Related papers: Convergence and the Length Spectrum

On a closed manifold, consider the space of all Riemannian metrics for which -Delta + kR is positive (nonnegative) definite, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature…

While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume…

We study compact and simply-connected Riemannian manifolds with positive sectional curvature $K\ge 1.$ For a non-trivial homology class of lowest dimension in the space of loops based at a point $p$ or in the free loop space one can define…

We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches…

We consider an elastic manifold of internal dimension $d$ and length $L$ pinned in a $N$ dimensional random potential and confined by an additional parabolic potential of curvature $\mu$. We are interested in the mean spectral density…

The Lagrange and Markov spectra are classical objects in Number Theory related to certain Diophantine approximation problems. Geometrically, they are the spectra of heights of geodesics in the modular surface. These objects were first…

The marked length spectrum (MLS) of a closed negatively curved manifold $(M, g)$ is known to determine the metric $g$ under various circumstances. We show that in these cases, (approximate) values of the MLS on a sufficiently large finite…

The index of a Riemannian symmetric space is the minimal codimension of a proper totally geodesic submanifold (Onishchik, 1980). There is a conjecture by the first two authors for how to calculate the index. In this paper we give an…

Let $M$ and $L$ be the Markov and Lagrange spectra, respectively. It is known that $L$ is contained in $M$ and Freiman showed in 1968 that $M\setminus L\neq \emptyset$. In 2018 the first region of $M\setminus L$ above $\sqrt{12}$ was…

Various results are proved giving lower bounds for the $m$th intrinsic volume $V_m(K)$, $m=1,\dots,n-1$, of a compact convex set $K$ in ${\mathbb{R}}^n$, in terms of the $m$th intrinsic volumes of its projections on the coordinate…

In this paper we prove a compactness theorem for a sequence of harmonic maps which are defined on a converging sequence of Riemannian manifolds.

One of the main purposes of this paper is to prove that on a complete K\"ahler manifold of dimension $m$, if the holomorphic bisectional curvature is bounded from below by -1 and the minimum spectrum $\lambda_1(M) \ge m^2$, then it must…

We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g_1,...,g_k of length at most 2D and relators of the form g_ig_m = g_j . In particular, we obtain an…

Let a compact Lie group act isometrically on a non-collapsing sequence of compact Alexandrov spaces with fixed dimension and uniform lower curvature and upper diameter bounds. If the sequence of actions is equicontinuous and converges in…

We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of compact manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then $Y$ has a universal cover. We then show that, for $i$…

Gauging $B-L$ symmetry provides a simple realization of the seesaw mechanism in a naturally anomaly free extension to the MSSM gauge group, $SU(3)_c\times SU(2)_L\times U(1)_Y \times U(1)_{B-L}$. However, as we discuss in here, it turns out…

We explore the distinctions between $L^p$ convergence of metric tensors on a fixed Riemannian manifold versus Gromov-Hausdorff, uniform, and intrinsic flat convergence of the corresponding sequence of metric spaces. We provide a number of…

We prove that if a family of metrics, $g_i$, on a compact Riemannian manifold, $M^n$, have a uniform lower Ricci curvature bound and converge to $g_\infty$ smoothly away from a singular set, $S$, with Hausdorff measure, $H^{n-1}(S) = 0$,…

We examine the limits of covering spaces and the covering spectra of oriented Riemannian manifolds, $M_j$, which converge to a nonzero integral current space, $M_\infty$, in the intrinsic flat sense. We provide examples demonstrating that…

We consider sequences of compact Riemannian manifolds with uniform Sobolev bounds on their metric tensors, and prove that their distance functions are uniformly bounded in the H\"{o}lder sense. This is done by establishing a general trace…