arXiv research

In this paper, we establish Buser type inequalities, i.e., upper bounds for eigenvalues in terms of Cheeger constants. We prove the Buser's inequality for an infinite but locally finite connected graph with Ricci curvature lower bounds. Furthermore, we derive that the graph with positive curvature is finite, especially…

Sharp inequalities proved for RCD spaces, showing equality conditions.

Paper connects probability density cuts to graph theory eigenfunctions.

The goal of the paper is to sharpen and generalise bounds involving the Cheeger's isoperimetric constant $h$ and the first eigenvalue $λ_{1}$ of the Laplacian. A celebrated lower bound of $λ_{1}$ in terms of $h$, $λ_{1}\geq h^{2}/4$, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on $λ_{1…

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t μ- P_t ν\|_1 \leq \frac{W_1(μ,ν)}{\sqrt{t}} \] for all probability measures $μ,ν$ where $P_t$ is the heat semigroup and $W_1$ is the $\ell_1$-Wasserstein distance. This turns out to be an equivalent formulation of a version of…

We introduce notions of Cheeger constants for graphons and graphings. We prove Cheeger and Buser inequalities for these. On the way we prove co-area formulae for graphons and graphings.

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.

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.

Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.

In this paper we study the systole growth of arithmetic locally symmetric spaces up congruence covers and show that this growth is at least logarithmic in volume. This generalizes previous work of Buser and Sarnak as well as Katz, Schaps and Vishne where the case of compact hyperbolic 2- and 3-manifolds was considered.

We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that $L+2 Ric$ is a positive operator where $L$ is the graph Laplacian. Assuming that the negative part of the Ricci curvature is small …

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 …

We define a hybrid between Ollvier and Bakry Emery curvature on graphs with dependence on a variable neighborhood. The hexagonal lattice is non-negatively curved under this new curvature notion. Bonnet-Myers diameter bounds and Lichnerowicz eigenvalue estimates follow from the standard arguments. We prove gradient esti…

We prove an inequality that must be satisfied by displacement of generators of free Fuchsian groups, which is the two-dimensional version of the $\log (2k-1)$ Theorem for Kleinian groups due to Anderson-Canary-Culler-Shalen. As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as …

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues $λ_k$ of conformal sub-Riemannian metrics that are asymptotically sharp as $k\to +\infty$. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for…

Simplified proof for Cheeger's isoperimetric constant.

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…

Lower bound on boundary injectivity radius for specific tubes.

Directly proves logarithmic systolic growth for all hyperbolic surfaces.

Improved Sobolev mappings in Carnot groups with weaker assumptions.

In this paper, we present a Lichnerowicz type estimate and (higher order) Buser type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential. These results relate eigenvalues, magnetic fields, Ricci curvature, and Cheeger type constants.

We discuss our recent work on small eigenvalues of surfaces. As an introduction, we present and extend some of the by now classical work of Buser and Randol and explain novel ideas from articles of Sévennec, Otal, and Otal-Rosas which are of importance in our line of thought.

In this paper, we study the spectrums of faithful dimension pairs on a closed Finsler manifold and obtain a Gromov type and a Buser type lower bounds for eigenvalues. Furthermore, for the Lusternik-Schnirelmann spectrum, we not only obtain a better lower bound, but also estimate the multiplicity of each eigenvalue.

We show that a general $n$-dimensional polarized abelian variety $(A,L)$ of a given polarization type and satisfying $h^0(A, L) \geq \dfrac{8^n}{2} \cdot \dfrac{n^n}{n !}$ 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…

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…

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 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.

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…

In this paper we prove that, for any arithmetic hyperbolic $n$-manifold $M$ of the first type, the systole of most of the principal congruence coverings $M_{I}$ satisfy $sys_{1}(M_{I})\geq \frac{8}{n(n+1)}\log(vol(M_{I}))-c,$ where $c$ is a constant independent of $I$. This generalizes previous work of Buser and Sarn…

The study proves optimal spectral gaps for hyperbolic surfaces.

In this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler-Laplacian operator, we introduce \textit{faithful dimension pairs} by means of which the spectrum of a compact reversible Finsler metric measure manifold is defined. Various upper and lower bounds of such eigen…

Paper shows surfaces can't be heard to be orientable.

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…

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 …

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…

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds wit…

We obtain upper bounds for the eigenvalues of the Schrödinger operator $L=Δ_g+q$ 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…

The isoperimetric inequality and related inequalities are explored.

New proof of Willmore inequality using geometric divergence inequality.

Lorentz-Finsler geometry reveals new and old inequalities.

The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.

The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.

New inequality on sphere generalizes circle inequality.

Paper proves anisotropic Minkowski inequality and related inequalities.

Explains geometric inequalities for minimal hypersurfaces.

The paper finds new inequalities for convex polygons.