arXiv subjects

Bobo Hua

Publications and source records attributed to Bobo Hua.

At least 19 recordsLinked to original sources

arXiv2026

Solvability of Semilinear Elliptic Equations on Infinite Graphs

We develop a constructive method for solving semilinear elliptic equations $\Delta u(x)=f(x,u(x))$ on locally finite, connected infinite graphs with layered structure. Using Eidelheit's theorem, we establish coupling criteria ensuring that arbitrary initial-layer data extend to global solutions for every $f$. We apply combinatorial criteria to prove solvability on leafless infinite trees, integer lattices, the triangular and hexagonal lattices, and a Cayley graph of the discrete Heisenberg group. We further establish solvability for a broad class of Cayley graphs of semidirect products $G\cong\mathbb Z\ltimes_\theta H$. In particular, $\Delta u=e^u$ has infinitely many solutions on $\mathbb Z^2$, but none of finite energy. We also extend the method to the bi-Laplacian under two-step coupling conditions, to the $p$-Laplacian under a unique-neighbor condition, and to magnetic Laplacians.

math.AP↗

arXiv2026

Curvature Diffusion of Inverse-weight Lin--Lu--Yau Ricci Flow on Finite Trees

We study the continuous Lin--Lu--Yau Ricci flow on a finite tree in the inverse-weight case. We investigate the diffusive structure of the curvature evolution equation and prove the convergence of the curvature along the Ricci flow. Moreover, we show that, in logarithmic coordinates, the Ricci flow can be characterized as the gradient flow of a convex potential.

math.CO↗

arXiv2026

Nonlinear parabolic characterizations of stochastic completeness at infinity on weighted graphs

We prove a nonlinear parabolic characterization of stochastic completeness at infinity for weighted graphs. For the filtration equation \[ (\partial_t + \Delta \Phi)u =0 \] where $\Delta$ is the non-negative formal graph Laplacian and $\Phi u =\phi \circ u$ with $\phi \colon \R\to\R$ nonconstant, continuous and increasing, stochastic completeness at infinity is equivalent to uniqueness of bounded pointwise solutions for every bounded initial datum. For $\Phi=\id$, this recovers the classical heat equation characterization of stochastic completeness at infinity, and of stochastic completeness when the killing term is trivial, i.e., when $\kappa=0$. If stochastic completeness at infinity fails, then every bounded initial datum admits infinitely many bounded pointwise solutions of the filtration equation. Admissible nonlinearities include the signed porous medium and fast diffusion powers $\phi(s)=s|s|^{m-1}$ for all $m>0$, as well as many others. Stochastic completeness at infinity is further characterized by a generalized mass balance: the total mass of a positive pointwise solution at time $t$, augmented by the mass $\int_0^t\sum_{x}\kappa(x)\phi(u(s,x)) \dd s$ dissipated by the killing term $\kappa$, equals the initial mass. This balance holds for every bounded positive solution on graphs of finite measure and for bounded finite-mass data on graphs of arbitrary measure under the sharp condition $\limsup_{r\to0^+}\phi(r)/r<\infty$. It also extends to positive pointwise solutions in $\ell^1$ that are bounded on every positive time interval. When the killing term is trivial, stochastic completeness at infinity reduces to stochastic completeness and generalized balance to conservation of mass.

math.PR↗

arXiv2026

The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation

We study the Cauchy problem for the generalized porous medium equation on infinite weighted graphs. For a general nonlinearity, we establish Dirichlet comparison and weak maximum principles on finite subgraphs and, through an exhaustion argument, construct minimal and maximal pointwise solutions for arbitrary $\ell^\infty$ initial data, controlled by explicit, possibly time-dependent, barriers. For the porous nonlinearity $\phi(s)=s|s|^{m-1}$, assuming a $\nu$-Sobolev inequality with $\nu>2$, we derive quantitative energy estimates for $\ell^1$-mild solutions. These yield finite-time extinction in the fast diffusion range $0 2/\nu$. Interestingly, we recover the Euclidean critical exponent for several model graphs. Finally, we prove an exact generalized mass balance for nonnegative $\ell^1$-mild solutions on graphs that are stochastically complete at infinity, allowing for an arbitrary killing term. The same balance is established for suitable classical and bounded pointwise solutions. In the absence of killing, these reduce to conservation of mass.

math.AP↗

arXiv2026

Generic properties of discrete Steklov eigenfunctions

Let $G=(V,E)$ be a finite connected graph with boundary $B$. We prove that for a generic positive edge weight function $w \in \mathbb{R}^{|E|}$, the Steklov eigenvalues of $(G,B,w)$ are simple and every Steklov eigenfunction does not vanish on the boundary. More precisely, the exceptional weights are contained in a zero set of a non-identically zero polynomial and hence form a set of Lebesgue measure zero and Hausdorff dimension at most $|E|-1$. Our results provide a discrete extension of the genericity theorem for the Steklov problem on compact manifolds.

