Infinitesimal Rigidity of Cyclic Surfaces and Alternating Surfaces

TL;DR

This paper establishes the infinitesimal rigidity of cyclic surfaces and introduces n-alternating surfaces in hyperbolic spaces, unifying and extending existing rigidity results through a Lie-theoretic framework.

Contribution

It develops a unified Lie-theoretic approach to prove infinitesimal rigidity of cyclic surfaces and introduces n-alternating surfaces in hyperbolic spaces, connecting them to cyclic harmonic bundles.

Findings

Proves infinitesimal rigidity for irreducible cyclic surfaces under smooth variations.

Establishes a correspondence between n-alternating surfaces and cyclic surfaces in hyperbolic spaces.

Unifies and extends known rigidity results for various classes of surfaces.

Abstract

We study the infinitesimal rigidity of equivariant minimal maps from the universal cover of a smooth oriented surface (possibly non-compact) into a Riemannian symmetric space, focusing on representations arising from cyclic harmonic bundles. By developing a unified Lie-theoretic framework that connects cyclic surfaces and cyclic harmonic bundles over Riemann surfaces, we prove the infinitesimal rigidity for irreducible cyclic surfaces under admissible smooth variations, including both compactly supported deformations and $L^p$-integrable variations on non-compact surfaces. As a geometric application, we introduce $n$-alternating surfaces in $\mathbb H^{p,q}$ and establish their correspondence with a special class of cyclic surfaces. This yields an infinitesimal rigidity theorem that conceptually unifies and extends known rigidity results for maximal space-like surfaces, alternating holomorphic curves, and $A$-surfaces in certain $\mathbb H^{p,q}$.