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 $r \geq 2$.
References
Alexandrov, A. D.: Uniqueness theorems for surfaces in the large V. Vestnik Leningrad Univ. Math. 13 (1958), 5-8. English translation: AMS Transl. 21 (1962), 412-416.
Mathematical Reviews (MathSciNet):
MR102114
Alías, L. J., López, R., Palmer, B.: Stable constant mean curvature surfaces with circular boundary. Proc. Amer. Math. Soc. 127 (1999), 1195-1200.
Alías, L. J., Herbert S. de Lira, J., Malacarne, J. M.: Constant higher order mean curvature hypersurfaces in Riemannian spaces. Preprint, 2002.
Alías, L. J., Palmer, B.: On the area of constant mean curvature discs and annuli with circular boundaries. Math. Z. 237 (2001), 585-599.
Barbosa, J. L.: Constant mean curvature surfaces bounded by a planar curve. Mat. Contemp. 1 (1991), 3-15.
Barbosa, J. L.: Hypersurfaces of constant mean curvature in $\mathbbR^n+1$ bounded by an Euclidean sphere. Geometry and topology of submanifolds, II, 1-9. World Sci. Publishing, 1990.
Barbosa, J. L., Colares, A. G.: Stability of hypersurfaces with constant $r$-mean curvature. Ann. Global Anal. Geom. 15 (1997), 277-297.
Barbosa, J. L., Jorge, L. P.: Stable $H$-surfaces spanning $\mathbbS^1(1)$. An. Acad. Brasil. Cienc. 61 (1994), 259-263.
Brito, F., Sá Earp, R., Meeks, W., Rosenberg, H.: Structure theorems for constant mean curvature surfaces bounded by a planar curve. Indiana Univ. Math. J. 40 (1991), 333-343.
Kapouleas, N.: Compact constant mean curvature surfaces in Euclidean three-space. J. Differential Geom. 33 (1991), 683-715.
Koiso, M.: Symmetry of hypersurfaces of constant mean curvature with symmetric boundary. Math. Z. 191 (1986), 567-574.
Mathematical Reviews (MathSciNet):
MR832814
Malacarne, J. M.: Fórmulas do fluxo, simetrias e $r$-curvatura média constante. Ph. D. Thesis. Universidade Federal do Ceará, Fortaleza, Brazil, 2001.
O'Neill, B.: Semi-Riemannian geometry with applications to relativity. Academic Press, 1983.
Mathematical Reviews (MathSciNet):
MR719023
Reilly, R. C.: Variational properties of functions of the mean curvature for hypersurfaces in space forms. J. Differential Geom. 8 (1973), 465-477.
Mathematical Reviews (MathSciNet):
MR341351
Ros, A.: Compact hypersurfaces with constant scalar curvature and a congruence theorem. J. Differential Geom. 27 (1988), 215-220.
Mathematical Reviews (MathSciNet):
MR925120
Ros, A.: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoamericana 3 (1987), 447-453.
Mathematical Reviews (MathSciNet):
MR996826
Rosenberg, H.: Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117 (1993), 211-239.
Yau, S. T.: Problem section. Seminar on Differential Geometry. Annals Math. Studies No. 102. Princeton University Press, 1982.
Mathematical Reviews (MathSciNet):
MR645728
