arXiv research

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

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…

Study groups with polynomial growth, finding structure and applications.

Liouville theorems extended to graphs with bounded geometry.

Sharp Liouville theorem for minimal graphs on manifolds with nonnegative Ricci curvature.

Study Liouville theorems for harmonic maps along ancient super Ricci flows.

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

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

A rigidity theorem for smooth Legendrian self-shrinkers is proven.

Bubbles are essential in certain economic models with high growth and low interest rates.

Fundamental groups of certain Kähler orbifolds have polynomial growth.

Proves effective linear volume growth for 3-manifolds with positive scalar curvature.

Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear volume growth…

The main results of this paper consists of two parts. Firstly, we obtain an almost rigidity theorem which says that on a RCD(0, N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Seco…

These short lecture notes provide a brief introduction to the field of homology growth. They are composed out of two lectures, which I have given at the Borel seminar 2017 in Les Diablerets. We give a proof of Lück's approximation theorem, discuss generalizations and mention some related open problems. Then we discuss …

A singularity theorem based on asymptotic volume growth

Ancient Lagrangian flows get limited convex solutions.

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…

Classifies gravitational instantons with quadratic volume growth.

We give a new proof of Gromov's theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.

The paper proves unique characterization of gravitational instantons with specific volume growth.

Defines signed quasiregular curves and proves growth theorem.

Paper proves stable minimal surfaces in 3D are flat.

In this paper, we study volume growth, Liouville theorem and the local gradient estimate for $f$-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.

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

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…

We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams asymptotic, (b) Fried's theorem on reconstructing the Alexander polynomial (c) Gordon's th…

In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fund…

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 …

Paper proves volume growth estimate for steady gradient Ricci solitons.

Paper proves flatness of anisotropic minimal graphs in half-spaces.

Optimal Liouville theorem for minimal disks in any codimension.

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 …

In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of Lück, dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invar…

Proves Calabi-Yau theorem for certain nonnegative curvature manifolds.

Minimal submanifolds confined in space are highly restricted.

We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We generalize Colding and Minicozzi's theorem [CM19] on manifolds, and the result [Hua19] on graphs with normalized Laplacians to the setting of graphs with unbounded Laplacians:…

Flat space for manifolds with tiny curvature.

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bi…

Let $M$ be a complete Kähler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and $M$ admits a nonconstant holomorphic function with polynomial growth, we prove $M$ must be of maximal volume growth. This confirms a conjecture of Ni. There are two essential ingredients in the p…

New inequality shows energy growth and decay in geometric problems.

The paper proves nonexistence results for certain parabolic inequalities on Riemannian manifolds.

The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions a…

We prove a Liouville theorem for the plurisubharmonic functions on complete Kaelher manifolds. As the applications, we prove a splitting theorem for complete Kaehler manifolds with nonnegative biscetional curvature in terms of the linear growth harmonic functions and a optomal gap theorem for such manifolds.

The paper extends a Liouville theorem to biharmonic functions on manifolds with nonnegative Ricci curvature.

We introduce a solvable model of randomly growing systems consisting of many independent subunits. Scaling relations and growth rate distributions in the limit of infinite subunits are analysed theoretically. Various types of scaling properties and distributions reported for growth rates of complex systems in a variety…

In this paper we develop the compactness theorem for $λ$-surface in $\mathbb R^3$ with uniform $λ$, genus, and area growth. This theorem can be viewed as a generalization of Colding-Minicozzi's compactness theorem for self-shrinkers in $\mathbb R^3$. As an application of this compactness theorem, we prove a rigidity th…

Study shows a central limit theorem for random coverings of manifolds with nilpotent groups.