arXiv research

Flips connect all triangulations of flat surfaces.

Study flat conical surfaces' Laplacian determinants, showing differences in holonomy.

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

We calculate finite volumes of moduli spaces of flat surfaces with conical singularities.

Derives formulas for determinant of Laplacian on curved surfaces.

We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization the…

Constructs scalar-flat Kähler metrics on toric symplectic manifolds.

Study shortest geodesics on flat cone spheres with conical singularities.

Study finds all local models for flat metrics with isolated singularities.

Study heat flow on collapsing K3 surfaces, handling conic singularities.

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.

The abstract proves spherical surface decompositions with conical singularities.

Bounds on saddle connections on flat spheres with conical singularities.

We prove a formula for the determinant of Laplacian on an arbitrary compact polyhedral surface of genus one. This formula generalizes the well-known Ray-Singer result for a flat torus. A special case of flat conical metrics given by the modulus of a meromorphic quadratic differential on an elliptic surface is also cons…

Study moduli spaces of flat tori using analytical and cohomological methods.

In this paper, we are interested in flat metric structures with conical singularities on surfaces which are obtained by deforming translation surface structures. The moduli space of such flat metric structures can be viewed as some deformation of the moduli space of translation surfaces. Using geodesic triangulations, …

We investigate the functional determinant of the laplacian on piece-wise flat two-dimensional surfaces, with conical singularities in the interior and/or corners on the boundary. Our results extend earlier investigations of the determinants on smooth surfaces with smooth boundaries. The differences to the smooth case a…

The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $Δ^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$. We define a regularized determinant for $Δ^{|df|^2}$ and study it as a functional on t…

This research examines anisotropic conformal transformations of pseudo-Finsler surfaces.

Constructs scalar-flat Kähler metrics with varying conical singularities.

Flat surfaces with finitely many singularities solve systolic extremal problem.

Regularized zeta function for polyhedra calculated from Riemann surface invariants.

The paper proves stability of Ricci de Turck flow near conical singularities.

Proves mass theorem for AF manifolds with conical singularities.

Compact polyhedral surfaces (or, equivalently, compact Riemann surfaces with conformal flat conical metrics) of an arbitrary genus are considered. After giving a short self-contained survey of their basic spectral properties, we study the zeta-regularized determinant of the Laplacian as a functional on the moduli space…

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

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

Proves positive mass theorem on conical manifolds with small angles.

Proves positive mass theorem for AF spin manifolds with conical singularities.

Study on conical Laplacian operators on Riemann surfaces, focusing on determinants and moduli spaces.

We prove an optimal systolic inequality for nonpositively curved Dyck's surfaces. The extremal surface is flat with eight conical singularities, six of angle theta and two of angle 9pi - theta, for a suitable theta with cos(theta) in Q(sqrt{19}). Relying on some delicate capacity estimates, we also show that the extrem…

The study proves an expansion theorem for scalar-flat asymptotically conical Kähler metrics.

The study computes indicial roots and metric convergence orders for Ricci-flat conifolds.

Formula derived for spectral determinant of sphere with conical singularities.

Using methods of A. Grigor'yan and L. Saloff-Coste we prove that on a manifold with a conical end the heat kernel has a Gaussian bound. This result is applied to asymptotically conical Kähler manifolds. It is a result of the author and R. Goto that a crepant resolution of a Ricci-flat Kähler cone admits a Ricci-flat Kä…

Study curve shortening flow on Riemann surfaces with conic singularities.

Constructs surfaces with conical singularities using variational methods.

We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.

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

Compact Ricci-flat manifolds with conical singularities are found.

Generalized Gauss-Bonnet formula for conical metrics on compact Riemann surfaces.

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric $|ω|^2$, where $ω$…

The study bounds and characterizes surfaces containing smooth conics and twistor fibers in a flag threefold.

Extended a formula to higher dimensions with singularities.

Study conical Ricci flow on surfaces with explicit asymptotic expansions.

Lecture notes introduce Abelian differentials and their flat surfaces, focusing on families and Teichmüller dynamics.

Gradient flow solves curvature problem on surfaces with conical points.

Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.