Searcharxiv⌕ Search

arXiv subjects

Bobo Hua

Publications and source records attributed to Bobo Hua.

At least 91 records · Page 5Linked to original sources
arXiv2018

3-dimensional Combinatorial Yamabe Flow in Hyperbolic Background Geometry

We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. For a triangulation of a 3-manifold, we prove that if the number of tetrahedra incident to each vertex is at least 23, then there exist real or virtual ball packings with vanishing (extended) combinatorial scalar curvature, i.e. the (extended) solid angle at each vertex is equal to 4π. In this case, if such a ball packing is real, then the (extended) combinatorial Yamabe flow converges exponentially fast to that ball packing. Moreover, we prove that there is no real or virtual ball packing with vanishing (extended) combinatorial scaler curvature if the number of tetrahedra incident to each vertex is at most 22.

math.DG↗
arXiv2018

Areas of spherical polyhedral surfaces with regular faces

For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of planar graphs which admit spherical polyhedral surfaces with the curvature bounded below by 1 in the sense of Alexandrov, i.e. the total angle at each vertex is at most $2π$. We classify all spherical tilings with regular spherical polygons, i.e. total angles at vertices are exactly $2π$. We prove that for any graph in this class which does not admit a spherical tiling, the area of the associated spherical polyhedral surface with the curvature bounded below by 1 is at most $4π- ε_0$ for some $ε_0 > 0$. That is, we obtain a definite gap between the area of such a surface and that of the unit sphere.

math.MG↗
arXiv2018

Sobolev-Type Inequalities and Eigenvalue Growth on Graphs with Finite Measure

In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds can not be established.

math.SP↗
arXiv2018

A note on Liouville type equations on graphs

In this note, we study the Liouville equation $Δu = -e^u$ on a graph G satisfying certain isoperimetric inequality. Following the idea of W. Ding, we prove that there exists a uniform lower bound for the energy, $Σ_G e^u$ of any solution $u$, to the equation. In particular, for the 2-dimensional lattice graph $Z^2$; the lower bound is given by 4.

math.AP↗
arXiv2018

Coverings and the heat equation on graphs: stochastic incompleteness, the Feller property and uniform transience

We study regular coverings of graphs and manifolds with a focus on properties of the heat equation. In particular, we look at stochastic incompleteness, the Feller property and uniform transience; and investigate the connection between the validity of these properties on the base space and its covering. For both graphs and manifolds, we prove the equivalence of stochastic incompleteness of the base and that of its cover. Along the way we also give some new conditions for the Feller property to hold on graphs.

math.FA↗
arXiv2017

Ricci curvature on birth-death processes

In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery's $CD(K,n)$ condition for linear graphs and prove the triviality of edge weights for every linear graph supported on the infinite line $\mathbb{Z}$ with non-negative curvature. Moreover, we show that linear graphs with curvature decaying not faster than $-R^2$ are stochastically complete. We deduce a type of Bishop-Gromov comparison theorem for normalized linear graphs. For normalized linear graphs with non-negative curvature, we obtain the volume doubling property and the Poincaré inequality, which yield Gaussian heat kernel estimates and parabolic Harnack inequality by Delmotte's result. As applications, we generalize the volume growth and stochastic completeness properties to weakly spherically symmetric graphs. Furthermore, we give examples of infinite graphs with a positive lower curvature bound.

math.DG↗
arXiv2017

Riesz transforms for bounded Laplacians on graphs

We study several problems related to the $\ell^p$ boundedness of Riesz transforms for graphs endowed with so-called bounded Laplacians. Introducing a proper notion of gradient of functions on edges, we prove for $p\in(1,2]$ an $\ell^p$ estimate for the gradient of the continuous time heat semigroup, an $\ell^p$ interpolation inequality as well as the $\ell^p$ boundedness of the modified Littlewood-Paley-Stein functions for all graphs with bounded Laplacians. This yields an analogue to Dungey's results in [Dungey08] while removing some additional assumptions. Coming back to the classical notion of gradient, we give a counterexample to the interpolation inequality hence to the boundedness of Riesz transforms for bounded Laplacians for $1<p<2$. Finally, we prove the boundedness of the Riesz transform for $1< p<\infty$ under the assumption of positive spectral gap.

math.MG↗
arXiv2016

A Note on Local Unitary Equivalence of Isotropic-like States

We consider the local unitary equivalence of a class of quantum states in bipartite case and multipartite case. The necessary and sufficient condition is presented. As special cases, the local unitary equivalent classes of isotropic state and werner state are provided. Then we study the local unitary similar equivalence of this class of quantum states and analyze the necessary and sufficient condition.

quant-ph↗
arXiv2016

Harmonic functions on metric measure spaces

In this paper, we study harmonic functions on metric measure spaces with Riemannian Ricci curvature bounded from below, which were introduced by Ambrosio-Gigli-Savaré. We prove a Cheng-Yau type local gradient estimate for harmonic functions on these spaces. Furthermore, we derive various optimal dimension estimates for spaces of polynomial growth harmonic functions on metric measure spaces with nonnegative Riemannian Ricci curvature.

math.AP↗
arXiv2015

Davies-Gaffney-Grigor'yan Lemma on Graphs

We prove a variant of the Davies-Gaffney-Grigor'yan Lemma for the continuous time heat kernel on graphs. We use it together with the Li-Yau inequality to obtain strong heat kernel estimates for graphs satisfying the exponential curvature dimension inequality.

math.DG↗