arXiv research

Study on geodesic surfaces in hyperbolic 3-orbifolds, proving overlaps in area sets.

Corners can be identified by a drum's sound spectrum.

Study of lengths of cycles in large genus random maps converging to Poisson process.

In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial geometric genus spectrum in which volume and 1-systole are controlled, and analyze th…

The study examines the flexibility of geometric and dynamical properties of metrics on surfaces.

We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric $g_{\mathrm{WP}}$ on $\mathcal M_γ$, the Riemann moduli space of surfaces of genus $γ> 1$. This space has a singular compactification with respect to $g_{\mathrm{WP}}$, and this metric has crossing…

The paper proves a unique orbit for a specific genus 3 curve.

Lower bounds for Hodge-Laplacian spectrum on orbifolds.

In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…

Study essential spectrum of differential operators on geometrically finite orbifolds.

Anosov surfaces with same length spectrum are isometric.

New method proves length spectrum rigidity in various geometric settings.

No Shimura-Teichmüller curves found in genus 5.

In this paper we study the behavior of the spectrum of a compact, connected Riemannian manifold $(M,g)$ of dimension $d \ge 2$, when we add an increasing number of increasingly small handles. No assumptions on any of the curvatures are needed.

Study counts minimal Lagrangians in hyperbolic surfaces with precise growth rate.

Global rigidity theorem for metrics on Σ×S1 without conjugate points.

In this article we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic surface, this spectrum determines the commensurability class. In addition, we show that any finite volume hyperbolic 3-manifold …

Study approximates lengths of random hyperbolic surfaces' geodesics.

The paper proves a bound on eigenvalues for surfaces embedded in 3D space.

Multivariate Poisson approximation of the length spectrum of random surfaces is studied by means of the Chen-Stein method. This approach delivers simple and explicit error bounds in Poisson limit theorems. They are used to prove that Poisson approximation applies to curves of length up to order $o(\log\log g)$ with $g$…

The study proves limitations on isospectral hyperbolic surfaces with discrete length spectra.

In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a sp…

Study the energy spectrum of metrics on surfaces and its relation to simple length spectrum.

We provide two examples of spectral analysis techniques of Schroedinger operators applied to geometric Laplacians. In particular we show how to adapt the method of analytic dilation to Laplacians on complete manifolds with corners of codimension 2 finding the absence of singular continuous spectrum for these operators,…

In this note, we consider the Dirac operator $D$ on a Riemannian symmetric space $M$ of noncompact type. Using representation theory we show that $D$ has point spectrum iff the $\hat A$-genus of its compact dual does not vanish. In this case, if $M$ is irreducible then $M = U(p,q)/U(p) \times U(q)$ with $p+q$ odd, and …

Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.

We prove that if the Lyapunov spectrum of the Kontsevich-Zorich cocycle over an affine SL$(2,\mathbb{R})$-invariant submanifold is completely degenerate, i.e. $λ_2 = \cdots = λ_g = 0$, then the submanifold must be an arithmetic Teichmueller curve in the moduli space of Abelian differentials over surfaces of genus three…

Reformulated Markov's conjecture in combinatorial terms.

The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating with respect to the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincid…

Study geodesics on random hyperbolic surfaces, finding variance similar to prime number theory.

A geometric argument is given to prove that the Seifert genus of a positive knot equals its slice genus. A combinatorial invariant, giving a lower bound for the slice genus, is formulated for arbitrary knots. Properties and applications of this invariant are discussed.

Closed hyperbolic manifolds and manifolds with nonpositive sectional curvature are geometrically rigid under certain curvature conditions.

Study on random hyperbolic surfaces with many cusps, focusing on tight geodesics.

Paper sharpens inequality linking curvature and spectrum on manifolds.

This paper presents a method to obtain geometric registrations between high-genus ($g\geq 1$) surfaces. Surface registration between simple surfaces, such as simply-connected open surfaces, has been well studied. However, very few works have been carried out for the registration of high-genus surfaces. The high-genus t…

First constructed genus 2 Cantor set in 3D space.

We construct universal geometric spaces over the real spectrum compactification $Ξ^{\mathrm{RSp}}$ of the character variety $Ξ$ of a finitely generated group $Γ$ in $\mathrm{SL}_n$, providing geometric interpretations of boundary points. For an algebraic set $Y(\mathbb{R})$ on which $\mathrm{SL}_n(\mathbb{R})$ acts by …

The paper classifies group-actions on surfaces of small genus, focusing on bounding and geometrically bounding cases.

Study geometric rigidity of surfaces in negative curvature manifolds.

The paper proves geometric bordisms for specific hyperbolic surfaces.

We determine the set of all genus g bridge numbers of many iterated torus knots, listing these numbers in a sequence called the bridge spectrum. In addition, we prove a structural lemma about the decomposition of a strongly irreducible bridge surface induced by cutting along a collection of essential surfaces.

We give an explicit description of the spectrum of the Hodge--Laplace operator on $p$-forms of an arbitrary lens space for any $p$. We write the two generating functions encoding the $p$-spectrum as rational functions. As a consequence, we prove a geometric characterization of lens spaces that are $p$-isospectral for e…

Study calculates geometric invariants from Navier-Lamé spectrum.

Localization reveals geometric and analytic properties of the Witten genus.

Study proves rigidity of marked length spectra in contracting group actions.

The paper describes a geometric Schottky group for a non-compact hyperbolic surface with infinite genus.

The paper identifies magnetic ground states and their role in determining the conformal class of a surface.

We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the…