arXiv subjects

Xinrong Zhao

Publications and source records attributed to Xinrong Zhao.

6 recordsLinked to original sources

arXiv2026

Discrete uniformization of polyhedral surfaces

The main result of the paper shows that each connected polyhedral surface with a hyperbolic background metric, or a Euclidean background metric with uniformly bounded circumdisk radii, is discrete conformal to a complete constant-curvature Riemannian surface equipped with a nonempty closed discrete subset. We also prove a discrete Riemann mapping theorem. The proofs are based on the recent work on the discrete Schwarz lemma, the discrete Liouville theorem for polyhedral surfaces, and a Weyl-type realization theorem for hyperbolic surfaces.

math.GT↗

arXiv2026

Rigidity Theorems for the Weyl Problem of Convex Surfaces in Hyperbolic 3-Space

In this paper, we study the rigidity of noncompact convex sets in hyperbolic 3-space. We prove that any intrinsic isometry between the boundaries of two closed, noncompact convex subsets of hyperbolic 3-space of dimension at least two extends to a global isometry of the ambient space, provided that their ideal boundaries are circle-type closed sets with countably many connected components. Moreover, the same conclusion holds if the ideal boundaries consist of finitely many mutually disjoint disks together with a set of one-dimensional Hausdorff measure zero. This result generalizes a recent rigidity theorem of Luo, Luo, and Rao by allowing the ideal boundaries to contain disk components. As a direct consequence, we establish a uniqueness result concerning the Weyl problem for convex surfaces in hyperbolic 3-space, as proposed by Luo and Wu. In particular, our approach provides an alternative proof of the discrete Schwarz lemma. The proof uses Pogorelov's rigidity theorem for compact convex bodies in $\mathbb{R}^3$, the Pogorelov map, and properties of locally convex surfaces in $\mathbb{R}^3$.

math.GT↗

arXiv2026

Leibenson's equation on graphs

In this paper we study on infinite graphs the Leibenson equation $$ \partial_t u = Δ_p u^q, $$ where $p>1$, $q>0$ and $Δ_p$ denotes the discrete $p$-Laplacian. We prove, for any integrable initial data $u_0$, the existence of a global solution, which is unique for a certain range of $p$ and $q$. Assuming a Faber--Krahn inequality, we obtain sharp $\ell^1$-$\ell^\infty$ smoothing estimates and quantitative bounds on the propagation of solutions with initially finite support. Under certain assumptions on $p$ and $q$, we also prove finite-time extinction results for solutions when the graph satisfies an \textit{isoperimetric inequality}. In particular, on Cayley graphs with polynomial volume growth, we establish the optimal large-time decay rate of the $\ell^\infty$-norm for nonnegative finite-mass solutions when $q(p-1)>1$, and demonstrate a sharp dichotomy regarding the finite-time extinction of exhaustion solutions.

math.AP↗

arXiv2026

Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact $3$-Manifolds with Boundary

In this paper, we study a combinatorial Ricci flow on closed pseudo $3$-manifolds $(M,\mathcal{T})$. We prove that if every edge in the triangulation $\mathcal{T}$ has valence at least $9$, then the combinatorial Ricci flow converges exponentially fast to a hyperbolic metric. As a consequence, for any compact $3$-manifold $N$ with boundary admitting an ideal triangulation $\mathcal{T}_N$ whose edges all have valence at least $9$, there exists a unique complete hyperbolic metric with totally geodesic boundary on $N$ such that $\mathcal{T}_N$ is isotopic to a geometric decomposition of $N$. This provides a partial solution to the conjecture of Costantino, Frigerio, Martelli and Petronio, and hence an affirmative answer of Thurston's geometric ideal triangulation conjecture for such manifolds. Moreover, we obtain explicit upper and lower bounds for the resulting hyperbolic metric.

math.GT↗

arXiv2025

Ground state solutions of $p$-Laplacian equations with nonnegative potentials on Lattice graphs

In this paper, we study the $p$-Laplacian equation $$ -Δ_p u + V(x)|u|^{p-2}u = f(x,u) $$ on the lattice graph $\mathbb{Z}^N$ with nonnegative potentials, where $Δ_p$ is the discrete $p$-Laplacian and $p\in(1,\infty)$. By employing the Nehari manifold method, we establish the existence of ground state solutions under suitable growth conditions on the nonlinearity $f(x,u)$, provided that the potential $V(x)$ is either periodic or bounded. Moreover, we prove that if $f$ is odd in $u$ and $p\geq2$, then the above equation admits infinitely many geometrically distinct solutions. Finally, we extend these results from $\mathbb{Z}^N$ to the more general setting of Cayley graphs.

math.AP↗

arXiv2025

A prescribed curvature flow on hyperbolic surfaces with infinite topological type

In this paper, we investigate the prescribed total geodesic curvature problem for generalized circle packing metrics in hyperbolic background geometry on surfaces with infinite cellular decompositions. To address this problem, we introduce a prescribed curvature flow-a discrete analogue of the Ricci flow on noncompact surfaces-specifically adapted to the setting of infinite cellular decompositions. We establish the well-posedness of the flow and prove two convergence results under certain conditions. Our approach resolves the prescribed total geodesic curvature problem for a broad class of surfaces with infinite cellular decompositions, yielding, in certain cases, smooth hyperbolic surfaces of infinite topological type with geodesic boundaries or cusps. Moreover, the proposed flow provides a method for constructing hyperbolic metrics from appropriate initial data.

math.GT↗