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Bart Dioos

Publications and source records attributed to Bart Dioos.

5 recordsLinked to original sources
arXiv2016

Lagrangian submanifolds in the nearly kaehler $s^3 \times s^3$

In this paper, we investigate Lagrangian submanifolds in the nearly Kaehler $S^3 \times S^3$. We construct a new example which is a at Lagrangian torus. We give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature in the nearly Kaehler $S^3\times S^3$.

math.DG↗
arXiv2014

Sequences of harmonic maps in the 3-sphere

We define two transforms between 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 non-conformal harmonic maps into the 3-sphere, $H$-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kähler manifold $S^3\times S^3$. As a consequence we can construct sequences of $H$-surfaces and almost complex surfaces.

math.DG↗
arXiv2014

On almost complex surfaces in the nearly Kähler $S^3\times S^3$

We study almost complex surfaces in the nearly Kähler $S^3\times S^3$. We show that there is a local correspondence between almost complex surfaces and solutions of the H-surface equation introduced by Wente. We find a global holomorphic differential on every almost complex surface, and show that when this differential vanishes, then the corresponding solution of the H-surface equation gives a constant mean curvature surface in $\mathbb{R}^3$. We use this, together with a theorem of Hopf, to classify all almost complex 2-spheres. In fact there is essentially only one, and it is totally geodesic. More details, as well as the proofs of the various theorems are given in [1]. Finally, we state two theorems, one of which states that locally there are just two almost complex surfaces with parallel second fundamental form.

math.DG↗
arXiv2013

Almost complex surfaces in the nearly Kähler $S^3\times S^3$

In this paper almost complex surfaces of the nearly Kähler $S^3\times S^3$ are studied in a systematic way. We show that on such a surface it is possible to define a global holomorphic differential, which is induced by an almost product structure on the nearly Kähler $S^3\times S^3$. We also find a correspondence between almost complex surfaces in the nearly Kähler $S^3\times S^3$ and solutions of the general $H$-system equation introduced by Wente, thus obtaining a geometric interpretation of solutions of the general $H$-system equation. From this we deduce a correspondence between constant mean curvature surfaces in $\mathbb R^3$ and almost complex surfaces in the nearly Kähler $S^3\times S^3$ with vanishing holomorphic differential. This correspondence allows us to obtain a classification of the totally geodesic almost complex surfaces. Moreover, we will prove that almost complex topological 2-spheres in $S^3\times S^3$ are totally geodesic. Finally, we also show that every almost complex surface with parallel second fundamental form is totally geodesic.

math.DG↗