Revista Matemática Iberoamericana

Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space

Luis J. Alías and J. Miguel Malacarne

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Abstract

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 $r \geq 2$.

Article information

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

Dates
First available in Project Euclid: 28 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1051544244

Mathematical Reviews number (MathSciNet)
MR1949835

Zentralblatt MATH identifier
1038.53060

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.rmi/1051544244


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