesic X-ray transform for asymptotically Euclidean or conic Riemannian metrics and show injectivity under non-trapping and no conjugate point assumptions. We also define a notion of lens data for such metrics and study the associated inverse problem.

Study local features of decorated representation spaces for spherical surfaces.

We give some non-existence results for Kähler-Einstein metrics with conical singularities along a divisor on Fano manifolds. In particular we show that the maximal possible cone angle is in general smaller than the invariant R(M). We study this discrepancy from the point of view of log K-stability.

Geodesic lines with specific boundaries found on a special type of manifold.

Defines Perelman's functionals on manifolds with non-isolated conical singularities.