# Sequences of harmonic maps in the 3-sphere

@article{Dioos2014SequencesOH,
title={Sequences of harmonic maps in the 3-sphere},
author={Bart Dioos and J. Veken and L. Vrancken},
journal={Mathematische Nachrichten},
year={2014},
volume={288},
pages={2001-2015}
}
• Published 2014
• Mathematics
• Mathematische Nachrichten
We define two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kahler manifold . As a consequence we can construct sequences of H-surfaces and almost complex surfaces.
10 Citations
Non-conformal harmonic maps into the 3-sphere
We present two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct from one non-conformal harmonic map a sequence of non-conformalExpand
The Bonnet problem for harmonic maps to the three-sphere
• Mathematics
• 2019
Abstract In previous work [15] the authors defined transforms for non-conformal harmonic maps from a Riemann surface into the 3-sphere. An observation from that study was that two invariants, a realExpand
Transforms for Non-conformal Harmonic Surfaces in $$\varvec{R^3}$$R3
For a non-conformal harmonic surface in $$R^3$$R3, we give transforms to get holomorphic maps to the 2-sphere, which is a generalization of the classical fact that the Gauss map of a minimal surfaceExpand
Rigidity of the almost complex surfaces in the nearly Kähler S3×S3
• Mathematics
• 2016
Abstract In this paper we first show that the well-known nearly Kahler manifold S 3 × S 3 is neither locally symmetric nor Chern flat. Then, by studying the rigidity of compact almost complexExpand
A representation formula for non-conformal harmonic surfaces in $R^3$
• Physics, Mathematics
• 2017
We discuss non-conformal harmonic surfaces in $R^3$ with prescribed ($\pm$)transforms, and we get a representation formula for non-conformal harmonic surfaces in $R^3$.
Lagrangian submanifolds in the 6-dimensional nearly Kähler manifolds with parallel second fundamental form
• Mathematics
• 2016
Abstract In this paper, we study the Lagrangian submanifolds in the homogeneous nearly Kahler S 3 × S 3 with parallel second fundamental form. We first prove that every Lagrangian submanifold withExpand
A Representation Formula for Non-conformal Harmonic Surfaces in $$\varvec{R}^{\mathbf{3}}$$R3
• Mathematics
• 2019
We discuss non-conformal harmonic surfaces in $$R^3$$R3 with prescribed (±)transforms, and we get a representation formula for non-conformal harmonic surfaces in $$R^3$$R3.
On some hypersurfaces of the homogeneous nearly Kähler S3×S3
• Mathematics
• 2018
In this paper, we study hypersurfaces of the homogeneous nearly Kahler manifold S3×S3 with typical properties. We first show that in the NK S3×S3 there exist neither totally umbilical hypersurfacesExpand
Transforms for minimal surfaces in 5-dimensional space forms
For a minimal surface in a 5-dimensional space form, we give transforms to get another minimal surface in another 5-or 4-dimensional space form.
Isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S3 × S3
• Mathematics
• 2016
We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Kähler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of theExpand

#### References

SHOWING 1-10 OF 19 REFERENCES
Transforms for minimal surfaces in the 5-sphere
• Mathematics
• 2005
We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of suchExpand
Rigidity of the almost complex surfaces in the nearly Kähler S3×S3
• Mathematics
• 2016
Abstract In this paper we first show that the well-known nearly Kahler manifold S 3 × S 3 is neither locally symmetric nor Chern flat. Then, by studying the rigidity of compact almost complexExpand
Minimal surfaces and the affine Toda field model.
• Mathematics
• 1995
For example, [15], harmonic maps of S into CP may be characterised äs being elements of Frenet frames of holomorphic curves in CP or, equivalently, in twistorial terms äs projections of suitableExpand
Conformal Geometry of Surfaces in S4 and Quaternions
• Mathematics
• 2002
Quaternions.- Linear algebra over the quaternions.- Projective spaces.- Vector bundles.- The mean curvature sphere.- Willmore Surfaces.- Metric and affine conformal geometry.- Twistor projections.-Expand
Sequences of minimal surfaces in S2n+1
• Mathematics
• 2010
For a minimal surface immersed into an odd-dimensional unit sphere S2n+1 with the first (n−2) higher-order ellipses of curvature being a circle, we construct a sequence of such surfaces andExpand
On harmonic maps
This work highlights the key questions of existence, uniqueness and regularity of harmonic maps between given manifolds, and surveys some of the main methods of global analysis for answering these questions. Expand
Harmonic Maps, Conservation Laws, And Moving Frames
Preface Introduction Acknowledgements Notations 1. Geometric and analytic setting 2. Harmonic maps with symmetries 3. Compensations and exotic function spaces 4. Harmonic maps without symmetries 5.Expand
Almost complex surfaces in the nearly K\"ahler $S^3\times S^3$
• Physics, Mathematics
• 2012
In this paper almost complex surfaces of the nearly K\"ahler $S^3\times S^3$ are studied in a systematic way. We show that on such a surface it is possible to define a global holomorphicExpand
Handbook of Global Analysis
• Mathematics
• 2011
This is a comprehensive exposition of topics covered by the American Mathematical Society's classification 'Global Analysis', dealing with modern developments in calculus expressed using abstractExpand