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.

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