Complex submanifolds of almost complex Euclidean spaces

Abstract

We prove that a compact Riemann surface can be realized as a pseudo-holomorphic curve of $(\mathbb{R}^4,J)$, for some almost complex structure $J$ if and only if it is an elliptic curve. Furthermore we show that any (almost) complex $2n$-torus can be holomorphically embedded in $(\mathbb{R}^{4n},J)$ for a suitable almost complex structure $J$. This allows us to embed any compact Riemann surface in some almost complex Euclidean space and to show many explicit examples of almost complex structure in $\mathbb{R}^{2n}$ which can not be tamed by any symplectic form.