JournalsrmiVol. 18, No. 2pp. 431–442

Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space

  • Luis J. Alías

    Universidad de Murcia, Spain
  • J. Miguel Malacarne

    Universidad Federal de Espírito Santo, Vitoria, Brazil
Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space cover
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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 R3\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 rr-mean curvature, with r2r \geq 2.

Cite this article

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

DOI 10.4171/RMI/325