The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1320–1355 | Cite as

On the Moduli Space of Isometric Surfaces with the Same Mean Curvature in 4-Dimensional Space Forms

  • Kleanthis Polymerakis
  • Theodoros VlachosEmail author


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 non-trivial 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 \(\mathbb {R}^4\) we prove that the moduli space is a finite set, under a condition on the Euler numbers of the tangent and normal bundles.


Mean curvature Bonnet problem Gauss map Gauss lift Holomorphic differential Associated family Superconformal surfaces 

Mathematics Subject Classification

53C42 53A10 


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© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of IoanninaIoanninaGreece

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