arXiv research

The paper extends decay estimates to graphs with positive spectrum.

ManifoldFlow relaxes fixed-spectrum Stiefel layers to learn a positive spectrum.

We show existence of solutions to the Poisson equation on Riemannian manifolds with positive essential spectrum, assuming a sharp pointwise decay on the source function. In particular we can allow the Ricci curvature to be unbounded from below. In comparison with previous works, we can deal with a more general setting …

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In particular, for any noncompact covering of a compact manifold, there is a metric on the b…

Paper bounds the lowest spectrum of manifolds with curvature constraints.

New inequality controls domain volume for manifolds with large spectrum.

We show how a suitably twisted Spin-cobordism spectrum connects to the question of existence of metrics of positive scalar curvature on closed, smooth manifolds by building on fundamental work of Gromov, Lawson, Rosenberg, Stolz and others. We then investigate this parametrised spectrum, compute its $mod~2$-cohomology …

Sharp bounds and parabolicity results for 3-manifolds with scalar curvature.

Study approximate marked length spectrum rigidity in non-positively curved groups.

Constructs a spectrum for knot Floer homology without holomorphic geometry.

Falsehood of Pólya's conjecture for spheres shown.

Formula derived for Laplace-Beltrami spectrum on homogeneous spaces.

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of $\mathbb{R}^{3}$ is discrete. This give…

Study magnetic potentials on Anosov manifolds using spectral data.

We show that the eigenspaces of the Laplacian $Δ_k$ on $k$-forms on a compact Kähler manifold carry Hodge and Lefschetz decompositions. Among other consequences, we show that the positive part of the spectrum of $Δ_k$ lies in the spectrum of $Δ_{k+1}$.

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

Study Dirichlet-to-Neumann maps on manifolds, focusing on covering and total spaces.

In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $(M,g)$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a completely new stabilit…

Geometric characterization of sub-Riemannian geodesics on frame bundles.

An adapted version of the proof (due to A. Weil) of the well-known de Rham Theorem allows us to compare uniformly the spectrum of the Hodge Laplacian acting on differential forms (on a compact Riemannian manifold) to the spectrum of the combinatorial Laplacian acting on cochains associated to an open cover (made of bal…

Study on signatures of positive braids with bounds derived.

Consider a quantum particle trapped between a curved layer of constant width built over a complete, non-compact, $\mathcal C^2$ smooth surface embedded in $\mathbb{R}^3$. We assume that the surface is asymptotically flat in the sense that the second fundamental form vanishes at infinity, and that the surface is not tot…

Paper extends noncompact Llarull's theorem to manifolds with boundary.

In this paper, we prove that Euclidean hypersurfaces with almost extremal extrinsic radius or $λ_1$ have a spectrum that asymptotically contains the spectrum of the extremal sphere in the Reilly or Hasanis-Koutroufiotis Inequalities. We also consider almost extremal hypersurfaces which satisfy a supplementary bound on …

The classical inverse problem of recovering a simply connected smooth planar domain from the Steklov spectrum \cite{E} is equivalent to the problem of recovering, up to a conformal equivalence, a positive function $a\in C^\infty({\mathbb S})$ on the unit circle ${\mathbb S}=\{e^{iθ}\}$ from the eigenvalue spectrum of t…

The study of spectral-tightness in Riemannian manifolds and its topological implications.

We establish a uniform comparison between the spectrum of the rough Laplacian (acting on sections of a vector bundle of complex rank one or of harmonic curvature) with the spectrum of a discrete operator (a generalization of a discrete magnetic Laplacian added with a potential) acting on a finite dimensional space comi…

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

Let $X$ be an infinite geodesically complete hyperbolic surface which can be decomposed into geodesic pairs of pants. We introduce Thurston's boundary to the Teichmüller space $T(X)$ of the surface $X$ using the length spectrum analogous to Thurston's construction for finite surfaces. Thurston's boundary using the leng…

We find a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the in…

TBIP uses texts to quantify lawmakers' political positions.

We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We de…

This paper studies the problem of spectrum shortage in an unmanned aerial vehicle (UAV) network during critical missions such as wildfire monitoring, search and rescue, and disaster monitoring. Such applications involve a high demand for high-throughput data transmissions such as real-time video-, image-, and voice- st…

Finite number of eigenvalues found in cylindrical surface.

The study examines arithmetic orbifolds and their length spectra, proving uniform discreteness and linear dependence of geodesic lengths.

Motivated by the theory of quantum waveguides, we investigate the spectrum of the Laplacian, subject to Dirichlet boundary conditions, in a curved strip of constant width that is defined as a tubular neighbourhood of an infinite curve in a two-dimensional Riemannian manifold. Under the assumption that the strip is asym…

Krein's formula for conic Laplacians on compact Riemann surfaces

Study shows no new eigenvalues in specific finite coverings.

Parreau compactified the Hitchin component of a closed surface $S$ of negative Euler characteristic in such a way that a boundary point corresponds to the projectivized length spectrum of an action of $π_1(S)$ on an $\mathbb R$-Euclidean building. In this paper, we use the positivity properties of Hitchin representatio…

We give a lower bound on the number of small positive eigenvalues of the p-form Laplacian in a certain type of collapse with curvature bounded below.

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

Dominant representations found via Fock-Goncharov coordinates.

The Hodge spectra can distinguish orbifolds from manifolds, especially in low dimensions.

The paper studies harmonic 1-forms on specific metric measure spaces.

The paper solves the Steklov spectral inverse problem for conformal metrics.

Study compares spectral properties of a specific tensor in geometry.

In this paper we investigate the existence of a solution to the Poisson equation on complete manifolds with positive spectrum and Ricci curvature bounded from below. We show that if a function $f$ has decay $f=O(r^{-1-\varepsilon})$ for some $\varepsilon >0,$ where $r$ is the distance function to a fixed point, then t…

We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In…