scholarly journals Classifications of Canal Surfaces with the Gauss Maps in Minkowski 3-Space

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1453
Author(s):  
Jinhua Qian ◽  
Xueqian Tian ◽  
Xueshan Fu ◽  
Young Ho Kim
Keyword(s):  
Minimal Surface ◽  
Gauss Map ◽  
Canal Surface ◽  
Surface Of Revolution ◽  
Canal Surfaces ◽  
Gauss Maps

In this work, we study the canal surfaces foliated by pseudo spheres S12 along a Frenet curve in terms of their Gauss maps in Minkowski 3-space. Such kind of surfaces with pointwise 1-type Gauss maps are classified completely. For example, the canal surface with proper pointwise 1-type Gauss map of the first kind if and only if it is a part of a minimal surface of revolution.

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.

Blending Cylinders and Cones using Canal Surfaces

2000 ◽  
Vol 5 ◽  
pp. 77-89 ◽  
Author(s):  
M. Kazakevičiūtė ◽  
R. Krasauskas
Keyword(s):  
Canal Surface ◽  
Variable Radius ◽  
Rolling Ball ◽  
Canal Surfaces ◽  
Rational Parameterization

There is reviewed the construction of a rational blending surface between cylinders and cones in some interlocation cases. This surface is constructed as a patch of rolling ball envelope, i.e. as a patch of tangent canal surface of rational-variable radius. This construction defines rational parameterization of a blending surface. The constructed surface is Laguerre invariant.

scholarly journals Local minima of the Gauss curvature of a minimal surface

1991 ◽  
Vol 44 (3) ◽  
pp. 397-404
Author(s):  
Shinji Yamashita
Keyword(s):  
Minimal Surface ◽  
Gauss Curvature ◽  
Gauss Map ◽  
Local Minima ◽  
Jordan Curves ◽  
The Third ◽  
Isolated Points ◽  
Set Of Points

Let D be a domain in the complex ω-plane and let x: D → R3 be a regular minimal surface. Let M(K) be the set of points ω0 ∈ D where the Gauss curvature K attains local minima: K(ω0) ≤ K(ω) for |ω – ω0| < δ(ω0), δ(ω0) < 0. The components of M(K) are of three types: isolated points; simple analytic arcs terminating nowhere in D; analytic Jordan curves in D. Components of the third type are related to the Gauss map.

scholarly journals Modified defect relations for the gauss map of minimal surfaces, III

1991 ◽  
Vol 124 ◽  
pp. 13-40 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Minimal Surface ◽  
Complex Projective Space ◽  
Meromorphic Functions ◽  
Total Curvature ◽  
Gauss Map ◽  
Holomorphic Maps ◽  
Complete Minimal Surface ◽  
Defect Relation ◽  
Definition Of ◽  
Complex Projective

In [5], the author proved that the Gauss map of a nonflat complete minimal surface immersed in R3 can omit at most four points of the sphere, and in [7] he revealed some relations between this result and the defect relation in Nevanlinna theory on value distribution of meromorphic functions. Afterwards, Mo and Osserman obtained an improvement of these results in their paper [11], which asserts that if the Gauss map of a nonflat complete minimal surface M immersed in R3 takes on five distinct values only a finite number of times, then M has finite total curvature. The author also gave modified defect relations for holomorphic maps of a Riemann surface with a complete conformai metric into the n-dimensional complex projective space Pn(C) and, as its application, he showed that, if the (generalized) Gauss map G of a complete minimal surface M immersed in Rm is nondegenerate, namely, the image G(M) is not contained in any hyperplane in Pm − 1(C), then it can omit at most m(m + 1)/2 hyperplanes in general position ([8]). Here, the number m(m + 1)/2 is best-possible for arbitrary odd numbers and some small even numbers m (see [6]). Recently, Ru showed that the “nondegenerate” assumption of the above result can be dropped ([13]). In this paper, we shall introduce a new definition of modified defect and prove a refined Modified defect relation. As its application, we shall give some improvements of the above-mentioned results in [5], [7], [8], [11] and [13].

scholarly journals On the Gauss map of minimal surfaces with finite total curvature

1991 ◽  
Vol 44 (2) ◽  
pp. 225-232 ◽  
Author(s):  
Min Ru
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
General Position ◽  
Total Curvature ◽  
Gauss Map ◽  
Finite Total Curvature

We prove that if a nonflat complete regular minimal surface immersed in Rn is of finite total curvature, then its Gauss map can omit at most (n – 1)(n + 2)/2 hyperplanes in general position in Pn–1 (ℂ).

The Gauss Map of a Minimal Surface

2021 ◽  
pp. 233-263
Author(s):  
Antonio Alarcón ◽  
Franc Forstnerič ◽  
Francisco J. López
Keyword(s):  
Minimal Surface ◽  
Gauss Map
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