Rotation hypersurfaces in spaces of constant curvature

Abstract

Rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed. A local characterization of such hypersurfaces, with dimensions greater than two, is given in terms of principal curvatures. Some special cases of rotation hypersurfaces, with constant mean curvature, in hyperbolic space are studied. In particular, it is shown that the well-known conjugation between the belicoid and the catenoid in euclidean three-space extends naturally to hyperbolic three-space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H cubed"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; in the latter case, catenoids are of three different types and the explicit correspondence is given. It is also shown that there exists a family of simply-connected, complete, embedded, nontotally geodesic stable minimal surfaces in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H cubed"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .