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June, 2002 Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space
Luis J. Alías, J. Miguel Malacarne
Rev. Mat. Iberoamericana 18(2): 431-442 (June, 2002).


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 $r \geq 2$.


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Luis J. Alías. J. Miguel Malacarne. "Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space." Rev. Mat. Iberoamericana 18 (2) 431 - 442, June, 2002.


Published: June, 2002
First available in Project Euclid: 28 April 2003

zbMATH: 1038.53060
MathSciNet: MR1949835

Primary: 53A10 , 53C42

Keywords: constant $r$-mean curvature , constant mean curvature , constant scalar curvature , Newton transformations

Rights: Copyright © 2002 Departamento de Matemáticas, Universidad Autónoma de Madrid


Vol.18 • No. 2 • June, 2002
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