arXiv research

New inequalities for spectral zeta kernels on spheres and manifolds.

Proves new inequality linking spectral numbers of Lagrangians and their reductions.

Study spectral properties on manifolds with conical singularities, proving new inequalities.

Proves spectral inequality and null-controllability for elliptic operators on closed manifolds.

Sharp Hardy and spectral gap inequalities found on special irreversible Finsler manifolds.

Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.

Proves $L^2$ Frölicher inequality on complex manifolds.

Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.

Derives generalizations of the long neck principle and spectral width inequality.

The paper proves optimal estimates and inequalities for spectral functions on certain manifolds.

This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This i…

New proof shows certain 3D spaces are essentially like infinite space.

Under a spectral assumption on the Laplacian of a Poincaré--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order $2γ\in(0,2)$ or $2γ\in(2,4)$ and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption…

Extends spectral torus band inequalities for compact manifolds with scalar curvature bounds.

Using McCann's transportation map, we establish a transport inequality on compact manifolds with positive Ricci curvature. This inequality contains the sharp spectral comparison estimates.

Study Bergman and spectral kernels for non-compact complex manifolds.

Develops a dynamical method to prove the sharp Berezin-Li-Yau inequality.

Seidel-Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith-type inequalities. Similar-looking spectral sequences have been defined by Lee, Bar-Natan, Ozsváth-Szabó, Lipshitz-Tre…

A reverse Riesz estimate and spectral gap imply a Poincaré inequality.

Study of WKB asymptotics of Stokes matrices and spectral curves, proving rhombus inequalities.

Study geometric properties and spectral estimates on warped products.

Paper connects probability density cuts to graph theory eigenfunctions.

The paper finds extremum values for mixed Laplacian eigenvalues on triangles and trapezoids.

Study on discrepancy principle for learning algorithms in nonparametric regression.

This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.

The paper studies isoperimetric inequalities on warped product manifolds.

Paper proves uniqueness of solutions to a geometric inequality problem.

Paper introduces magnetic Steklov operator on differential forms and its properties.

We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric construction related to the discrete isoperimetric problem on the surface of a cone. In…

Establishes a rank inequality between knot Floer homologies of freely 2-periodic knots and their quotients.

The study proves inequalities on curved spaces without global curvature bounds.

The study proves Lieb-Thirring inequalities on hyperbolic manifolds.

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

Proves Hölder-type inequality for Lagrangians' distance.

Let φ(G) be the minimum conductance of an undirected graph G, and let 0=λ_1 <= λ_2 <=... <= λ_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, φ(G) = O(k) λ_2 / \sqrt{λ_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. …

PAC-Bayesian matrix completion with a spectral scaled Student prior offers efficient inference.

Graph Laplacians and machine learning predict properties of finite graphs.

A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci curvature or the Bakry--Emery curvature-dimension inequalities. We show that when this …

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

Let $\widetilde{K}$ be a 2-periodic knot in $S^3$ with quotient $K$. We prove a rank inequality between the knot Floer homology of $\widetilde{K}$ and the knot Floer homology of $K$ using a spectral sequence of Hendricks, Lipshitz and Sarkar. We also conjecture a filtered refinement of this inequality, for which we giv…

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…

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…

We obtain a Bonnet-Myers theorem under a spectral condition: a closed Riemannian manifold $(M^n,g)$ for which the lowest eigenvalue of the Ricci tensor $ρ$ is such that the Schrödinger operator $(n-2)Δ+ ρ$ is positive has finite fundamental group. As a continuation of our earlier results, we obtain isoperimetric inequa…

The purpose of this paper is to give a self-contained proof that a complete manifold with more than one end never supports an $L^{q,p}$-Sobolev inequality ($2 \leq p$, $q\leq p^{*}$), provided the negative part of its Ricci tensor is small (in a suitable spectral sense). In the route, we discuss potential theoretic pro…

We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in the high tensor powers of the line bund…

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…

We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying $\mathrm{Ric}_{\infty} \ge K>0$. Assuming equality holds, we show that the $1$-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap …

The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.