Surfaces of Rotation with Constant Extrinsic Curvature in a Conformally Flat 3-Space

Abstract

In this work, 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 E3-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 E3. We obtain a one-parameter family of two-sheeted hyperboloids that are complete surfaces with zero extrinsic curvature in E3. 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 E3. Moreover, we show that there exist complete surfaces with constant negative extrinsic curvature in E3. As an application we prove that there exist complete surfaces with Gaussian curvature K ≤ −ε < 0, in contrast with Efimov's Theorem for the Euclidean space, and Schlenker's Theorem for the hyperbolic space.Mathematics Subject Classification (2010). Primary 53C21; Secondary 53C42.

Helicoidal Minimal Surfaces in a Conformally Flat 3-Space

“…The following result (see Theorem 1 in [4]) establishes some relations between concepts defined in (R 3 , , ) and (R 3 , , g ), respectively.…”

“…We denote this space by (R 3 , , ). In a similar case, Beneki, Kaimakamis and Papantoniou [2], studied helicoidal surfaces with prescribed Gaussian and mean curvature in Minkowski space R In the recent work [4], Corro, Pina and Souza, began to study the surface theory for a three-dimensional complete manifold (R 3 , , g ), where , g is a conformal metric to Euclidean metric , , which means that there exists a positive differentiable function F : R 3 → R such that…”

“…The study of immersed surfaces in spaces that are conformal to the Euclidean space is natural, because they include very important spaces of constant curvature as the sphere S 3 and the hyperbolic space H 3 . The authors in [4] studied surfaces of rotation with constant extrinsic curvature in E 3 = (R 3 , , g1 ) whose the components of the metric , g1 are given by…”

Complete surfaces with zero curvatures in conformally flat spaces

“…The following result (refer to [7,Theorem 1]) establishes some relations between the concepts defined in (R 3 , •, • ) and (R 3 , •, • g ).…”

“…The authors in [7] studied surfaces of rotation with constant extrinsic curvature in (R 3 , •, • g ), where…”