math.CO↗

arXiv2026

The Ollivier Ricci flow with prescribed curvature on infinite graphs

In this paper, we consider the Ricci flow with prescribed curvature on infinite graphs, which reads as \begin{equation*}\label{flow-equation3} \frac{d}{dt}\omega(t)=-(\kappa(t)-\kappa^*)\omega(t),~~ t>0, \end{equation*} where $\omega$ is the edge weight, $\kappa$ and $\kappa^*$ are Lin-Lu-Yau Ricci curvature and the prescribed curvature on the set of edges, respectively. First, we establish the existence and uniqueness of the solution to the Ricci flow. Furthermore, we prove the convergence of the Ricci flow for graphs with girth at least 6 under two different conditions. Our convergence result aligns with the conclusion of Rodin and Sullivan (J Differ Geom, 26(2) 1987) that a circle packing in the plane with the hexagonal pattern is the regular hexagonal packing.

math.DG↗

arXiv2026

Edge Subdivision and the Perron Eigenvalue of Tree Ricci Matrices

The Ricci matrix $R_T$ of a finite tree encodes its discrete Einstein metrics via the Perron eigenvector, with Lin-Lu-Yau's Ollivier Ricci curvature: $\kappa = -\lambda_{\max}(R_T)$. We show that edge subdivision, the natural operation of lengthening a tree, can decrease, preserve, or increase $\lambda_{\max}$. Compressing each branch into a scalar feedback function via the Schur complement reduces the spectral problem to a one-dimensional Chebyshev equation. We obtain an exact one-step trichotomy, a scalar transmission equation for arbitrary length, and the long-chain limit. Examples on double stars, including an asymmetric case where subdivision strictly increases $\lambda_{\max}$, illustrate the theory.

math.DG↗

arXiv2026

Spectral Monotonicity under Leaf Attachment and Limiting Behavior in Discrete Einstein Trees

Let $R_T$ be the Ricci matrix of a finite tree $T$ introduced in \cite{BaiChengHua2026}, the largest eigenvalue $\lambda_{\max}(R_T)$ determines the sign of a discrete Einstein metric curvature on the tree. This paper investigates the asymptotic behavior of the sequence $\lambda_k = \lambda_{\max}(R_{T_k})$ obtained by repeatedly adding pendant edges at a fixed vertex. We prove that $\lambda_k$ converges to a limit $\lambda_\infty$ that depends only on the local branch data of $T$, and establish a first-order asymptotic expansion: \[ \lambda_k = \lambda_\infty + \frac{\alpha}{d+k} + O\!\left(\frac{1}{(d+k)^2}\right), \] where $d$ is the degree of the original vertex, and the coefficient $\alpha$ is given by a spectral projection. As a corollary, when $\alpha \neq 0$, $\lambda_k$ is eventually strictly monotonic (increasing or decreasing). This theory reveals the fine influence of local leaf addition on the global spectrum.

math.DG↗

arXiv2026

Discrete Einstein metrics on trees

We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. We establish a sharp upper bound for the largest eigenvalue of the associated Ricci matrix in terms of the maximum degree. Turning to structural properties, notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge.

math.DG↗

arXiv2026

Nonlocal Characterizations of Stochastic Completeness on Complete Riemannian Manifolds

In this paper, we first prove that the following generalized conservation principle holds on complete Riemannian manifolds: for every $0 0$, \[ T_t^{(s)}\mathbf 1+\int_0^t T_\tau^{(s)}\mathcal R_s\,d\tau=1 \qquad\text{on }M, \] where $\mathcal R_s$ is the intrinsic killing term measuring the loss of mass of the subordinate semigroup, and the condition $\mathcal R_s\equiv0$ is equivalent to the stochastic completeness of $M$. We then provide several new nonlocal characterizations of stochastic completeness. In particular, we show that stochastic completeness is equivalent to genuinely nonlocal conditions, including the zero-mean identity \[ \int_M (-\Delta)^s\varphi\,dV_g=0 \qquad\forall\,\varphi\in C_c^\infty(M), \] as well as the uniqueness of bounded distributional solutions to the associated fractional elliptic and parabolic equations. We also revisit the equivalent $L^1$-core characterization for the generator of the heat semigroup, which plays an important role in our approach. In addition, we prove $L^p$-contractivity and smoothing properties of the subordinate semigroup, establish both short-time and long-time asymptotic results for the fractional heat kernel, derive the short-time asymptotics of jump probabilities for the associated Markov process, and study the variational characterization and minimality properties of the fractional resolvent. Together, these results provide a unified analytic and probabilistic framework for the fractional Laplacian on complete Riemannian manifolds.

math.AP↗

arXiv2025

Isocapacitary constants associated with $p$-Laplacian on graphs

In this paper, we introduce isocapacitary constants for the $p$-Laplacian on graphs and apply them to derive estimates for the first eigenvalues of the Dirichlet $p$-Laplacian, the Neumann $p$-Laplacian, and the $p$-Steklov problem.

