View
7
Download
0
Category
Preview:
Citation preview
Presented at Kyungpook National University
Drawing a Maximal Surface in L3
Young Wook Kim + Seong-Deog Yang
Department of Mathematics
Korea University
Nov. 12, 2004
1
Introduction
We introduce a known technic of drawing minimal
surfaces using Mathematica and show how we use
this to draw a family of maximal surfaces in L3.
Using graphics technics we may visualize candi-
dates of nice maximal surfaces and find symetry
properties of them which enables us to find a closed
maximal surface.
2
Background
• (R. Schoen, 1982) Catenoids in E3 are the only
complete minimal surfaces of finite total curvature
with 2 embedded ends.
• (Kobayashi, 1983) Construction of catenoid in
L3 using Weierstrass-type representation formula.
3
• (Hoffman-Meeks, 1990) Construction of gener-
alized Costa surface with many handles.
• (Weber, 90’s) Drawing of Costa surface (genus
1) in E3. (cf. Rossman)
Question: Is there a maximal surface in L3 similar
to the Costa-Hoffman-Meeks surfaces?
4
Weierstrass Representation Formulaein E3 and in L3
Let M be a Riemann surface, and f, g : M → C be analytic
functions. Then, the following map
Re
{∫ z
z0
((1− g(w)2)f(w), i (1 + g(w)2)f(w),2g(w)f(w)
)dw
}
is a minimal immersion into R3, and the following map
Re
{∫ z
z0
((1 + g(w)2)f(w), i (1− g(w)2)f(w),2 g(w)f(w)
)dw
}
is a space-like maximal immersion into L3. (The last coordiante
is the time component.)
5
Examples of minimal surfaces in E3
Surfaces Riemann Surface M f dz g
catenoid S2 r {0,∞}1
z2dz z
helicoid S2 r {0,∞}i
z2dz z
Enneper’s S C dz z
Trinoid S2 r {1, e2πi/3, e4πi/3}1
(z3 − 1)2dz z2
Costa’s S{w2 = z(z2 − 1)} r {3 pts.}⊂ S2 × S2
w
z2 − 1dz
c
w
6
Examples of maximal surfaces in L3:
Surfaces Riemann Surface M f dz g
catenoid S2 r {0,∞}1
z2dz z
helicoid S2 r {0,∞}i
z2dz z
Enneper’s surface C dz z
Trinoid S2 r {1, e2πi/3, e4πi/3}1
(z3 − 1)2dz z2
Costa’s surface ? ? ?
7
How to draw.
• Predraw and analyze the surface
• Parametrize the fundamental domain
• Do the integration
• Extract the data and Plot - Mathematica
• Analyze the results8
Domain of Definition
Closed Riemann surface of genus k (k a positive integer.):
Mk = {(α, β) : βk+1 = αk(α + 1)(α− 1)} ⊂ S2 × S2
(k = 1,2,3, · · · )
We parametrize the surface with the parameter α and over k+1
copies of C except the points α = 0 which together with α = ∞represent the ends.
9
The W-data(1)
Data of Costa-Hoffman-Meeks minimal surfaces in E3:
wk+1 = zk(z2 − 1), η =(
z
w
)kdz, g =
ρ
w.
Our data for maximal surfaces in L3:
wk+1 = zk(z2 − 1), η =1
z
(z
w
)kdz, g = ρ
z
w
(Ends at (z, w) = (±1,0))10
The W-data(2)
Use conformal transform
z =1− α
1 + α, w = k+1√4
α(1− α)
β(1 + α).
to get
βk+1 = αk(α2 − 1), g =ρ
k+1√4
β
α, η =
k+1√4
2
dα
β.
11
The W-data(3)
η =dα
β,
g = σβ
α,
Mk = Mk \ {(0,0), (∞,∞)},
(α0, β0) = (1,0) (base point of integration)
12
Integration
The real part of the following integration defines the maximal
surfaces.
∫ (α,β)
(α0,β0)
((1 + g2)η, i(1− g2)η,2gη
)
The image of (α0, β0) = (1,0) is the origin (0,0,0). Denote
φ = (φ1, φ2, φ0) =((1 + g2)η, i(1− g2)η,2gη
)13
Periods(1)
On the Riemann surface there is one homology cycle γ which
poses period problem.
14
Periods(2)
The t-component of the period around γ is 0.
Re∫γ2gη = 0
The period problem around γ for the xy-components are 0 iff
σ =
√1
2
A
B
where
A =∫ 1
0
dt
k+1√
tk(1− t2), B =
∫ 1
0
k+1√
tk(1− t2)dt
1− t2.
15
Automorphisms of Mk
The following symmetries are conformal automorphisms of the
Riemann surface:
κ(α, β) := (α, β),
λ(α, β) := (−α, ckβ),
µ(α, β) := (α−1, cα−2β)
where c = eπ
k+1i. They satisfy
κ2 = λ2(k+1) = µ2(k+1) = id
16
Symetries of the Maximal Surfaces
κ, λ, µ induce isometries K, L, M of the maximal surface, which
generate a group of order 8(k + 1):
K =
1 0 00 −1 00 0 1
,
L =
− cos kπ
k+1 sin kπk+1 0
− sin kπk+1 − cos kπ
k+1 0
0 0 1
,
M =
− cos π
k+1 sin πk+1 0
− sin πk+1 − cos π
k+1 0
0 0 −1
.
17
Fundamental Domain
18
Analysis – Time levels
|α1| = |α2| ⇐⇒ t(α1) = t(α2)
The concentric quater circles in the fundamental do-main lies in the same time levels.
19
Analysis – Singularities
The metric is
ds2 = (1− |g|2)2|η|2
and the singularities occur at the points where |g| = 1.
In polar coordinate α = r eiθ, they are on the curve
r2 + r−2 = σ−2(k+1) + 2cos2θ.
20
Analysis – Topology
How does topology show up in the surface?
21
References
Schoen, R., Uniqueness, symmetry, and embeddedness of mini-
mal surfaces, J. Diff. Geom. 18 (1983), 791–809.
Kobayashi, O., Maximal surfaces in the 3-dimensional Minkowski
space L3, Tokyo J. Math., 6 (1983), 297–309.
Hoffman, D. and W. H. Meeks, III, Embedded minimal surfaces
of finite topology, Ann. of Math., 131 (1990), 1–34.
Weber, M., Costa’s Minimal Surface (http://php.indiana.edu/
∼matweber/)
Rossman, W., Personal communications.
22
Recommended