Analytic regularity of CR maps into spheres

Let $M$ be a connected real-analytic hypersurface in \C^N and $\S$ the unit real sphere in \C^{N'}, $N'> N\geq 2$. Assume that $M$ does not contain any complex-analytic hypersurface of \C^N and that there exists at least one strongly pseudoconvex point on $M$. We show that any CR map $f\colon M\to \S$ of class $C^{N'-N+1}$ extends holomorphically to a neighborhood of $M$ in \C^N.Comment: 11 page

Superloop Space

In this paper will construct and analyse the superloop space formulation of a $\mathcal{N} =1$ supergauge theory in three dimensions. We will obtain expressions for the connection and the curvature in this superloop space in terms of ordinary supergauge fields. This curvature will vanish, unless there is a monopole in the spacetime. We will also construct a quantity which will give the monopole charge in this formalism. Finally, we will show how these results even hold for a deformed superspace.Comment: 10 pages, 0 figures, accepted for publication in EP

Formal biholomorphic maps of real analytic hypersurfaces

Let f : (M,p)\to (M',p') be a formal biholomorphic mapping between two germs of real analytic hypersurfaces in \C^n, p'=f(p). Assuming the source manifold to be minimal at p, we prove the convergence of the so-called reflection function associated to f. As a consequence, we derive the convergence of formal biholomorphisms between real analytic minimal holomorphically nondegenerate hypersurfaces. Related results on partial convergence of formal biholomorphisms are also obtained.Comment: 15 pages, Late

Algebraic approximation in CR geometry

We prove the following CR version of Artin's approximation theorem for holomorphic mappings between real-algebraic sets in complex space. Let M\subset \C^N be a real-algebraic CR submanifold whose CR orbits are all of the same dimension. Then for every point $p\in M$, for every real-algebraic subset S'\subset \C^N\times\C^{N'} and every positive integer $\ell$, if f\colon (\C^N,p)\to \C^{N'} is a germ of a holomorphic map such that {\rm Graph}\, f \cap (M\times \C^{N'})\subset S', then there exists a germ of a complex-algebraic map f^\ell \colon (\C^N,p)\to \C^{N'} such that {\rm Graph}\, f^\ell \cap (M\times \C^{N'})\subset S' and that agrees with $f$ at $p$ up to order $\ell$.Comment: To appear in J. Math. Pures App