arXiv research

The existence of Kahler-Einstein metrics on a Fano manifold is characterized in terms of a uniform gap between 0 and the first positive eigenvalue of the Cauchy-Riemann operator on smooth vector fields. It is also characterized by a similar gap between 0 and the first positive eigenvalue for Hamiltonian vector fields. …

Characterizes Anosov representations and strongly convex cocompact groups with eigenvalue gaps.

Estimates gaps between eigenvalues for elliptic operators on manifolds.

Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.

Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose fi…

Study on spectral gaps of hyperbolic surfaces as genus increases.

Study gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.

Study shows optimal spectral gaps diminish in large genus surfaces.

The paper provides estimates for eigenvalues of elliptic differential problems.

We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator in the case when the potential is strongly convex. In particular, if the Hessian of the potential is bounded from below by a positive constant, the gap has a lower bound independent of the dimension. We also estimate the gap wh…

This note removes technical assumptions and characterizes relatively dominated representations.

We prove the existence of metrics maximizing the first eigenvalue normalized by area on closed, non-orientable surfaces assuming two spectral gap conditions. These spectral gap conditions are proved by the authors in \cite{MS3}.

Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.

In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which affirmatively answers to a conjecture proposed by Chen-Zheng-Yang. In addition,…

The paper studies eigenvalues in gaps of the essential spectrum of a Bochner-Schrödinger operator.

Sharp upper bounds found for Steklov eigenvalues of warped products.

Study shows fundamental gap of horoconvex domains in hyperbolic space has no positive lower bound.

In this paper, we investigate eigenvalues of the Dirichlet problem and the closed eigenvalue problem of drifting Laplacian on the complete metric measure spaces and establish the corresponding general formulas. By using those general formulas, we give some upper bounds of consecutive gap of the eigenvalues of the eigen…

The article explores the fundamental gap in Bakry-Emery geometry.

We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator with a nonconvex potential in terms of a distance associated with the potential. The results here can be applied to the double well potential.

This paper studies eigenvalues of the clamped plate problem on a bounded domain in an $n$-dimensional Euclidean space. We give an estimate for the gap between $\sqrt {Γ_{k+1}-Γ_{1}}$ and $\sqrt {Γ_{k}-Γ_{1}}$, for any positive integer $k$. According to the asymptotic formula of Agmon and Pleijel, we know, the gap betwe…

The study proves optimal spectral gaps for hyperbolic surfaces.

The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.

Estimates the mass gap for domains with integral Ricci curvature bounds.

By the calculation of the gap of the consecutive eigenvalues of $\Bbb S^n$ with standard metric, using the Weyl's asymptotic formula, we know the order of the upper bound of this gap is $k^{\frac{1}{n}}.$ We conjecture that this order is also right for general Dirichlet problem of the Laplace operator, which is optimal…

The paper derives upper bounds on eigenvalues of Laplace-Beltrami operator on hyperbolic surfaces.

Random feature model shows slow self-correction of generalization gap.

Paper improves volume gap between minimal submanifolds and unit spheres.

New bounds on hyperbolic surfaces' properties using linear programming.

Study on Kähler manifolds shows rigidity of eigenvalues with positive Ricci bound.

Negative curvature restricts the gap between the first and second eigenvalues of convex domains.

The paper constructs noncompact hyperbolic surfaces with uniform spectral gaps using random graph models.

Our main result is that if a generic convex domain in $\R^n$ collapses to a domain in $\R^{n-1}$, then the difference between the first two Dirichlet eigenvalues of the Euclidean Laplacian, known as the fundamental gap, diverges. The boundary of the domain need not be smooth, merely Lipschitz continuous. To motivate th…

Estimates spectral gap for Brownian motion on sticky-reflecting domains.

Using expander graphs, we construct a sequence of smooth compact surfaces with boundary of perimeter N, and with the first non-zero Steklov eigenvalue uniformly bounded away from zero. This answers a question which was raised in [9]. The genus grows linearly with N, this is the optimal growth rate.

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We give a new lower bound for the first gap $λ_2 - λ_1$ of the Dirichlet eigenvalues of the Schr{ö}dinger operator on a bounded convex domain $Ω$ in R$^n$ or S$^n$ and greatly sharpens the previous estimates. The new bound is explicit and computable.

Three models are shown to be isometrically equivalent, with a gapless first eigenvalue.

We study the spectral gap of the Erdős--Rényi random graph through the connectivity threshold. In particular, we show that for any fixed $δ> 0$ if $p \ge \frac{(1/2 + δ) \log n}{n},$ then the normalized graph Laplacian of an Erdős--Rényi graph has all of its nonzero eigenvalues tightly concentrated around $1$. We est…

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero poten…

Explicitly bounds the spectral gap for Schottky subgroups of SL(2,Z).

We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model centers of the tube. The bounds are sharp on these models. Moreover, when the mo…

We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with $C^2$ boundaries. We show that for an $n$-dimensional geometry, the spectral gap is bounded above by $(n-1)^2/4$, which we prove to be the infimum of the essential spectrum. We also construct examples of c…

New bounds for private matrix approximation using Gaussian noise and Dyson Brownian Motion.

In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space and conjectured similar results holds for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely wh…

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…

Random covers of hyperbolic surfaces have a spectral gap with polynomial rate.

The paper proves Simon's conjecture for closed Einstein spaces, setting eigenvalue bounds.