## Journal of Mathematics of Kyoto University

### 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 $\mathscr{M}$.

#### Article information

Source
J. Math. Kyoto Univ., Volume 39, Number 4 (1999), 597-606.

Dates
First available in Project Euclid: 17 August 2009

https://projecteuclid.org/euclid.kjm/1250517816

Digital Object Identifier
doi:10.1215/kjm/1250517816

Mathematical Reviews number (MathSciNet)
MR1740193

Zentralblatt MATH identifier
0964.53039

#### Citation

Cho, Chung-Ki; Han, Chong-Kyu. Ellipticity of certain conformal immersions. J. Math. Kyoto Univ. 39 (1999), no. 4, 597--606. doi:10.1215/kjm/1250517816. https://projecteuclid.org/euclid.kjm/1250517816