The paper estimates magnetic Laplacian eigenvalues using Ricci curvature and magnetic fields.
problem Estimating eigenvalues of the magnetic Laplacian on manifolds.
method Lichnerowicz and Buser type estimates related to Ricci curvature and magnetic fields.
result Relates eigenvalues, magnetic fields, Ricci curvature, and Cheeger type constants.
Lower bound on boundary injectivity radius for specific tubes.
problem Estimating the boundary injectivity radius of Margulis tubes.
method Using curvature bounds to derive a lower bound.
result A lower bound on the boundary injectivity radius is provided.
The paper proves Buser's inequality for graphs with positive curvature.
problem Establishing bounds for eigenvalues in terms of Cheeger constants for infinite graphs.
method Proved Buser's inequality for graphs with Ricci curvature lower bounds and derived a lower bound on Cheeger constant in terms of positive curvature.
result Graphs with positive curvature are finite, especially for unbounded Laplacians.
Graphs with non-negative curvature have improved heat semigroup estimates.
problem Estimating heat semigroup norms on graphs with curvature.
method Proved a reverse Poincaré inequality for graphs with non-negative Ollivier curvature.
result Heat semigroup norm estimates lead to Buser inequality, Liouville property, and eigenvalue estimates.
Sharp bounds on Cheeger's isoperimetric constant and Laplacian eigenvalue in metric measure spaces.
problem Sharp bounds on Cheeger's isoperimetric constant and Laplacian eigenvalue in metric measure spaces.
method Sharp bounds on Cheeger's isoperimetric constant and Laplacian eigenvalue in metric measure spaces.
result Sharp bounds on Cheeger's isoperimetric constant and Laplacian eigenvalue in metric measure spaces.
Simplified proof for Cheeger's isoperimetric constant.
problem Cheeger's isoperimetric constant
method Simplified proof of Buser's result
result Simplified proof for Cheeger's isoperimetric constant
Explains new insights on small surface eigenvalues.
problem Understanding small eigenvalues of surfaces.
method Extends classical work and introduces new ideas.
result Novel approaches to small eigenvalues.
Sharp inequalities proved for RCD spaces, showing equality conditions.
problem Proving sharp inequalities for RCD spaces and identifying equality conditions.
method Analyzing RCD(1,∞) and RCD(K,∞) spaces to prove inequalities and identify equality conditions. result Equality conditions for Buser's and Cheeger's inequalities in RCD spaces.
Paper connects probability density cuts to graph theory eigenfunctions.
problem Developing sparse cuts for probability densities.
method Defines sparse cuts and principal eigenfunctions for probability densities, proving Cheeger and Buser inequalities.
result No such inequalities hold for prior definitions, proving new inequalities for probability densities.
Directly proves logarithmic systolic growth for all hyperbolic surfaces.
problem Proving logarithmic systolic growth for all hyperbolic surfaces.
method Using original Brooks/Buser-Sarnak surfaces through a direct approach.
result Directly proves logarithmic systolic growth for all hyperbolic surfaces.
The paper sets lower bounds for eigenvalues on Finsler manifolds.
problem Understanding the spectrum of Finsler manifolds.
method Analysis of faithful dimension pairs and application of Gromov and Buser types of lower bounds.
result Improved lower bounds for eigenvalues and estimates of multiplicity.
New curvature measure on graphs improves diameter and eigenvalue bounds.
problem Improving curvature bounds on graph structures.
method Hybrid curvature definition on variable neighborhoods.
result Gradient estimates and curvature bounds proven.
Buser's inequality gives an upper bound on the first non-zero eigenvalue of the Laplacian of a closed manifold M in terms of the Cheeger constant h(M). Agol later gave a quantitative improvement of Buser's inequality. Agol's result is less transparent since it is given implicitly by a set of equations, one of which is …
This paper studies graph curvature and its geometric implications.
problem Analyzing non-constant Ricci curvature bounds on graphs.
method Proves eigenvalue estimates, finiteness of fundamental group, diameter bounds, Harnack inequality, and Buser inequality under specific curvature conditions.
result Establishes spectral positive Bakry-Émery Ricci curvature on graphs, providing new geometric insights.
