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.

arXiv comment7 pages. To be appear in Q. J. Math

Cited by in corpus (2)

• On embeddings of almost complex manifolds in almost complex Euclidean spaces
• Embedding almost-complex manifolds in almost-complex Euclidean spaces