Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

Green function and self-adjoint Laplacians on polyhedral surfaces

Using Roelcke formula for the Green function, we explicitly construct a basis in the kernel of the adjoint Laplacian on a compact polyhedral surface $X$ and compute the $S$-matrix of $X$ at the zero value of the spectral parameter. We apply these results to study various self-adjoint extensions of a symmetric Laplacian on a compact polyhedral surface of genus two with a single conical point. It turns out that the behaviour of the $S$-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.

Desingularization of Coassociative 4-folds with Conical Singularities

Given a coassociative 4-fold N with a conical singularity in a varphi-closed 7-manifold M (a manifold endowed with a distinguished closed 3-form varphi), we construct a smooth family, {N'(t): t\in(0,tau)} for some tau>0, of (smooth, nonsingular,) compact coassociative 4-folds in M which converge to N in the sense of currents, in geometric measure theory, as t tends to 0. This realisation of desingularizations of N is achieved by gluing in an asymptotically conical coassociative 4-fold in R^7, dilated by t, then deforming the resulting compact submanifold of M to the required coassociative 4-fo…

Calabi-Yau metrics with cone singularities along intersecting complex lines: the unstable case

We produce local Calabi-Yau metrics on $\mathbf C^2$ with conical singularities along three or more complex lines through the origin whose cone angles strictly violate the Troyanov condition. The tangent cone at the origin is a flat polyhedral Kähler cone with conical singularities along two intersecting lines: one with cone angle corresponding to the line with smallest cone angle, while the other forms as the collision of the remaining lines into a single conical line. Using a branched covering argument, we can construct Calabi-Yau metrics with cone singularities along cuspidal curves with co…

Fuchsian polyhedra in Lorentzian space-forms

Let S be a compact surface of genus >1, and g be a metric on S of constant curvature K\in\{-1,0,1\} with conical singularities of negative singular curvature. When K=1 we add the condition that the lengths of the contractible geodesics are >2π. We prove that there exists a convex polyhedral surface P in the Lorentzian space-form of curvature K and a group G of isometries of this space such that the induced metric on the quotient P/G is isometric to (S,g). Moreover, the pair (P,G) is unique (up to global isometries) among a particular class of convex polyhedra, namely Fuchsian polyhedra. This e…

为多面体定义正则化的ζ_Δ(1)

Regularized $ζ_Δ(1)$ for Polyhedra

本文提出了一种为多面体定义正则化的ζ_Δ(1)的方法，通过全纯不变量推导出谱不变量的显式表达式，并研究了其渐进行为及四面体上的自共轭扩展的计算。

Epstein-Penner凸包的边界度量与离散共形性

Boundary metric of Epstein-Penner convex hull and discrete conformality

本文研究Epstein-Penner凸包在Minkowski等空间中诱导的边界度量，证明装饰Teichmüller空间与特定度量模空间间存在双射，并给出离散共形性的新3维解释。