A Weierstrass Representation for Minimal Surfaces in 3-Dimensional Manifolds

2011 ◽  
Vol 60 (1-4) ◽  
pp. 311-323 ◽  
Author(s):  
J. H. Lira ◽  
M. Melo ◽  
F. Mercuri
Keyword(s):  
Minimal Surfaces ◽  
Weierstrass Representation ◽  
3 Dimensional

Description of a 3-periodic minimal surface family with trigonal symmetry

1994 ◽  
Vol 209 (1) ◽  
Author(s):  
A. Fogden
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Weierstrass Representation ◽  
Systematic Analysis ◽  
Trigonal Symmetry ◽  
Triply Periodic Minimal Surfaces ◽  
Periodic Minimal Surface ◽  
Triply Periodic ◽  
The Family

AbstractA systematic analysis of a family of triply periodic minimal surfaces of genus seven and trigonal symmetry is given. The family is found to contain five such surfaces free from self-intersections, three of which are previously unknown. Exact parametrisations of all surfaces are provided using the Weierstrass representation.

scholarly journals C∞-Convergence of Circle Patterns to Minimal Surfaces

2009 ◽  
Vol 194 ◽  
pp. 149-167 ◽  
Author(s):  
Shi-Yi Lan ◽  
Dao-Qing Dai
Keyword(s):  
Minimal Surface ◽  
Existence Theorem ◽  
Minimal Surfaces ◽  
Weierstrass Representation ◽  
Circle Pattern ◽  
Vertex Set ◽  
Circle Patterns ◽  
Simply Connected Region ◽  
Simply Connected ◽  
The Relationship

AbstractGiven a smooth minimal surface F: Ω → ℝ3 defined on a simply connected region Ω in the complex plane ℂ, there is a regular SG circle pattern . By the Weierstrass representation of F and the existence theorem of SG circle patterns, there exists an associated SG circle pattern in ℂ with the combinatoric of . Based on the relationship between the circle pattern and the corresponding discrete minimal surface F∊: → ℝ3 defined on the vertex set of the graph of , we show that there exists a family of discrete minimal surface Γ∊: → ℝ3, which converges in C∞(Ω) to the minimal surface F: Ω → ℝ3 as ∊ → 0.

Stability of minimal surfaces in a 3-dimensional hyperbolic space

1981 ◽  
Vol 36 (1) ◽  
pp. 554-557 ◽  
Author(s):  
J. L. Barbosa ◽  
M. do Carmo
Keyword(s):  
Hyperbolic Space ◽  
Minimal Surfaces ◽  
3 Dimensional

Parametrization of a Family of Minimal Surfaces Bounded by the Broken Lines in

1997 ◽  
Vol 4 (3) ◽  
pp. 201-219
Author(s):  
R. Abdulaev
Keyword(s):  
Minimal Surfaces ◽  
Weierstrass Representation ◽  
Coordinate Plane ◽  
Parameter Domain

Abstract Consideration is given to a family of minimal surfaces bounded by the broken lines in which are locally injectively projected onto the coordinate plane. The considered family is bijectively mapped by means of the Enepper–Weierstrass representation onto a set of circular polygons of a certain type. The parametrization of this set is constructed, and the dimension of the parameter domain is established.

scholarly journals Minimal surfaces in euclidean 3-space and their mean curvature 1 cousins in hyperbolic 3-space

2003 ◽  
Vol 75 (3) ◽  
pp. 271-278
Author(s):  
Shoichi Fujimori
Keyword(s):  
Minimal Surface ◽  
Hyperbolic Space ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
3 Dimensional

We show that the Hopf differentials of a pair of isometric cousin surfaces, a minimal surface in euclidean 3-space and a constant mean curvature (CMC) one surface in the 3-dimensional hyperbolic space, with properly embedded annular ends, extend holomorphically to each end. Using this result, we derive conditions for when the pair must be a plane and a horosphere.

scholarly journals Helicoidal minimal surfaces in hyperbolic space

1989 ◽  
Vol 114 ◽  
pp. 65-75 ◽  
Author(s):  
Jaime B. Ripoll
Keyword(s):  
Hyperbolic Space ◽  
Minimal Surfaces ◽  
3 Dimensional ◽  
Helicoidal Surfaces ◽  
Helicoidal Surface ◽  
Sectional Curvatures ◽  
The One ◽  
Group Of Isometries

Denote by H3 the 3-dimensional hyperbolic space with sectional curvatures equal to – 1, and let g be a geodesic in H3 Let {ψt} be the translation along g (see § 2) and let {φt} be the one-parameter subgroup of isometries of H3 whose orbits are circles centered on g. Given any α ∊ R, one can show that λ = {λt} = ψt ∘ φαt} is a one-parameter subgroup of isometries of H3 (see § 2) which is called a helicoidal group of isometries with angular pitch α. Any surface in H3 which is λ-invariant is called a helicoidal surface.In this work we prove some results concerning minimal helicoidal surfaces in H3. The first one reads:

scholarly journals Minimal surfaces in 3-dimensional solvable Lie groups II

2006 ◽  
Vol 73 (3) ◽  
pp. 365-374 ◽  
Author(s):  
Jun-Ichi Inoguchi
Keyword(s):  
Integral Representation ◽  
Lie Groups ◽  
Minimal Surfaces ◽  
Representation Formula ◽  
Gauss Map ◽  
Invariant Metric ◽  
Solvable Lie Groups ◽  
3 Dimensional ◽  
Integral Representation Formula

An integral representation formula in terms of the normal Gauss map for minimal surfaces in 3-dimensional solvable Lie groups with left invariant metric is obtained.

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