On the moduli space of isometric surfaces with the same mean curvature in 4-dimensional space forms

TL;DR

This paper investigates the classification and deformation of isometric surfaces with identical mean curvature in 4D space forms, revealing conditions for finiteness and continuous families within their moduli space.

Contribution

It establishes bounds on the number of congruence classes and characterizes the structure of the moduli space for such surfaces, including special cases with harmonic Gauss lifts.

Findings

At most three nontrivial congruence classes for certain surfaces.

Existence of a holomorphic quadratic differential for vertically harmonic Gauss lifts.

Moduli space is finite or a circle depending on the surface's properties.

Abstract

We study the moduli space of congruence classes of isometric surfaces with the same mean curvature in 4-dimensional space forms. Having the same mean curvature means that there exists a parallel vector bundle isometry between the normal bundles that preserves the mean curvature vector fields. We prove that if both Gauss lifts of a compact surface to the twistor bundle are not vertically harmonic, then there exist at most three nontrivial congruence classes. We show that surfaces with a vertically harmonic Gauss lift possess a holomorphic quadratic differential, yielding thus a Hopf-type theorem. We prove that such surfaces allow locally a one-parameter family of isometric deformations with the same mean curvature. This family is trivial only if the surface is superconformal. For such compact surfaces with non-parallel mean curvature, we prove that the moduli space is the disjoint union of two sets, each one being either finite, or a circle. In particular, for surfaces in $\R^4$ we prove that the moduli space is a finite set, under a condition on the Euler numbers of the tangent and normal bundles.