Revista Matemática Iberoamericana

Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space

Luis J. Alías and J. Miguel Malacarne

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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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Rev. Mat. Iberoamericana, Volume 18, Number 2 (2002), 431-442.

First available in Project Euclid: 28 April 2003

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Zentralblatt MATH identifier

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Constant mean curvature constant scalar curvature constant $r$-mean curvature Newton transformations


Alías, Luis J.; Malacarne, J. Miguel. Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space. Rev. Mat. Iberoamericana 18 (2002), no. 2, 431--442.

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