Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.05023

Rigidity on compact surfaces through hyperbolic symmetries

Sean Dewar·Alison La Porta·Rebecca Monks·Anthony Nixon·Klara Stokes·Joannes Vermant
View original source ↗Read source document ↗Export metadata ↓

Abstract

Generically the rigidity of bar-joint structures admits combinatorial characterisations in the Euclidean plane and, more generally, for frameworks on the sphere and the torus. The remaining case of compact surfaces of genus at least two has remained open. Using the hyperbolic geometry of their universal covers, we develop a theory of infinitesimal rigidity for frameworks on compact surfaces of genus at least two. By the uniformisation theorem, every such surface is a quotient of the hyperbolic plane by a surface group, allowing frameworks on the surface to be represented as infinite symmetric frameworks in the hyperbolic plane. Encoding the symmetry through gain graphs, we prove that infinitesimal rigidity is determined entirely by finite combinatorial data. Specifically, a framework is generically rigid if and only if its associated gain graph contains a spanning (2,3,1,0)-gain tight subgraph. This yields the first combinatorial characterisation of generic rigidity for frameworks on compact surfaces of genus at least two.

Explore related subjects

math.DGmath.COmath.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sean Dewar, Alison La Porta, Rebecca Monks, Anthony Nixon, Klara Stokes, Joannes Vermant. 2026-07-10. Rigidity on compact surfaces through hyperbolic symmetries. https://arxiv.org/abs/2607.05023

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Ancient mean curvature flow asymptotic to a minimal quadratic cone

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an $O(n)\times O(m)$ symmetric minimal quadratic cone for $n +m \geq 10$, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a $\textbf{singular minimal cone as the asymptotic model}$.

math.DG↗
arXiv2026

Log-Concavity of First Dirichlet Eigenfunctions on $\mathbb{CP}^2$

We study the log-concavity property of first Dirichlet eigenfunctions on domains in $\mathbb{CP}^2$. For every smooth $1$-convex domain $Ω\subset\mathbb{CP}^2$, we prove the quantitative estimate \[ \nabla^2(-\log u) > \max\left\{ψ(s),\frac85\right\}g, \text{ where } ψ(|\grad f|^2) = \frac{s}{\sqrt{1+s}}-\log(1+s), \] for its first Dirichlet eigenfunction $u$. In particular, $u$ is strictly log-concave. As consequences, we obtain a uniform convexity estimate for the regular level sets of $u$ and the fundamental gap bound $λ_2-λ_1>46/5$.

math.DG↗
arXiv2026

Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications

We prove a Pólya--Szegö inequality for functions defined on an $n$-dimensional submanifold $Σ$ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on $\mathbb R^n$, and the constant depends on the $L^n$-norm of the mean curvature of $Σ$ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo--Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser--Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya--Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.

math.DG↗