Improved Sobolev mappings in Carnot groups with weaker assumptions.
problem Improving Sobolev mappings in Carnot groups with weaker conditions.
method Using Buser-Karcher center-of-mass and polynomial expressions in moments.
result Rigidity and structural results hold under weaker Sobolev exponents.
In this paper, we consider Cheeger's constant and the first eigenvalue of the nonlinear Laplacian on closed Finsler manifolds. Being based on these, we establish Cheeger's inequality and Buser's inequality for closed Finsler manifolds.
Cheeger inequalities defined for graph limits, proving key inequalities.
problem Defining and proving Cheeger inequalities for graph limits.
method Introducing graphon and graphing concepts, proving inequalities.
result Proved Cheeger and Buser inequalities for graphons and graphings.
We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove the versions of these results for the Dirichlet and Neumann boundary value problems. Eigenvalue multiplicity bounds and related open problems are also discussed.
In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a bounded domain (with smooth boundary) in a given complete (not compact a priori) Riemannian manifold with Ricci bounded below . For this, we use test functions for the Rayleigh quotient subordinated to a family of open se…
Study shows logarithmic growth in systole for arithmetic spaces.
problem Understanding systole growth in arithmetic locally symmetric spaces.
method Examined congruence covers and showed logarithmic growth in systole.
result Logarithmic growth in systole is at least as large as volume.
We show that a general n-dimensional polarized abelian variety (A,L) of a given polarization type and satisfying h0(A,L)≥28n⋅n!nn is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely one-dimensional complex analytic subvariety i…
This article is dedicated to prove Buser's conjecture about Bers' constants for spheres with cusps (or marked points) and for hyperelliptic surfaces. More specifically, our main theorem states that any hyperbolic sphere with n cusps has a pants decomposition with all of its geodesics of length bounded by a constant r…
The systole of most congruence coverings of arithmetic hyperbolic manifolds is at least a certain value.
problem Estimating the systole of congruence coverings of arithmetic hyperbolic manifolds.
method Proving a lower bound on the systole of most principal congruence coverings using logarithmic and constant factors.
result The systole of most congruence coverings satisfies a specific lower bound.
We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues λk of conformal sub-Riemannian metrics that are asymptotically sharp as k→+∞. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for…
It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in ge…
We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). Th…
Proves inequality for Fuchsian groups, improving surface geometry.
problem Displacement of generators in free Fuchsian groups.
method Analyzes displacement inequalities for Fuchsian groups.
result Quantitative results on hyperbolic surface geometry.
P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus s, of the least conformal length of a nonseparating loop, is logarithmic in s. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic i…
Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality on Finsler metric measure manifolds.
method Defining volume entropy and second Cheeger constant, proving sharp inequality.
result Sharp isoperimetric inequality involving volume entropy and weighted Ricci curvature.
We examine the large systole problem, which concerns compact hyperbolic Riemannian surfaces whose systole, the length of the shortest noncontractible loops, grows logarithmically in genus. The generalization of a construction of Buser and Sarnak by Katz, Schaps, and Vishne, which uses principal "congruence" subgroups o…
Constructs non-isometric iso-length-spectral surfaces.
problem Creating non-isometric surfaces with identical geodesic lengths.
method Combining Sunada's construction with amalgams of hyperbolic surfaces.
result Found non-isometric surfaces with the same geodesic lengths.
Our main result is that for all sufficiently large x0>0, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field k and systole bounded below by x0 has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds wit…
The study proves optimal spectral gaps for hyperbolic surfaces.
problem Proving optimal spectral gaps for hyperbolic surfaces.
method Proving the absence of eigenvalues in a specific range for random covers of hyperbolic surfaces.
result The first non-zero eigenvalue of the Laplacian on a sequence of closed hyperbolic surfaces tends to 1/4.
