We provide a method to obtain linear Weingarten surfaces from a given such surface, by imposing a one parameter algebraic condition on a Ribaucour transformation. Our main result extends classical results for surfaces of constant Gaussian or mean curvature. By applying the theory to the cylinder, we obtain a two-parameter family of complete linear Weingarten surfaces (hyperbolic, elliptic and tubular), asymptotically close to the cylinder, which have constant mean curvature when one of the parameters vanishes. The family contains n-bubble Weingarten surfaces which are 1-periodic, have genus zero and two ends of geometric index m, where n/m is an irreducible rational number. Their total curvature vanishes, while the total absolute curvature is 8πn. We also apply the method to obtain families of complete constant mean curvature surfaces, associated to the Delaunay surfaces, which are 1-periodic for special values of the parameter.Introduction.
We construct examples of flat surfaces in H 3 which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in H 3 with only one end and at most two isolated singularities.
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.
We present surfaces with prescribed normal Gauss map. These surfaces are obtained as the envelope of a sphere congruence where the other envelope is contained in a plane. We introduce classes of surfaces that generalize linear Weingarten surfaces, where the coe cients are functions that depend on the support function and the distance function from a xed point (in short, DSGW-surfaces). The linear Weingarten surfaces, Appell's surfaces and Tzitzeica's surfaces are all DSGW-surfaces. From them we obtain new classes of DSGWsurfaces applying inversions, dilatations and parallel surfaces. For a special class of DSGW-surfaces, which is invariant under dilatations and inversions, we obtain a Weierstrass type representation (in short, EDSGWsurfaces). As applications we classify the EDSGW-surfaces of rotation and present a 2-parameter family of complete cyclic EDSGW-surfaces with an isolated singularity and foliated by non-parallel planes.
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