相关论文: Spectrum of compact manifolds with high genus

It was shown by Seaman that if a compact, oriented 4-dimensional riemannian manifold (M, g) of positive sectional curvature admits a harmonic 2-form of constant length, its intersection form is definite and such a harmonic form is unique up…

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension at least 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for…

Let (M,g) a compact Riemannian $n$-dimensional manifold with umbilic boundary. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean…

This paper establishes quantitative high-probability bounds on the eigenvalues and eigenfunctions of $\epsilon$-neighborhood graph Laplacians constructed from i.i.d. random variables on $m$-dimensional closed Riemannian manifolds $(M,g)$…

A pair of points in a riemannian manifold $M$ is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in $M$…

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In…

We consider the spectrum of the Laplace operator acting on $\mathcal{L}^p$ over a conformally compact manifold for $1 \leq p \leq \infty$. We prove that for $p \neq 2$ this spectrum always contains an open region of the complex plane. We…

In \cite{GGKM-SSS} we examined the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We showed that…

We show the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a quotient manifold of $\mathbb{S}^{n-1}\times \mathbb{R}$…

We study the length, weak length and complex length spectrum of closed geodesics of a compact flat Riemannian manifold, comparing length-isospectrality with isospectrality of the Laplacian acting on p-forms. Using integral roots of the…

We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called riedrichs and Neumann extensions. We introduce a new…

We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable "metric" category of spectral triples over commutative…

The mapping class group of a compact oriented surface of genus greater than one with boundary acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group, for every choice of…

To study a noncompact Riemannian manifold, it is often useful to find a compactification. We discuss several common compactifications and survey some recent results.

We consider a closed Riemannian manifold $M$ of negative curvature and dimension at least 3 with marked length spectrum sufficiently close (multiplicatively) to that of a locally symmetric space $N$. Using the methods of Hamenst\"adt, we…

Let (M,g) be a compact Riemannian manifold of dimension n. For k \in {0,...,n}, we denote Gr_{k}(M) the set of compact, connected and oriented submanifolds of M of dimension k. This set is called the non-linear Grassmannian. In this…

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$.…

An $(a,b)$-biregular bipartite graph is a bipartite graph with bipartition $(X, Y)$ such that each vertex in $X$ has degree $a$ and each vertex in $Y$ has degree $b$. By the bipartite expander mixing lemma, biregular bipartite graphs have…

We show the existence of a natural Dirichlet-to-Neumann map on Riemannian manifolds with boundary and bounded geometry, such that the bottom of the Dirichlet spectrum is positive. This map regarded as a densely defined operator in the…

We consider uniformly semi-locally 1-connected sequences of closed connected Riemannian 2-manifolds. In particular, we assume that the manifolds are homeomorphic to each other and that their total absolute curvature is uniformly bounded.…