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 Surfaces of Finite III-type

2021 ◽  
Vol 15 ◽  
pp. 190-194
Author(s):  
Hassan Al-Zoubi
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Principal Curvature ◽  
Gauss Curvature ◽  
Dimensional Euclidean Space ◽  
Surfaces Of Revolution ◽  
3 Dimensional ◽  
The Third ◽  
Third Fundamental Form ◽  
Special Case

In this paper, we consider surfaces of revolution in the 3-dimensional Euclidean space E3 with nonvanishing Gauss curvature. We introduce the finite Chen type surfaces with respect to the third fundamental form of the surface. We present a special case of this family of surfaces of revolution in E3, namely, surfaces of revolution with R is constant, where R denotes the sum of the radii of the principal curvature of a surface.

A class of generalized special Weingarten surfaces

2019 ◽  
Vol 30 (14) ◽  
pp. 1950075
Author(s):  
Armando M. V. Corro ◽  
Diogo G. Dias ◽  
Carlos M. C. Riveros
Keyword(s):  
Fixed Point ◽  
Holomorphic Functions ◽  
Gauss Map ◽  
Type Representation ◽  
The Third ◽  
Weingarten Surfaces ◽  
Lines Of Curvature ◽  
Third Fundamental Form ◽  
Pass Through ◽  
Weierstrass Type Representation

In [Classes of generalized Weingarten surfaces in the Euclidean 3-space, Adv. Geom. 16(1) (2016) 45–55], the authors study a class of generalized special Weingarten surfaces, where coefficients are functions that depend on the support function and the distance function from a fixed point (in short EDSGW-surfaces), this class of surfaces has the geometric property that all the middle spheres pass through a fixed point. In this paper, we present a Weierstrass type representation for EDSGW-surfaces with prescribed Gauss map which depends on two holomorphic functions. Also, we classify isothermic EDSGW-surfaces with respect to the third fundamental form parametrized by planar lines of curvature. Moreover, we give explicit examples of EDSGW-surfaces and isothermic EDSGW-surfaces.

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).

scholarly journals Purely entropic self-assembly of the bicontinuous Ia 3 d gyroid phase in equilibrium hard-pear systems

2017 ◽  
Vol 7 (4) ◽  
pp. 20160161 ◽  
Author(s):  
Philipp W. A. Schönhöfer ◽  
Laurence J. Ellison ◽  
Matthieu Marechal ◽  
Douglas J. Cleaver ◽  
Gerd E. Schröder-Turk
Keyword(s):  
Phase Diagram ◽  
Minimal Surface ◽  
Lipid Bilayers ◽  
Self Assembly ◽  
Particle Shape ◽  
Voronoi Tessellation ◽  
Gauss Curvature ◽  
Constant Density ◽  
Continuous Change ◽  
Link Type

We investigate a model of hard pear-shaped particles which forms the bicontinuous Ia d structure by entropic self-assembly, extending the previous observations of Barmes et al. (2003 Phys. Rev. E 68 , 021708. ( doi:10.1103/PhysRevE.68.021708 )) and Ellison et al. (2006 Phys. Rev. Lett. 97 , 237801. ( doi:10.1103/PhysRevLett.97.237801 )). We specifically provide the complete phase diagram of this system, with global density and particle shape as the two variable parameters, incorporating the gyroid phase as well as disordered isotropic, smectic and nematic phases. The phase diagram is obtained by two methods, one being a compression–decompression study and the other being a continuous change of the particle shape parameter at constant density. Additionally, we probe the mechanism by which interdigitating sheets of pears in these systems create surfaces with negative Gauss curvature, which is needed to form the gyroid minimal surface. This is achieved by the use of Voronoi tessellation, whereby both the shape and volume of Voronoi cells can be assessed in regard to the local Gauss curvature of the gyroid minimal surface. Through this, we show that the mechanisms prevalent in this entropy-driven system differ from those found in systems which form gyroid structures in nature (lipid bilayers) and from synthesized materials (di-block copolymers) and where the formation of the gyroid is enthalpically driven. We further argue that the gyroid phase formed in these systems is a realization of a modulated splay-bend phase in which the conventional nematic has been predicted to be destabilized at the mesoscale due to molecular-scale coupling of polar and orientational degrees of freedom.

scholarly journals Surfaces of Finite III-type in the Euclidean 3-Space

2021 ◽  
Vol 20 ◽  
pp. 729-735
Author(s):  
Hassan Al-Zoubi ◽  
Farhan Abdel-Fattah ◽  
Mutaz Al-Sabbagh
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Gauss Curvature ◽  
Surfaces Of Revolution ◽  
Laplace Operators ◽  
The Third ◽  
Third Fundamental Form

In this paper, we firstly investigate some relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E3. Then, we introduce the finite Chen type surfaces of revolution with respect to the third fundamental form which Gauss curvature never vanishes.

Sign in / Sign up

Export Citation Format

Share Document