arXiv research

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…

Study on polynomial growth functions and forms on gradient Ricci solitons.

Study growth rates of harmonic functions on curved surfaces.

The study examines polynomial growth functions on gradient shrinking Ricci solitons.

Classifies polynomial growth solutions to drift-harmonic equations on asymptotically paraboloidal manifolds.

Study polynomial growth harmonic functions on infinite penny graphs.

The paper studies stability and area growth of $λ$-hypersurfaces.

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 harmonic functions on submanifolds and their cones.

Study harmonic function growth on curved spaces, proving inequalities.

Sharp heat kernel and Green's function estimates on Euclidean volume growth manifolds.

In this article we have studied some properties of subharmonic functions in a strongly symmetric Riemannian manifold with a pole. As a generalization of polynomial growth of a function we have introduced the notion of polynomial growth of some degree of a function with respect to a real function and proved that any non…

Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.

In this paper, we consider the formal power series whose n-th coefficient is the number of copies of a given finite graph in the ball of radius n centred at the identity element in the Cayley graph of a finitely generated group and call it the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that the growth…

Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this resul…

Covering spaces with exponential growth have non-constant positive harmonic functions.

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…

Study on Kähler manifolds connects curvature decay with growth of holomorphic functions.

Study ancient solutions on graphs with unbounded Laplacians, generalizing previous results.

Two rigidity theorems for manifolds with nonnegative Ricci curvature and specific volume growth.

We prove a Liouville property for any $f$-harmonic function with polynomial growth on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ when the Bakry-Émery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth.

The filling volume functions of the n-th quaternionic Heisenberg group grow, up to dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n+1, which is strictly faster than the growth rate of the (n+1)-dimensional filling volume function of the E…

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

New algorithms optimize private convex optimization with faster rates for functions with κ-growth.

On a complete Calabi-Yau manifold $M$ with maximal volume growth, a harmonic function with subquadratic polynomial growth is the real part of a holomorphic function. This generalizes a result of Conlon-Hein. We prove this result by proving a Liouville type theorem for harmonic $1$-forms, which follows from a new local …

Let $G$ be a finitely generated group with a finite generating set $S$. For $g\in G$, let $l_S(g)$ be the length of the shortest word over $S$ representing $g$. The growth series of $G$ with respect to $S$ is the series $A(t) = \sum_{n=0}^\infty a_n t^n$, where $a_n$ is the number of elements of $G$ with $l_S(g)=n$. If…

We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of Carnot groups. This class contains groups for which it is known that their Dehn function grows no faster than $n^2\log n$. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functio…

Study groups with polynomial growth, finding structure and applications.

Ancient caloric functions on manifolds with polynomial growth are studied under volume doubling barrier.

The paper splits manifolds using infinity harmonic functions with linear growth.

We introduce a new statistical test of the hypothesis that a balanced panel of firms have the same growth rate distribution or, more generally, that they share the same functional form of growth rate distribution. We applied the test to European Union and US publicly quoted manufacturing firms data, considering functio…

Study of harmonic functions on infinite penny graphs.

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

Growth functions of Coxeter groups and the Poincare series of Kleinian and Fuchsian singularities are $2$-tangle $q$-fractions.

We study minimal graphic functions on complete Riemannian manifolds $\Si$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with …

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

In this paper we study volume growth of gradient steady Ricci solitons. We show that if the potential function satisfies a uniform condition, then the soliton has at most Euclidean volume growth.

We show that every non-decreasing function $f\colon \mathbb N\to \mathbb N$ bounded from above by $a^n$ for some $a\ge 1$ can be realized (up to a natural equivalence) as the conjugacy growth function of a finitely generated group. We also construct a finitely generated group $G$ and a subgroup $H\le G$ of index 2 such…

Discretizes diffusions and harmonic functions on covering spaces.

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

Optimizes dimension estimate for holomorphic functions on Kähler manifolds.

Study shows convergence rate for empirical minimizer of unbounded functions with fast growth.

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

Strict concavity proven for growth indicator function of certain groups.

Holomorphic functions grow polynomially on Kähler-Ricci shrinkers, proving ring finitely generated.

Paper proves existence of nonconstant CR-holomorphic functions in Sasakian manifolds.

Study projection in acylindrically hyperbolic groups, proving sublinear tracking and growth bounds.

The study bounds harmonic functions on manifolds with nonnegative Ricci curvature.