arXiv subjects

John Bolton

Publications and source records attributed to John Bolton.

3 recordsLinked to original sources

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↗

arXiv2006

Lagrangian submanifolds attaining equality in the improved Chen's inequality

Recently Oprea gave an improved version of Chen's inequality for Lagrangian submanifolds of $\mathbb CP^n(4)$. For minimal submanifolds this inequality coincides with the original previously proved version. We consider here those non minimal 3-dimensional Lagrangian submanifolds in $\mathbb CP^3 (4)$ attaining at all points equality in the improved Chen inequality. We show how all such submanifolds may be obtained starting from a minimal Lagrangian surface in $\mathbb CP^2(4)$ and a suitable horizontal curve in $S^3(1)$.

math.DG↗