Complete constant mean curvature hypersurfaces in Euclidean space of dimension four or higher

In this paper, we study harmonic evolutes of [Formula: see text]-scrolls, that is, of ruled surfaces in Lorentz–Minkowski space having no Euclidean counterparts. Contrary to Euclidean space where harmonic evolutes of surfaces are surfaces again, harmonic evolutes of [Formula: see text]-scrolls turn out to be curves. In particular, we show that the harmonic evolute of a [Formula: see text]-scroll of constant mean curvature together with its base curve forms a null Bertrand pair. This allows us to characterize [Formula: see text]-scrolls of constant mean curvature and reconstruct them from a given null curve which is their harmonic evolute.

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Isometric Immersions of Constant Mean Curvature and Triviality of the Normal Connection

Nagoya Mathematical Journal ◽

10.1017/s0027763000014719 ◽

1972 ◽

Vol 45 ◽

pp. 139-165 ◽

Cited By ~ 17

Author(s):

Joseph Erbacher

Keyword(s):

Euclidean Space ◽

Scalar Curvature ◽

Mean Curvature ◽

Additional Assumption ◽

Constant Mean Curvature ◽

Constant Scalar Curvature ◽

Normal Connection ◽

Isometric Immersions ◽

The Mean ◽

Negative Sectional Curvature

In a recent paper [2] Nomizu and Smyth have determined the hypersurfaces Mn of non-negative sectional curvature iso-metrically immersed in the Euclidean space Rn+1 or the sphere Sn+1 with constant mean curvature under the additional assumption that the scalar curvature of Mn is constant. This additional assumption is automatically satisfied if Mn is compact. In this paper we extend these results to codimension p isometric immersions. We determine the n-dimensional submanifolds Mn of non-negative sectional curvature isometrically immersed in the Euclidean Space Rn+P or the sphere Sn+P with constant mean curvature under the additional assumptions that Mn has constant scalar curvature and the curvature tensor of the connection in the normal bundle is zero. By constant mean curvature we mean that the mean curvature normal is paral lel with respect to the connection in the normal bundle. The assumption that Mn has constant scalar curvature is automatically satisfied if Mn is compact. The assumption on the normal connection is automatically sa tisfied if p = 2 and the mean curvature normal is not zero.

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The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space

Mathematics ◽

10.3390/math7121211 ◽

2019 ◽

Vol 7 (12) ◽

pp. 1211 ◽

Cited By ~ 2

Author(s):

Rafael López

Keyword(s):

Dirichlet Problem ◽

Euclidean Space ◽

Minkowski Space ◽

Mean Curvature ◽

Elliptic Equations ◽

Constant Mean Curvature ◽

Euclidean Case ◽

Curvature Equation ◽

Mean Curvature Equation ◽

The Mean

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the Lorentz-Minkowski space allows dropping the hypothesis on the mean convexity, which is required in the Euclidean case.

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