arXiv research

Survey on bottom of spectrum of Hodge Laplacian on complete noncompact Kähler manifolds

The study finds the maximum spectrum of 3D manifolds with lower scalar curvature.

The spectrum of certain manifolds matches that of hyperbolic space if the bottom spectrum is maximal.

Sharp spectral estimates for negatively curved foliations.

We study the bottom of the spectrum in Hilbert geometries, we show that it is zero if and only if the geometry is amenable, in other words if and only if it admits a Fölner sequence. We also show that the bottom of the spectrum admits an upper bound, which depends only on the dimension and which is the bottom of the sp…

We prove that if a Riemannian covering preserves the bottom of the spectrum of a Schrödinger operator, which belongs to the discrete spectrum of the operator on the base manifold, then the covering is amenable.

Paper sharpens inequality linking curvature and spectrum on manifolds.

Paper bounds the lowest spectrum of manifolds with curvature constraints.

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

We discuss the behaviour of the bottom of the spectrum of scalar Schrödinger operators under Riemannian coverings.

Closed hyperbolic manifolds and manifolds with nonpositive sectional curvature are geometrically rigid under certain curvature conditions.

We consider a class of complete Kahler manifolds with a strictly pseudoconvex boundary at infinity. After studying its asymptotic geometry, we formulate a conjecture in the Kahler-Einstein case relating the bottom of spectrum to the CR geometry on the boundary. We prove some partial results.

Study bottom of spectra on orbifolds via coverings.

Sharp bounds on scalar curvature spectrum and rigidity theorems.

Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.

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

In this paper, we study self-expanders for mean curvature flows. First we show the discreteness of the spectrum of the drifted Laplacian on them. Next we give a universal lower bound of the bottom of the spectrum of the drifted Laplacian and prove that this lower bound is achieved if and only if the self-expander is th…

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

Study on spectral properties of Riemannian submersions with special fibers.

New invariant extends curvature estimates to noncompact manifolds.

We prove an upper bound on the bottom of the essential spectrum of a diffusion in term of the growth of the volume of $X$, generalizing a result by R. Brooks.

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

In \cite{LiWang2001complete1,LiWang2001complete2}, Li-Wang proved a splitting theorem for an n-dimensional Riemannian manifold with $Ric\geqslant -(n-1)$ and the bottom of spectrum $λ_0(M)=\frac{(n-1)^2}{4}$. For an n-dimensional compact manifold $M$ with $Ric\geqslant-(n-1)$ with the volume entropy $h(M)=n-1$, Ledrapp…

Estimates area and spectrum of stable minimal surfaces in Euclidean and hyperbolic spaces.

On a complete noncompact Kähler manifold we prove that the bottom of the spectrum for the Laplacian is bounded from above by $m^2$ if the Ricci curvature is bounded from below by $-2(m+1)$. Then we show that if this upper bound is achieved then the manifold has at most two ends. These results improve previous results o…

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

Upper bounds for essential spectrum of minimal submanifolds linked to volume growth.

In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that $4λ_1 > h^2$, where $λ_1$ is the bottom of the $L^2$-spectrum and $h$ the…

For a Riemannian covering $π\colon M_1\to M_0$, the bottoms of the spectra of $M_0$ and $M_1$ coincide if the covering is amenable. The converse implication does not always hold. Assuming completeness and a lower bound on the Ricci curvature, we obtain a converse under a natural condition on the spectrum of $M_0$.

We prove that the Hilbert geometry of a convex domain in the plane is Gromov hyperbolic, if, and only if, the bottom of its spectrum is not zero

Study on spectral stability of Riemannian coverings.

We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex. A sufficient condition for non-uniqueness is also presented. Furthermore, we …

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

We give a lower bound for the bottom of the $L^2$ differential form spectrum on hyperbolic manifolds, generalizing thus a well-known result due to Sullivan and Corlette in the function case. Our method is based on the study of the resolvent associated with the Hodge-de Rham Laplacian and leads to applications for the (…

Study estimates hypersurface areas in curved spaces, with applications to spectrum bounds.

Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.

We obtain upper estimates for the bottom (that is, greatest lower bound) of the essential spectrum of weighted Laplacian operator of a weighted manifold under assumptions of the volume growth of their geodesic balls and spheres. Furthermore, we find examples where the equality occurs in the estimates obtained. As a con…

We consider noncompact complete manifolds with Spin(9) holonomy and proved an one end result and a splitting type theorem under different conditions on the bottom of the spectrum. We proved that any harmonic functions with finite Dirichlet integral must be Cayley-harmonic, which allowed us to conclude an one end result…

We prove a sharp integral gradient estimate for harmonic functions on noncompact Kähler manifolds. As application, we obtain a sharp estimate for the bottom of spectrum of the p-Laplacian and prove a splitting theorem for manifolds achieving this estimate.

Let X=G/K be a symmetric space of noncompact type and let L be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of L admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane w…

New inequality controls domain volume for manifolds with large spectrum.

We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmueller geodesic flow restricted to the hyperelliptic connected components, settling a question of Farb. We stress that all proofs a…

From a graph $G$ with constant valency $v$ and a (non-compact) manifold $C$ with $v$ boundary components, we build a $G$-periodic manifold $M$. This process gives a class of topologically infinite manifolds which generalizes periodic manifolds and includes all riemannian coverings with finitely generated deck-group. Ou…

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

Global solutions found for certain reaction-diffusion equations on specific manifolds.

The paper examines properties and rigidity of self-expanders in Euclidean space.

The purpose of this paper is to point out that `supremum' in two inequalities of Brooks should be replaced with `infimum'. The results of this paper are already known by Professor Higuci. Hence, I want to delete this paper from this preprint server.

Consider a compact Kähler manifold $M^m$ with Ricci curvature lower bound $Ric_M\geq -2(m+1) .$ Assume that its universal cover $$ has maximal bottom of spectrum $λ_1(\widetilde{M}$ Then we prove that $\widetilde{M}$ is isometric to the complex hyperbolic space $\Bbb{CH}^m.$