Paper shows surfaces can't be heard to be orientable.
problem Determining orientability from spectral data.
method Applied Sunada's and Buser's methods to orbifolds.
result Constructed isospectral flat surfaces with different orientability.
Investigates spectral problems in Finsler geometry with novel dimension pairs.
problem Spectral problems in Finsler geometry due to nonlinearity of the Finsler-Laplacian operator.
method Introduced 'faithful dimension pairs' to define the spectrum of compact reversible Finsler metrics.
result Provided upper and lower bounds for eigenvalues, extending previous results.
We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-Émery Ricci curvature. We derive some universal inequalities among eigenvalues of the weighted Laplacian on such manifolds. These inequalities are quantitative versions of the previous t…
Researchers approximate spectral targets on manifolds with constant negative curvature.
problem Prescribing an arbitrary finite portion of the Laplace-Beltrami spectrum on manifolds of constant negative curvature.
method Constructing macroscopically heterogeneous hyperbolic covering manifolds in d≥3 and using discrete spectral limit theorems in d=2. result Any finite strictly increasing target list can be approximated to arbitrary precision by a closed manifold of constant negative curvature.
To a compact Riemann surface of genus g can be assigned a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Schottky problem is to discern the Jacobians among the PPAVs. Buser and Sarnak showed, that the square of the first successive minimum, the squared norm of the …
We obtain upper bounds for the eigenvalues of the Schrödinger operator L=Δg+q depending on integral quantities of the potential q and a conformal invariant called the min-conformal volume. Moreover, when the Schrödinger operator L is positive, integral quantities of q which appear in upper bounds, can be repla…
We give a brief literature review of the isoperimetric problem and discuss its relationship with the Cheeger constant of Riemannian n-manifolds. For some non-compact, finite area 2-manifolds, we prove the existence and regularity of subsets whose isoperimetric ratio is equal to the Cheeger constant. To do this, we us…
New curvature tensor and matrices for connection graphs derived from Bakry-Émery curvature.
problem Deriving Buser-type bounds on eigenvalues of connection Laplacians.
method Reformulation of Bakry-Émery curvature through curvature matrices and tensor representations.
result Extension of curvature matrices to connection graphs, addressing eigenfunction challenges.
Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the lengths of homologically independent curves on closed Riemannian surfaces. As a conseq…
New estimators outperform maximum likelihood without hyper-parameter estimation.
problem Improving system identification performance without hyper-parameter estimation.
method Developed generalized Bayes and closed-form biased estimators using excess MSE.
result New estimators have comparable performance to empirical-Bayes-based regularized estimator.
Dual Bayesian Affine Estimators for Wiener-type state-space models
problem Estimating parameters in Wiener-type state-space models
method Fixed-point architecture combining two affine estimators
result Dual basis-parameter estimator achieves comparable parameter MSE to purely affine estimator
New estimator reduces kernel mean estimation error.
problem Kernel mean estimation in reproducing kernel Hilbert spaces.
method Corrupt data with known distributions and estimate kernel mean under the corrupted distribution.
result The marginalized kernel mean estimator achieves lower estimation error.
Enhances gradient estimates for Hermitian Monge-Ampère equations.
problem Improving estimates for Hermitian Monge-Ampère equations.
method Improves gradient estimates using Evans-Krylov and third derivatives estimates.
result Enhanced estimates for second and third order derivatives.
Paper proposes robust estimators for GANs under Wasserstein contamination.
problem Robust estimation of distributions under contamination.
method Wasserstein GAN-based estimators for location, covariance, and regression.
result Proposed estimators are minimax optimal in many scenarios.
New framework converts offline to online estimation using black-box offline estimators.
problem Convert offline estimation algorithms to online estimation algorithms.
method Oracle-Efficient Online Estimation (OEOE) framework.
result Achieves near-optimal online estimation error via black-box offline estimators.