arXiv research

The paper studies the growth of the Weil-Petersson inradius in moduli space.

Study collapsing manifolds with boundary and inradius convergence to zero.

Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds

Study inradius collapsed manifolds with lower Ricci curvature bounds, proving properties of their limits.

Sharp inradius estimates for convex domains in 2D Alexandrov spaces.

The paper finds lower bounds on curve length with constraints on width and inradius.

Sharp stability results for reverse isoperimetric inequalities in 2D.

Optimal bounds for Laplacian eigenvalues on weighted graphs.

We bound two global invariants of cusped hyperbolic manifolds: the length of the shortest closed geodesic (the systole), and the radius of the biggest embedded ball (the inradius). We give an upper bound for the systole, expressed in terms of the dimension and simplicial volume. We find a positive lower bound on the in…

The paper confirms a conjecture about convex bodies and their properties.

The paper examines critical points of solutions to a surface equation in 3D spacelike spaces.

The study examines nodal geometry, heat diffusion, and Brownian motion on Riemannian manifolds.

We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equ…

In this paper, we define and study the Weil-Petersson geometry. Under the framework of the Weil-Petersson geometry, we study the Weil-Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil-Petersson metric and the Ricci curvature of the Weil-Petersson metric f…

The paper examines limit sets of Weil-Petersson geodesics and their properties.

Harmonic extension of Weil-Petersson circle homeomorphisms

In this paper, we proved that the Weil-Petersson volume of Calabi-Yau moduli is a rational number. We also proved that the integrations of the invariants of the Ricci curvature of the Weil-Petersson metric with respect to the Weil-Petersson volume form are all rational numbers.

Convexity properties of Weil-Petersson geodesics on the Teichmüller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considera…

Study develops Weil-Petersson geometry on stability space.

The paper solves reverse isoperimetric problems for convex bodies with curvature constraints.

We extend Weil-Petersson theory to infinite type Teichmüller spaces.

The Weil-Petersson metric's curvature vanishes for surfaces with short geodesics.

Study abelian varieties' Weil-Petersson metric asymptotics.

The sectional curvature of the Weil-Petersson metric on Teichmuller space is known to be negative. We show that this Weil-Petersson sectional curvature is not pinched from above by any negative constants, i.e., there is no negative upper bound.

A summary introduction of the Weil-Petersson metric space geometry is presented. Teichmueller space and its augmentation are described in terms of Fenchel-Nielsen coordinates. Formulas for the gradients and Hessians of geodesic-length functions are presented. Applications are considered. A description of the Weil-Peter…

We give a new proof that the completion of the Weil-Petersson metric on Teichmüller space is Gromov-hyperbolic if the surface is a five-times punctured sphere or a twice-punctured torus. Our methods make use of the synthetic geometry of the Weil-Petersson metric.

Two metrics on graph spaces compared to Weil-Petersson.

Dual volume bound linked to Weil-Petersson distance in quasi-Fuchsian manifolds.

In this paper we proved that the Weil-Petersson volume of the Chern class of any order over the moduli space of Calabi-Yau manifolds is a rational number. We also found the necessary and sufficient condition of the incompleteness of Weil-Petersson metric in several variables case.

The study proves curvature implications for Weil-Petersson metrics.

The study creates examples of geodesics with unique limit sets.

Defines a new metric on Fano Kaehler-Ricci solitons.

The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.

Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures established.

Maximal discs in Anti-de Sitter space linked to Teichmüller space.

New proof of Weil-Petersson metric for Calabi-Yau varieties.

New interpretation of complex hyperbolic form as Weil-Petersson form.

The paper calculates super Weil-Petersson volumes for large genus.

Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil-Petersson form. Mirzakhani proved that Weil-Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers on moduli spaces of curves. In this survey article, we discuss these results as w…

We consider inverse curvature flows in the $(n+1)$-dimensional Euclidean space, $n\geq 2,$ expanding by arbitrary negative powers of a 1-homogeneous, monotone curvature function $F$ with some concavity properties. We obtain asymptotical roundness, meaning that circumradius minus inradius of the flow hypersurfaces decay…

We compute curvatures of a three-manifold formed by a Weil-Petersson geodesic in Teichmuller space.

Study calculates volumes of Fano K-moduli spaces in various dimensions.

Let $S_g$ be a closed surface of genus $g$ and $\mathbb{M}_g$ be the moduli space of $S_g$ endowed with the Weil-Petersson metric. In this paper we investigate the Weil-Petersson curvatures of $\mathbb{M}_g$ for large genus $g$. First, we study the asymptotic behavior of the extremal Weil-Petersson holomorphic sectiona…

The paper connects Weil-Petersson homeomorphisms to maximal surfaces in anti-de Sitter space.

The universal Liouville action equals the renormalized volume of a hyperbolic 3-manifold.

Weil-Petersson volumes vary continuously with weighted points on a projective line.

We show that the strong asymptotic class of Weil-Petersson (WP) geodesics with narrow end invariant and bounded annular coefficients is determined by the forward ending lamination. This generalizes the Recurrent Ending Lamination Theorem of Brock-Masur-Minsky. As an application we provide a symbolic condition for diver…

Estimates scalar curvature on moduli space as genus grows.