Weingarten and Linear Weingarten Type Tubular Surfaces in E³

Abstract

In this paper we study a large class of Weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3-space. We show that these surfaces can be parametrized by holo-morphic data like minimal surfaces in the Euclidean 3-space and we use it to study their completeness. We also establish some existence and uniqueness theorems by studing the Plateau problem at infinity: when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family? and, how many embedded solutions are there?

Surfaces at a constant distance from the edge of regression on a surface of revolution in E^3

“…where ( ) = ( ) + 1̇( )− 3 ℎ̇( ) and ( ) = ℎ( ) + 1 ℎ̇( ) + 3̇( ). If we substitute coefficients ℎ , given into equations (4), (5), (6), (7), (8), (9) into equations (12), (13), we can easily obtain the equations (10) and (11). Then, we obtain first and second fundamental forms, respectively as follows:…”

The new study of some characterization of canal surfaces with Weingarten and linear Weingarten types according to Bishop frame

“…Also ϒ is said to be a (x, y)-linear Weingarten surface if for a pair (x, y), x = y of the curvatures K, H, and K II of a surface ϒ satisfies ax + by = c, where a, b, c ∈ R and (a, b, c) = (0, 0, 0) (for more details see [1][2][3][4][5][6][7]). In 1863, Julius Weingarten was able to make a major step forward in the topic when he gave a class of surfaces isometric to a given surface of revolution.…”

“…Especially in recent years, the geometry of the second fundamental form II has become an important issue in terms of investigating intrinsic and extrinsic geometric properties of the surfaces. Very recent results concerning the curvature properties associated with II and other variational aspects can be found in [7][8][9][10].…”

Linear Weingarten rotational surfaces in pseudo-Galilean 3-space

“…For instance, W. Kühnel [14] investigated ruled Weingarten surfaces in E 3 . Following this line of reasoning, F. Dillen and W. Kühnel [6] were studied ruled Weingarten [8], [12], [16], [20] and [22], etc. Recently, when the ambient is (pseudo-) Galilean space, tubular surfaces and ruled surfaces studied in [5], [13] and [18].…”