GAUSSIAN CURVATURE AND UNICITY PROBLEM OF GAUSS MAPS OF VARIOUS CLASSES OF SURFACES

2019 ◽  
Vol 240 ◽  
pp. 275-297
Author(s):  
PHAM HOANG HA
Keyword(s):  
Gaussian Curvature ◽  
Riemann Surfaces ◽  
Constant Mean Curvature ◽  
Holomorphic Maps ◽  
Flat Surfaces ◽  
Gauss Maps ◽  
Improper Affine Spheres ◽  
Space Constant ◽  
Euclidean Three Space ◽  
Three Space

In this article, we establish a new estimate for the Gaussian curvature of open Riemann surfaces in Euclidean three-space with a specified conformal metric regarding the uniqueness of the holomorphic maps of these surfaces. As its applications, we give new proofs on the unicity problems for the Gauss maps of various classes of surfaces, in particular, minimal surfaces in Euclidean three-space, constant mean curvature one surfaces in the hyperbolic three-space, maximal surfaces in the Lorentz–Minkowski three-space, improper affine spheres in the affine three-space and flat surfaces in the hyperbolic three-space.

scholarly journals Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

2015 ◽  
Vol 67 (6) ◽  
pp. 1411-1434 ◽  
Author(s):  
Yu Kawakami
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Forms ◽  
Flat Surfaces ◽  
Immersed Surfaces ◽  
Gauss Maps ◽  
Improper Affine Spheres ◽  
Euclidean Three Space ◽  
Three Space ◽  
Three Dimensional Space

AbstractWe elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

scholarly journals Quaternionic representation of the moving frame for surfaces in Euclidean three-space and Lax pair

2004 ◽  
Vol 2004 (15) ◽  
pp. 755-762 ◽  
Author(s):  
Paul Bracken
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Lax Pair ◽  
Moving Frame ◽  
Lax Representation ◽  
Euclidean Three Space ◽  
Three Space ◽  
Codazzi Equations

The moving frame and associated Gauss-Codazzi equations for surfaces in three-space are introduced. A quaternionic representation is used to identify the Gauss-Weingarten equation with a particular Lax representation. Several examples are given, such as the case of constant mean curvature.

scholarly journals Delaunay surfaces expressed in terms of a Cartan moving frame

2020 ◽  
Vol 26 (1) ◽  
pp. 153-160
Author(s):  
Paul Bracken
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Elliptic Functions ◽  
Moving Frame ◽  
Jacobi Elliptic Functions ◽  
Surfaces Of Revolution ◽  
Constant Mean Curvature Surfaces ◽  
The Mean ◽  
Euclidean Three Space ◽  
Three Space

AbstractDelaunay surfaces are investigated by using a moving frame approach. These surfaces correspond to surfaces of revolution in the Euclidean three-space. A set of basic one-forms is defined. Moving frame equations can be formulated and studied. Related differential equations which depend on variables relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case in which the mean curvature is constant, these functions can be expressed in terms of Jacobi elliptic functions.

scholarly journals Surfaces of Revolution and Canal Surfaces with Generalized Cheng–Yau 1-Type Gauss Maps

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1728
Author(s):  
Jinhua Qian ◽  
Xueshan Fu ◽  
Xueqian Tian ◽  
Young Ho Kim
Keyword(s):  
Gauss Map ◽  
Profile Curve ◽  
Surfaces Of Revolution ◽  
Canal Surface ◽  
Open Part ◽  
Canal Surfaces ◽  
Gauss Maps ◽  
Unit Speed ◽  
Euclidean Three Space ◽  
Three Space

In the present work, the notion of generalized Cheng–Yau 1-type Gauss map is proposed, which is similar to the idea of generalized 1-type Gauss maps. Based on this concept, the surfaces of revolution and the canal surfaces in the Euclidean three-space are classified. First of all, we show that the Gauss map of any surfaces of revolution with a unit speed profile curve is of generalized Cheng–Yau 1-type. At the same time, an oriented canal surface has a generalized Cheng–Yau 1-type Gauss map if, and only if, it is an open part of a surface of revolution or a torus.

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