A representation for pseudoholomorphic surfaces in spheres

TL;DR

This paper introduces a local representation for pseudoholomorphic surfaces in spheres using holomorphic data, extending the classical Weierstrass representation to higher-dimensional spheres.

Contribution

It provides a novel method to represent pseudoholomorphic surfaces in spheres via holomorphic functions, generalizing existing representations.

Findings

Representation formula for pseudoholomorphic surfaces in spheres

Extension of Weierstrass representation to higher dimensions

Holomorphic data characterizes the surfaces

Abstract

We give a local representation for the pseudoholomorphic surfaces in Euclidean spheres in terms of holomorphic data. Similar to the case of the generalized Weierstrass representation of Hoffman and Osserman, we assign such a surface in $\Sf^{2n}$ to a given set of $n$ holomorphic functions defined on a simply-connected domain in $\C$.