Ellipticity of certain conformal immersions

Abstract

We study prolongation of the conformal embedding equations. Let $(\mathscr{M}, g)$ be a $C^{\infty}$ Riemannian manifold of dimension $n \geq 3$ and $(\tilde{\mathscr{M}}, \tilde{g})$ be a $C^{\infty}$ Riemannian manifold of dimension $n + d$, $d <\frac{1}{2}n(n-1)$. Suppose that $f : \mathscr{M} \to \tilde{\mathscr{M}}$ is a conformal immersion with conformal factor $v$. If the conformal 1-nullity off at a point $P \in \mathscr{M}$ does not exceed $n - 2$, we prove that the system of conformal embedding equations admits a prolongation to a system of nonlinear partial differential equations of second order which is elliptic at the solution $(f, v)$. In particular, if $(\mathscr{M}, g)$ and $(\tilde{\mathscr{M}}, \tilde{g})$ are analytic and $f$ and $v$ are of differentiability class $C^{2}$ then $f$ and $v$ are analytic on a neighborhood of $P$ in $\mathscr{M}$.