It is still an open question whether a compact embedded hypersurface in the
Euclidean space with constant mean curvature and spherical boundary is necessarily
a hyperplanar ball or a spherical cap, even in the simplest case of a compact
constant mean curvature surface in $\mathbb{R}^3$ bounded by a circle. In this
paper we prove that this is true for the case of the scalar curvature. Specifically
we prove that the only compact embedded hypersurfaces in the Euclidean space with
constant scalar curvature and spherical boundary are the hyperplanar round balls
(with zero scalar curvature) and the spherical caps (with positive constant scalar
curvature). The same applies in general to the case of embedded hypersurfaces with
constant $r$-mean curvature, with $r \geq 2$.
Publié le : 2002-03-14
Classification:
Constant mean curvature,
constant scalar curvature,
constant $r$-mean curvature,
Newton transformations,
53A10,
53C42
@article{1051544244,
author = {Al\'\i as, Luis J. and Malacarne, J. Miguel},
title = {Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space},
journal = {Rev. Mat. Iberoamericana},
volume = {18},
number = {1},
year = {2002},
pages = { 431-442},
language = {en},
url = {http://dml.mathdoc.fr/item/1051544244}
}
Alías, Luis J.; Malacarne, J. Miguel. Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space. Rev. Mat. Iberoamericana, Tome 18 (2002) no. 1, pp. 431-442. http://gdmltest.u-ga.fr/item/1051544244/