SOME APPLICATIONS OF THE EXTRINSIC DISTANCE TO IMMERSED
AND EMBEDDED SUBMANIFOLDS

Vicente  Miquel

Key words: Isometric immersion, bounded mean curvature,
constant mean curvature, geodesic ball, complex space form,
estimate of first Dirichlet eigenvalue, immersion with bounded
Ricci and scalar curvatures, Hopf hypersurface.

1991 Math. Subject Classification: 53C42, 58G25.

Introduction.
We survey here some recent results about embedded or immersed submanifolds of
complex or real space forms obtained by F.J. Carreras, F. Gim\'enez and the
author
(cfr
\cite{CGM1,2}, \cite {GM2} and \cite{Mi}). The common thread of these results
is in their proofs: A basic tool used in all of them  is a
computation of some laplacians (in the submanifold) of some well chosen
functions of the extrinsic distance to a point of the ambient space.