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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
Article
  • 1.1k Downloads

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 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.

Keywords

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

Mathematics Subject Classification

53C42 53A10 

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of IoanninaIoanninaGreece

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