Surfaces of rotation with constant extrinsic curvature in a conformally flat 3-space. (English) Zbl 1251.53037
Summary: We relate the extrinsic curvature of surfaces with respect to the Euclidean metric and any metrics that are conformal to the Euclidean metric. We introduce the space \({\mathbb{E}_3}\) – the 3-dimensional real vector space equipped with a conformally flat metric that is a solution of the Einstein equation. We characterize the surfaces of rotation with constant extrinsic curvature in the space \({\mathbb{E}_3}\). We obtain a one-parameter family of two-sheeted hyperboloids that are complete surfaces with zero extrinsic curvature in \({\mathbb{E}_3}\). Moreover, we obtain a one-parameter family of cones and show that there exists another one-parameter family of complete surfaces of rotation with zero extrinsic curvature in \({\mathbb{E}_3}\). Moreover, we show that there exist complete surfaces with constant negative extrinsic curvature in \({\mathbb{E}_3}\). As an application we prove that there exist complete surfaces with Gaussian curvature \(K \leq - \varepsilon < 0\), in contrast with Efimov’s theorem for the Euclidean space, and Schlenker’s theorem for the hyperbolic space.
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
Keywords:
extrinsic curvature; conformal metrics; Efimov’s embedding theorem; Schlenker’s embedding theoremReferences:
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