arXiv research

Study conic singular manifolds, proving Lipschitz normal embedding.

We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1.

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.

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

Joachimsthal integrals characterize conics in various geometries.

Proof of Tait-Kneser theorem and related variations using Lorentzian geometry.

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

Let $(M,g)$ be a surface with Riemannian metric and curved conic singularities. More precisely, a neighbourhood of a singularity is isometric to $(0,1)\times S^1$ with metric $g_{\text{conic}}=dr^2+f(r)^2dθ^2, r\in(0,1)$. We study the spectral geometry of $(M,g)$ using the heat trace expansion. We express the first few…

We prove a convergence theorem on the moduli space of constant $σ_{2}$ metrics for conic 4-spheres. We show that when a numerical condition is convergent to the boundary case, the geometry of conic 4-spheres converges to the boundary case while preserving capacity.

Survey on metrics with conic singularities on Riemann surfaces.

We construct infinitely many new 1-parameter families of simply connected complete noncompact G_2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter…

Unique circle patterns on spheres found for spherical conical metrics.

We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities, and hence they generalize the classical results of Kondratiev on domains with c…

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.

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…

The paper explores cone structures and their connections to parabolic geometries in complex manifolds.

Study shows flow convergence to smooth K-Ricci outside a divisor with cusp singularity.

We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross-section we show well posedness of the short time solution in the $L^q$-setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asym…

As part of the general investigation of Ricci flow on complete surfaces with finite total curvature, we study this flow for surfaces with asymptotically conical (which includes as a special case asymptotically Euclidean) geometries. After establishing long-time existence, and in particular the fact that the flow preser…

In this paper we define a new convergence called "asymptotically conic convergence" in which a smooth family of Riemannian metrics on a fixed compact manifold degenerate to a metric with isolated conic singularity. Our results are: convergence of the spectrum of the geometric Laplacians and uniform convergence of the c…

As an application of the method of [4], we find the metric and connection on the space of conics in $\mathbb{CP}^2$ determined as the solution space of the ODE eqn(1). These calculations underpin the twistor construction of the Radon transform on conics in $\mathbb{CP}^2$ described in [5]. Two further examples of the m…

We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical.

Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.

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

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

We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley [BS]. In the subsequent paper we investigate how the terms in the expansion reflect the geometry …

The authors study a generalized notion of null geodesic defined by the Legendrian dynamics of a regular conical subbundle of the tangent bundle on a manifold. A natural extension of the Weyl tensor is shown to exist, and to depend only on this conical subbundle. Given a suitable defining function of the conical bundle,…

This is an expository paper describing the geometry of certain Sasakian-Einstein manifolds. Such manifolds have recently become of interest due to Maldacena's AdS/CFT conjecture. They describe near-horizon geometries of branes at conical singularities.

We derive parametrizations of the Delaunay constant mean curvature surfaces of revolution that follow directly from parametrizations of the conics that generate these surfaces via the corresponding roulette. This uniform treatment exploits the natural geometry of the conic (parabolic, elliptic or hyperbolic) and leads …

Study shows Kähler-Einstein metric singularities linked to curvature.

Proves Verdier duality for sheaves on stratified spaces.

Quiver varieties' geometry at infinity studied using Nakajima metric.

We study the deformation of spherical conical metrics with at least some of the cone angles larger than $2π$. We show in this note via synthetic geometry that for one family of such metrics, there is local rigidity in the choice of cone positions if angles are fixed. This gives an evidence of the analytic obstruction c…

Study trisections on rational elliptic surfaces to find new Zariski pairs.

Bounds on saddle connections on flat spheres with conical singularities.

We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an …

In this paper we consider the isoptic curves on the 2-dimensional geometries of constant curvature $\bE^2,~\bH^2,~\cE^2$. The topic is widely investigated in the Euclidean plane $\bE^2$ see for example \cite{CMM91} and \cite{Wi} and the references given there, but in the hyperbolic and elliptic plane there are few resu…

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 …

Solves complex equation for specific geometric solitons.

New one-parameter families of $SU(2)^2$-invariant instantons found on Calabi-Yau 3-folds.

We show that conically smooth stratified spaces embed fully faithfully into $\infty$-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each $\infty$-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions…

Infinite circle packings on surfaces with conical singularities are possible.

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

Uniform elliptic theory for Dirac operators on orbifold resolutions.

The paper studies metrics on hyperkähler manifolds using sub-twistor constraints.

Paper solves curvature assignment on surfaces with sharp points and edges.

This is a continuation of paper \cite{Li}. On any toric Fano manifold, we discuss the behavior of limit metric of a sequence of metrics, which are solutions to a continuity family of complex Monge-Ampere equations in Kahler-Einstein problem. We show that the limit metric satisfies a singular complex Monge-Ampere equati…

New Spin(7) manifolds created from Calabi-Yau bundles.