arXiv research

The paper generalizes a mean value theorem for solutions of the ultrahyperbolic equation.

The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic to the fundamental group of a complement of a line arrangement, has some remarkab…

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.

The fundamental group of the complement of a plane curve is a very important topological invariant. In particular, it is interesting to find out whether this group is determined by the combinatorics of the curve or not, and whether it is a direct sum of free groups and a free abelian group, or it has a conjugation-free…

Paper shows X-ray transform invertible on certain curved spaces.

New discretizations of principal curvature lines discovered.

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

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

Paper shows invertibility of tensor X-ray transform on certain manifolds.

Study dispersive estimates for Schrödinger and wave equations on a cone with specific metric.

The paper connects geodesic flows and limit sets on visibility manifolds.

Study examines wave equation decay and Strichartz estimates on conic manifolds.

Geometric focusing affects dispersive estimates for Schrödinger and wave equations.

We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent …

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…

The paper classifies isoparametric hypersurfaces in conic Finsler spaces.

Supercyclides are surfaces with a characteristic conjugate parametrization consisting of two families of conics. Patches of supercyclides can be adapted to a Q-net (a discrete quadrilateral net with planar faces) such that neighboring surface patches share tangent planes along common boundary curves. We call the result…

Two unique conic-line arrangements with degree 9 are found.

Study conic singular manifolds, proving Lipschitz normal embedding.

Desingularizes conically singular Cayley submanifolds.

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

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

Confocal conics form an orthogonal net. Supplementing this net with one of the following: 1) the net of Cartesian coordinate lines aligned along the principal axes of conics, 2) the net of Apollonian pencils of circles whose foci coincide with the foci of conics, 3) the net of tangents to a conic of the confocal family…

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

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…

Proves smoothness of conical singularities in mean curvature flow.

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

Study describes ALF instantons with conical singularities.

We establish a parabolic version of Tian's $C^{2,α}$-estimate for conical complex Monge-Ampere equations, which includes conical Kähler-Einstein metrics. Our estimate will complete the proof of the existence of unnormalized conical Kähler-Ricci flow in arXiv:1411.7284.

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…

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 conic-line arrangements of degree 7, finding their topology and components.

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

The paper explores conic-line arrangements via Poncelet's theorem and finds families of reducible curves.

Constructs surfaces with conical singularities using variational methods.

Paper proves unique conical tangent flows in all dimensions.

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

Paper shows regularizing flow for conical Kähler-Ricci equations.

Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.

Study curve shortening flow on Riemann surfaces with conic singularities.

Proves positive mass theorem on conical manifolds with small angles.

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…

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

The abstract proves spherical surface decompositions with conical singularities.

Paper proves Whitney stratified spaces can be given a conically smooth structure.

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

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.