arXiv research

Study bounds growth rate of irreducible meanders, showing proportion vanishes.

For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions…

Constructs manifolds with infinite Betti numbers and close to quadratic volume growth.

Liouville theorems extended to graphs with bounded geometry.

We study ancient solutions of polynomial growth to heat equations on graphs, and extend Colding and Minicozzi's theorem [CM19] on manifolds to graphs: For a graph of polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dime…

Study finds minimum growth rate for surface solutions.

We characterize functions which are growth types of Riemannian manifolds of bounded geometry.

Upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.

We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.

In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if…

Polynomial growth bounds for eigenfunctions on non-compact spaces.

In this paper, We define a $\mathcal{F}$-functional and study $\mathcal{F}$-stability of $λ$-hypersurfaces, which extend a result of Colding-Minicozzi. Lower bound growth and upper bound growth of area for complete and non-compact $λ$-hypersurfaces are studied.

Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any suc…

The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.

Model predicts growth competition on curved surfaces.

Introduced by Gromov in the nineties, the systolic growth of a Lie group gives the smallest possible covolume of a lattice with a given systole. In a simply connected nilpotent Lie group, this function has polynomial growth, but can grow faster than the volume growth. We express this systolic growth function in terms o…

Upper bound on geodesic ball volume in Riemannian manifolds.

Paper establishes a universal growth rate for smooth surrogate losses in classification.

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

For covering spaces and properly discontinuous actions with compatible diffusion operators, we discuss Lyons-Sullivan discretizations of the associated diffusions and harmonic functions of bounded growth.

Grimaldi-Pansu metrics are constructed for manifolds with multiple ends.

We establish new strong lower bounds on the (subnormal) subgroup growth of a large class of groups. This includes the fundamental groups of all finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic groups. The lower bound is nearly exponential, which should be compared with the fastest possible subg…

Study Hilbert's projective metric for bounded growth functions leading to Sinkhorn's algorithm convergence.

This paper, focusing on the growth rate of the measure, gives pointwise bounds of solutions of eigenvalue equations of the Laplace-Beltrami operator on noncompact Riemannian manifolds.

This paper considers the growth in the length of one-dimensional trajectories as they are passed through deep ReLU neural networks, which, among other things, is one measure of the expressivity of deep networks. We generalise existing results, providing an alternative, simpler method for lower bounding expected traject…

Proves optimal volume growth for certain nonnegative Ricci curvature manifolds.

We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manif…

The study proves that certain manifolds can have metrics with specific volume growth.

We provide a lower bound for the uniform exponential growth rate of closed nonflat nonpositively curved 3-manifold groups. A detailed study of the uniform exponential growth rate of closed 3-manifold groups is also presented.

Minimal graphs over simply connected domains grow at most exponentially.

We use spectral embeddings to give upper bounds on the spectral function of the Laplace--Beltrami operator on homogeneous spaces in terms of the volume growth of balls. In the case of compact manifolds, our bounds extend the 1980 lower bound of Peter Li for the smallest positive eigenvalue to all eigenvalues. We also i…

Let p and l be two distinct prime numbers and let G be a group. We study the asymptotic behaviour of the mod-l Betti numbers in p-adic analytic towers of finite index subgroups. If X is a finite l-group of automorphisms of G, our main theorem allows to lift lower bounds for the mod-l cohomology growth in the fixed poin…

Paper proves volume growth estimate for steady gradient Ricci solitons.

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

We make some improvements to our previous results. First, we prove a version of our volume growth theorem which does not require any assumption on the first Betti number. Second, we show that our local regularity theorem only requires a lower volume growth assumption, not a full Sobolev constant bound. These results al…

The study provides volume growth estimates for specific types of manifolds.

Recently, Sogge-Zelditch and Colding-Minicozzi gave new power law lower bounds on the size of the nodal sets of eigenfunctions. The purpose of this short note is to point out a third method to obtain a power law lower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growth bound for …

The paper explores curvature-free effects in manifolds with volume growth and ends-counting.

We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is "large".

Directly proves logarithmic systolic growth for all hyperbolic surfaces.

In this paper we study the growth rates of Artin monoids and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given.

We give an upper bound for the growth of homology torsions of finite coverings of irreducible 3-manifolds with tori boundary in terms of hyperbolic volume.

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.

Study of harmonic functions on infinite penny graphs.

Study saddle connections on hyperelliptic surfaces, finding growth rates.

New proof linking scalar curvature to volume growth on 3-manifolds.

We give lower bounds for the growth of the number of Reeb chords and for the volume growth of Reeb flows on spherizations over closed manifolds M that are not of finite type, have virtually polycyclic fundamental group, and satisfy a mild assumption on the homology of the based loop space. For the special case of geode…

Study growth of systoles in arithmetic manifolds, focusing on $k$-dimensional cases.