186 M. Monastyrsky
For arbitrary genus it is extremely difficult and remains an unsolved prob-
lem. There exist some partial results and interesting hypotheses. However for
surfaces of genusg=0andg= 1 we have more complete results.
The following results due to Blaschke and Thomsen [11] are important.
Proposition 9.LetM ̃^2 be a minimal surface inS^3 andγa stereographic
projectionS^3 →R^3 .Then
γ(M ̃^2 )=M^2 andF(M^2 )=γ(M ̃^2 ),whereδ(M ̃^2 )is the area of minimal surface.
Unfortunately, membranes are in general not exhausted by the projection of
minimal surfaces. For example, there exists an infinite set of torical membranes
that are not equivalent to minimal tori inS^3. These examples were constructed
by Pinkal and used Hopf fibrations [12]. We called themPinkal Hopf tori.
10.3.4 Construction of Hopf tori
We identifyS^3 with the set of unit quaternions{a∈H,qq ̄=1}andS^2 with
the unit sphere in the subspace ofH, spanned by 1,j,kbut sendsito−i.
Define
π:S^3 →H byπ(q)= ̃qq.
Thenπhas the following properties:
(a)π(S^3 )=S^2
(b)π(eiφq)=π(q) for allq∈S^3 ,φ∈R
(c) the groupS^3 acts isometrically onS^3 by right multiplication and on
S^2 via
q→rqr, r ̃ ∈S^3
πintertwines these two actions, i.e., for allq, r∈S^3 we have
π(qr)= ̃rπ(q)r.Letρ:[a, b]→S^2 be an immersed curve. Chooseη:[a, b]→S^3 such that
π◦η=ρ. Let us consider an immersionχof the cylinder [a, b]×S^1 intoS^3
by
χ(t, φ)=eiφη(t). (10.16)
Ifρis a closed curve, i.e.,ρ(t+L)=ρ(t), equation (10.16) determined a
torus inS^3. This torus will be called theHopf toruscorresponding toρ.
The main curvatureH ̃ of a Hopf torus coincides with the curvaturekof
the curveρ.
The functionalFis reduced to the integral
π∫L
0(1 +k^2 (s))ds.