math.AP↗

arXiv2025

On positive solutions of Lane-Emden equations on the integer lattice graphs

In this paper, we investigate the existence and nonexistence of positive solutions to the Lane-Emden equations $$ -\Delta u = Q |u|^{p-2}u $$ on the $d$-dimensional integer lattice graph $\mathbb{Z}^d$, as well as in the half-space and quadrant domains, under the zero Dirichlet boundary condition in the latter two cases. Here, $d \geq 2$, $p > 0$, and $Q$ denotes a Hardy-type positive potential satisfying $Q(x) \sim (1+|x|)^{-\alpha}$ with $\alpha \in [0, +\infty]$. \smallskip We identify the Sobolev super-critical regions of the parameter pair $(\alpha, p)$ for which the existence of positive solutions is established via variational methods. In contrast, within the Serrin sub-critical regions of $(\alpha, p)$, we demonstrate nonexistence by iteratively analyzing the decay behavior at infinity, ultimately leading to a contradiction. Notably, in the full-space and half-space domains, there exists an intermediate regions between the Sobolev critical line and the Serrin critical line where the existence of positive solutions remains an open question. Such an intermediate region does not exist in the quadrant domain.

math.AP↗

arXiv2025

On the Ricci flow on Trees

In this paper, we study the evolution of metrics on finite trees under continuous-time Ricci flows based on the Lin-Lu-Yau version of Ollivier Ricci curvature. We analyze long-time dynamics of edge weights and curvatures, providing precise characterizations of their limiting behaviors. We prove that the Ricci flow converges to metric with zero curvature on edges whose normalized weights converge to positive values only if the tree is a caterpillar tree.

math.DG↗

arXiv2025

A generalized Cheeger inequality and the Steklov Problem on finite graphs

We prove generalized Cheeger inequalities for eigenvalues of Laplacians for reversible Markov chains. Then we apply Hassannezhad and Miclo's convergence result to obtain Jammes Cheeger inequalities for Steklov eigenvalues. In particular, we get a sharp estimate for the first non-trivial Steklov eigenvalue via Escobar Cheeger constant. At the end, we extend Hassannezhad and Miclo's convergence result to non-reversible Markov chains via a different method based on resolvent convergence, answering one of their questions.

math.DG↗

arXiv2025

Spectral gaps for noncompact hyperbolic surfaces with linearly many cusps

We construct complete finite-area noncompact hyperbolic surfaces with linearly many cusps and a uniform spectral gap. More precisely, for every $\theta>0$, we construct a sequence $S_{g,n(g)}\in\mathcal{M}_{g,n(g)}$ such that $\lim\limits_{g\to\infty}\frac{n(g)}{g}=\theta$ and the spectrum of the Laplacian has a uniform gap above zero. The construction is based on explicit expanding $(1,3)$-graphs, viewed as combinatorial skeletons for pants decompositions. We also establish a Steklov-type upper bound showing that expansion cannot persist when the number of boundary vertices is much larger than the genus.

math.DG↗

arXiv2025

On a magneto-spectral invariant on finite graphs

In this paper, we introduce a magneto-spectral invariant for finite graphs. This invariant vanishes on trees and is maximized by complete graphs. We compute this invariant for cycles, complete graphs, wheel graphs, hypercubes, complete bipartite graphs and suspensions of trees and derive various lower and upper bounds. In particular, we provide a sharp upper bound for regular bipartite graphs and derive a direct relation between the class of graphs assuming this upper bound and the class of unit weighing matrices, which are generalizations of complex Hadamard matrices. Moreover, this class of bipartite graphs has non-negative magnetic Bakry-\'Emery curvature and is preserved under both the Cartesian product and a partial tensor product for bipartite graphs. The study of our invariant for certain pairs of cospectral graphs indicates also that this invariant allows us to distinguish between them. Finally, we discuss the behaviour of this invariant under various graph operations and investigate relations to the spectral gap.

math.SP↗

arXiv2025

Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

In his seminal work \cite{Ri96}, Rivin characterized finite ideal polyhedra in three-dimensional hyperbolic space. However, the characterization of infinite ideal polyhedra, as proposed by Rivin, has remained a long-standing open problem. In this paper, we introduce the combinatorial Ricci flow for infinite ideal circle patterns, a discrete analogue of Ricci flow on non-compact Riemannian manifolds, and prove a characterization of such circle patterns under certain combinatorial conditions. Our results provide affirmative solutions to Rivin's problem.

math.GT↗

arXiv2025

Existence theory for elliptic equations of general exponential nonlinearity on finite graphs

We study semilinear elliptic equations on finite graphs with fully general exponential nonlinearities, thereby extending classical equations such as the Kazdan-Warner and Chern-Simons equations. A key contribution of this work is the development of new techniques for deriving a priori estimates in this generalized setting, which reduce the original finite graph to a graph with only two vertices. This reduction enables us to explicitly compute the Brouwer degree and to establish the existence of solutions when the degree is nonzero. Furthermore, using the method of sub- and supersolutions, we also prove the existence of solutions in cases where the Brouwer degree vanishes.

math.AP↗