Metrics of positive curvature with conic singularities on the sphere

On metrics of constant positive curvature with four conic singularities on the sphere

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.

Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic metric with conical singularities of positive singular curvature on a compact surface of genus greater than one. We prove that these metrics are actually realised by exactly one convex Fuchsian polyhedron (up to global isometries). This extends a famous theorem of A.D. Alexandro…

Metrics with conic singularities and spherical polygons

A spherical n-gon is a bordered surface homeomorphic to a closed disk, with n distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these polygons and enumerate them in the case that two angles at the corners are not multiples of pi. The problem is equivalent to classification of some second order linear differential equations with regular singularities, with real parameters and unitary monodromy.

On surfaces with conical singularities of arbitrary angle

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.

关于全局锥形奇异流形上Lipschitz几何的注记

Remarks on Lipschitz geometry on globally conic singular manifolds

该研究探讨带锥形奇异性的流形的度量性质，建立度量锥形与渐近锥形行为的关联，证明满足奇点锥形、端渐近锥形条件的环境奇异流形中的奇异子流形是Lipschitz正规嵌入的，解决了此前提出的相关问题。

Metrics with four conic singularities and spherical quadrilaterals

A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that two angles at the corners are multiples of pi. The problem is equivalent to classification of Heun's equations with real parameters and unitary monodromy.