arXiv research

Study volume growth in Milnor fibers using real Lagrangians.

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

We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.

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…

The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{po…

Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.

Efficiently samples arbitrary compact bodies with polynomial complexity.

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…

The paper studies Q-curvature and volume entropy on manifolds, proving polynomial growth polyharmonic function finiteness and rigidity.

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finit…

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

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…

We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the K-theory of the uniform Roe algebra. As an application we will discuss non-vanishing of rough index classes of Dirac operators over such mani…

Efficient algorithm for sampling from arbitrary compact bodies.

In this paper we provide a lower bound for the long time on-diagonal heat kernel of minimal submanifolds in a Cartan-hadamard ambient manifold assuming that the submanifold is of polynomial volume growth. In particular cases, that lower bound is related with the number of ends of the submanifold.

The Volume conjecture claims that the hyperbolic Volume of a knot is determined by the colored Jones polynomial. The purpose of this article is to show a Volume-ish theorem for alternating knots in terms of the Jones polynomial, rather than the colored Jones polynomial: The ratio of the Volume and certain sums of coeff…

Study of harmonic functions on infinite penny graphs.

Uniqueness found for elliptic equations with drift on manifolds.

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:…

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 …

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

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…

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…

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

It is our purpose to study complete self-shrinkers in Euclidean space. First of all, we show some examples of complete self-shrinkers without polynomial volume growth. By making use of the generalized maximum principle for $\mathcal{L}$-operator, we give a complete classification for 2-dimensional complete self-shrinke…

In this paper, we show that there exists a nonconstant CR holomorphic function of polynomial growth in a complete noncompact Sasakian manifold of nonnegative pseudohermitian bisectional curvature with the CR maximal volume growth property. This is the very first step toward the CR analogue of Yau uniformization conject…

The volume conjecture and its generalization state that the series of certain evaluations of the colored Jones polynomials of a knot would grow exponentially and its growth rate would be related to the volume of a three-manifold obtained by Dehn surgery along the knot. In this paper, we show that for the figure-eight k…

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…

Optimizes dimension estimate for holomorphic functions on Kähler manifolds.

It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for $\mathcal{L}$-operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on \emph{polynomial volume growth}, which is…

Characterizes metrics with finite total Q-curvature and introduces new volume entropy.

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…

In this paper, we introduce a definition of $λ$-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that $λ$-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $λ$-hypersurfaces with …

In this paper, we study the generalized volume conjecture for the colored Jones polynomials of links with complements containing more than one hyperbolic piece. First of all, we construct an infinite family of prime links by considering the cabling on the figure eight knot by the Whitehead chains. The complement of the…

Suppose $(M^{n},g)$ is a Riemannian manifold with nonnegative Ricci curvature, and let $h_{d}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $d$. Colding and Minicozzi proved that $h_{d}(M)$ is finite. Later on, there are many researches which give better estimate…

Study on colored Jones polynomial of figure-eight knot for complex parameters.

Quantum approach to volume computation from colored Jones polynomials.

We give a definition of convergence of differential of Lipschitz functions with respect to measured Gromov-Hausdorff topology. As their applications, we give a characterization of harmonic functions with polynomial growth on asymptotic cones of manifolds with nonnegative Ricci curvature and Euclidean volume growth, and…

The study counts ends on shrinkers using geometric covering methods.

We study the uniformization conjecture of Yau by using the Gromov-Haudorff convergence. As a consequence, we confirm Yau's finite generation conjecture. More precisely, on a complete noncompact Kähler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely genera…

We will prove that \emph{there are no stable complete hypersurfaces of $\mathbb{R}^4$ with zero scalar curvature, polynomial volume growth and such that $\dfrac{(-K)}{H^3}\geq c>0$ everywhere, for some constant $c>0$}, where $K$ denotes the Gauss-Kronecker curvature and $H$ denotes the mean curvature of the immersion. …

The volume conjecture and its generalizations say that the colored Jones polynomial corresponding to the N-dimensional irreducible representation of sl(2;C) of a (hyperbolic) knot evaluated at exp(c/N) grows exponentially with respect to N if one fixes a complex number c near 2*Pi*I. On the other hand if the absolute v…

We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete Riemannian manifolds with non-negative Ricci curvature and maximal volume growt…

We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a $\mathbb{Z}$-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley…

In this paper, we firstly verify that if $M$ is a complete self-shrinker with polynomial volume growth in $\mathbb{R}^{n+1}$, and if the squared norm of the second fundamental form of $M$ satisfies $0\leq|A|^2-1\leq\frac{1}{18}$, then $|A|^2\equiv1$ and $M$ is a round sphere or a cylinder. More generally, let $M$ be a …

The study classifies complete self-shrinkers in Euclidean space.

The study examines complete Kähler manifolds with nonnegative Ricci curvature and discovers rigidity properties.

Study on Lane-Emden equation on curved spaces, revealing new existence and non-existence phenomena.