arXiv research

Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.

Study compares spectral properties of a specific tensor in geometry.

Constructs Serre spectral sequence for bounded cohomology.

Study approximates sub-Riemannian structures with Riemannian metrics and analyzes spectral convergence.

Paper proves new theorems about curvature in weighted manifolds.

We use spectral embeddings to give upper bounds on the spectral function of the Laplace--Beltrami operator on homogeneous spaces in terms of the volume growth of balls. In the case of compact manifolds, our bounds extend the 1980 lower bound of Peter Li for the smallest positive eigenvalue to all eigenvalues. We also i…

Sub-Riemannian spectral distance defined using eigenfunctions of sub-Laplacian

In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectra…

The study compares spectral volumes of manifolds with weakly convex boundaries.

The study examines hyperbolic 3-manifolds with uniform spectral gaps for coclosed 1-forms.

Study spectral functionals on manifolds with torsion.

Infinite volume requires no atoms at the bottom of the spectrum for certain groups.

New volume functions for random hyperbolic surfaces link to spectral gaps.

Researchers calculate spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.

The study examines spectral properties of the Laplacian on forms for open Riemannian manifolds.

Study shows eigenvalue of Hodge Laplacian on coexact 1-forms in hyperbolic 3-manifolds is related to isoperimetric ratio.

We prove two-sided inequalities for the $L^p$-norm of a pushforward or pullback (with respect to an orientation-preserving diffeomorphism) on oriented volume and Riemannian manifolds. For a function or density on a volume manifold, these bounds depend only on the Jacobian determinant, which arises through the change of…

Study on variance of Laplace eigenfunctions on manifolds.

Let $M$ be a connected, noncompact, complete Riemannian manifold, consider the operator $L=\DD +\nn V$ for some $V\in C^2(M)$ with $\exp[V]$ integrable w.r.t. the Riemannian volume element. This paper studies the existence of the spectral gap of $L$. As a consequence of the main result, let $\rr$ be the distance functi…

In a noncommutative torus, effect of perturbation by inner derivation on the associated quantum stochastic process and geometric parameters like volume and scalar curvature have been studied. Cohomological calculations show that the above perturbation produces new spectral triples. Also for the Weyl C^*-algebra, the La…

Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.

The paper calculates spectral invariants of magnetic Steklov eigenvalues on Riemannian manifolds.

Study of large-$n$ asymptotics for Weil-Petersson volumes of hyperbolic surfaces with cusps.

In recent years, several families of hyperbolic knots have been shown to have both volume and $λ_1$ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist nu…

In this paper, we prove (1): for any closed contact three-manifold with a $C^\infty$-generic contact form, the union of periodic Reeb orbits is dense, (2): for any closed surface with a $C^\infty$-generic Riemannian metric, the union of closed geodesics is dense. The key observation is $C^\infty$-closing lemma for 3D R…

A simple analytically solvable model exhibiting a 1/f spectrum in an arbitrarily wide frequency range was recently proposed by Kaulakys and Meskauskas (KM). Signals consisting of a sequence of pulses show that inherent origin of the 1/f noise is Brownian fluctuations of the average intervent time between subsequent pul…

New cutoff phenomenon found for geodesic paths on hyperbolic manifolds.

Dirichlet-Neumann duality for Riemannian submersions

Paper derives inequalities for eigenvalues of Witten-Laplacian under fixed volume constraint.

The paper extends spectral estimates to hyperbolic surfaces with hyperbolic ends.

We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collecti…

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eige…

At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the deepest results of this theory is the Alexandrov-Fenchel inequality, which subsumes…

On curved spaces, viscous fluids reach equilibrium quickly.

Magnitude of geometric shapes studied for smooth manifolds, revealing spectral geometry insights.

Let $(M^m,g_M)$ be a closed, connected manifold with positive scalar curvature and $(T^k,g)$ some flat $k$-Torus of unit volume. By a result of F. Dobarro and E. Lami Dozo, there exists a unique $f: M \rightarrow \mathbf{R}_{>0}$ such that the warped product $M\times_f T^k$ has constant scalar curvature and unit volume…

The paper analyzes graph Laplacians on manifolds with curvature bounds and applies to non-collapsed spaces.

Unified view of spectral networks linking geometry and gauge theory.

We obtain an Euclidean volume growth results for complete Riemannian manifolds satisfying a Euclidean Sobolev inequality and a spectral type condition on the Ricci curvature. We also obtain eigenvalue estimates, heat kernel estimates, Betti number estimates for closed manifolds whose Ricci curvature is controlled in th…

We show that a bi-invariant metric on a compact connected Lie group $G$ is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric $g_0$ on $G$ there is a positive integer $N$ such that, within a neighborhood of $g_0$ in the class of left-invariant metrics of a…

Sharp spectral theorems and isoperimetric inequalities for manifolds with nonnegative Ricci curvature.

Single-particle electron cryomicroscopy is an essential tool for high-resolution 3D reconstruction of proteins and other biological macromolecules. An important challenge in cryo-EM is the reconstruction of non-rigid molecules with parts that move and deform. Traditional reconstruction methods fail in these cases, resu…

The class of Riemannian orbifolds of dimension n defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is …

We prove a number of results relating various measures (volume, Legendrian index, stability index, and spectral curve genus) of the geometric complexity of special Lagrangian $T^2$-cones. We explain how these results fit into a program to understand the "most common" three-dimensional isolated singularities of special …

Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.

Randomly glued tetrahedra form connected 3-manifolds with a single boundary.

The paper examines properties and rigidity of self-expanders in Euclidean space.

The paper studies essential spectra of submanifolds in Euclidean spaces.