Assignment 4 (corrected), due 23 May

Summer 2016
Technische Universität Berlin
Institut für Mathematik
Pinkall/Lam
Due: 23 May 2016
http://www3.math.tu-berlin.de/geometrie/Lehre/SS16/DGI
Differentialgeometrie I: Kurven und Flächen
Homework 4
Problem 1
Let c ∈ R and γ : R → R2 be a curve with kγ 0 k = 1 and curvature
κ (s) =
(4 points)
c
.
cosh s
Show that γ is an elastic curve for some suitable choice of c.
Problem 2
(4 points)
The oriented area A(γ) of a regular closed plane curve (γ, τ ) is defined as
Z τ
1
A(γ) := 2
det (γ, γ 0 ) .
0
We denote MRL the space of closed plane curves (γ, τ ) with length L > 0, i.e.
τ
ML = {(γ, τ )| 0 |γ 0 | = L}. Determine the critical points of the area functional
A : ML → R,
γ 7→ A(γ).
Problem 3
Let γ : [a, b] × R → R2 be a flow of plane curves given by
γ̇ =
(4+2+2 points)
κ0
JT + 21 κ2 T
|γ 0 |
where T is the unit vector and κ is the curvature of the curve γt : [a, b] → R2 , γt (x) :=
d
and · = dtd . We also use the sign convecntion that
γ(x, t). Here we write 0 = dx
T 0 = κvN . Show:
(a) γ0 is an arc-length parametrization if and only if γt is an arc-length parametrization.
(b) If γ0 is an arc-length parametrization, then κ satisfies the modified Korteweg-de
Vries equation
κ̇ = κ000 + 23 κ2 κ0 .
(mKdV)
(c) Let γ̂ : [a, b] → R2 be an arc-length parametrization of an elastic curve. Show
that there exists a constant c ∈ R such that the curvature of the curve
x 7→ γ(x, t) := γ̂(x + ct)
solves the mKdV equation.

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