Chapter 2
Surfaces: local theory
2.1 The first fundamental form
LetS⊂R3be a surface. Thefirst fundamental formofSis just the restrictionI of the dot product ofR3to the tangent spaces ofS:
Ip=h·,·i|TpS×TpS.
Ifϕ: U ⊂R2 → ϕ(U) =V ⊂ Sis a parametrization, we already know that {ϕu(u, v), ϕv(u, v)} is a basis ofTϕ(u,v)S. Then any tangent vector to S at a point inV can be writtenw=aϕu+bϕvand thus
I(w, w) =a2I(ϕu, ϕu)
| {z }
E(u,v)
+2ab I(ϕu, ϕv)
| {z }
F(u,v)
+b2I(ϕv, ϕv)
| {z }
G(u,v)
.
E,F,Gare smooth functions onU, the so called coefficients of the first fun- damental form. If{du, dv}denotes the dual basis of{ϕu, ϕv}, thena=du(w), b=dv(w)and we can write
I=Edu2+ 2F dudv+Gdv2.
This is the local expression ofIwith respect toϕ. The matrix associated to this bilinear form is
(I) =
E F
F G
.
Examples 2.1 1. A plane can be parametrized byϕ(u, v) = p+uw1 +vw2, wherepis a point inR3,{w1, w2}is an orthonormal set of vectors inR3, and (u, v)∈R2. We haveϕu=w1,ϕv=w2, soE=G= 1,F = 0andI=du2+dv2.
2. The (right circular) cilinder can be parametrized byϕ(u, v) = (cosu,sinu, v), where uo < u < u0 + 2πand v ∈ R. In this case, ϕu = (−sinu,cosu,0), ϕv= (0,0,1), soE=G= 1,F = 0andI=du2+dv2.
11
3. The helicoid is the union of horizontal lines joining the z-axis to the points of an helix, namely, it is the image of the parametrizationϕ(u, v) = (vcosu, vsinu, au)(a >0). We see thatI= (a2+v2)du2+dv2.
4. Spherical coordinatesϕ(u, v) = (sinu,cosv,sinusinv,cosu), where0 <
u < π/2,v0< v < v0+ 2π, yield thatI=du2+ sin2udv2on the sphere.
Remark 2.2 If two surfacesS1,S2can be covered by open sets which are im- ages of parametrizations such that the local expressions of the first fundamen- tal forms ofS1,S2coincide on corresponding open sets (like in the cases of the plane and the cilinder), thenS1andS2are calledlocally isometric.
Using the first fundamental form, we can define:
Lengthof a smooth curveγ: (a, b)→Sby L(γ) =
Z b a
I( ˙γ(t),γ(t))˙ 1/2dt.
Anglebetween two vectorsw1,w2∈TpSby cos∠(w1, w2) = Ip(w1, w2)
Ip(w1, w1)1/2Ip(w2, w2)1/2.
Surface integral of a compactly supported functionf : S → R. If the supportDoff is contained in the image of a parametrizationϕ:U →S, then
Z Z
D
f dS= Z Z
ϕ−1(D)
f(ϕ(u, v))||ϕu×ϕv||dudv,
where the left hand side is a double integral. Taking another parametrization
¯
ϕ : ¯U → S, the change of paramters(u, v) = (ϕ−1◦ϕ)(¯¯ u,¯v)is smooth with Jacobian determinant ∂(u,v)∂(¯u,¯v). Note that
||ϕ¯u¯×ϕ¯v¯||=
∂(u, v)
∂(¯u,v)¯
||ϕu×ϕv||dudv.
The formula of change of variables in the double integral yields that Z Z
¯ ϕ−1(D)
f( ¯ϕ(¯u,¯v))||ϕ¯u¯×ϕ¯v¯||d¯ud¯v
= Z Z
¯ ϕ−1(D)
f( ¯ϕ(¯u,v))¯ ||ϕu×ϕv||
∂(u, v)
∂(¯u,¯v) d¯ud¯v
= Z Z
ϕ−1(D)
f(ϕ(u, v))||ϕu×ϕv||dudv,
so the definition is independent of the choice of parametrization. In general, onde needs to cover the support off by finitely many parametrizations and
2.2. THE GAUSS MAP AND THE SECOND FUNDAMENTAL FORM 13 define the surface integral as a sum of double integrals. The proof of indepen- dence of the choices involved is more complicated in this case, and we will not go into details. The relation to the first fundamental form is that
||ϕu×ϕv||2=||ϕu||2||ϕv||2− hϕu, ϕvi2, so
||ϕu×ϕv||=p
EG−F2. Areaof a compact domainD⊂Sis
area(D) = Z Z
D
1dS.
In particular, ifDis contained in the image ofϕ, area(D) =
Z Z
ϕ−1(D)
pEG−F2dudv.
2.2 The Gauss map and the second fundamental form
LetS ⊂R3be a surface. For eachp∈S, we want to assign a unit vectorν(p) which is normal toTpS; note that there are exactly two possible choices. If it is possible to make such an assigment continuously along the whole ofS, we say thatSisorientable. The resulting map
ν:S→S2
into the unit sphere is called the Gauss map. We will always assume that the Gauss map is continuous.
Examples 2.3 1. Ifϕ:U →Sis a parametrization, then we can take ν = ϕu×ϕv
||ϕu×ϕv||.
This construction shows that every surface is locally orientable. It also shows that the Gauss map is smooth.
2. IfSis given as the inverse image underF of regular value, then we can take
ν= ∇F
||∇F||. The differential
dνp:TpS→Tν(p)S2=TpS
is an operator onTpS, sinceTqS2is always normal toq, forq∈S2. The operator
−dνp :TpS→TpS is called theWeingarten operator.
Proposition 2.4 The Weingarten operator is symmetric:
hdνp(w1), w2i=hw1, dνp(w2)i, wherew1,w2∈TpS.
Proof. By linearity, it suffices to check the relation for a basis ofTpS. Let ϕ:U →Sbe a parametrization; then{ϕu, ϕv}is a tangent frame. SetN =ν◦ϕ.
Then
dν(ϕu) =dν
∂ϕ
∂u
=∂u∂ (ν◦ϕ) = ∂N∂u.
Similarlydν(ϕv) = ∂N∂v. Sinceh∂ϕ∂v, Ni= 0onU, differentiating with respect to v,
h ∂2ϕ
∂u∂v, Ni+h∂ϕ∂v,∂N∂ui= 0, and similarly
h∂2ϕ
∂v∂v, Ni+h∂ϕ∂v,∂N∂vi= 0, Taking the difference of these equations,
h∂ϕ∂v,∂N∂ui − h∂ϕ∂u,∂N∂vi= 0, which says that
hϕv, dν(ϕv)i − hϕu, dν(ϕv)i= 0,
as we wished.
The associated symmetric bilinear form
IIp(w1, w2) =−hdνp(w1), w2i is called thesecond fundamental form.
Proposition 2.5 Letγ: (−ǫ, ǫ)→Sbe a smooth curve parametrized by arc-length, γ(0) =p,γ′(0) =w∈TpS. Then
hγ′′(0), ν(p)i=IIp(w, w).
Proof. Start with the equationhγ′(s), ν(γ(s))i= 0and differentiate it ats= 0 to obtain
hγ′′(0), ν(p)i+hw, d
ds|s=0ν(γ(s))i= 0.
Thenhγ′′(0), ν(p)i=−hw, dνp(w)i=IIp(w, w), as desired.
There is a geometric interpretation of last proposition (Meusnier). Given a unit vectorw∈TpS, the affine plane throughpspanned bywandν(p)meetsS transversally along a curve which is called thenormal section of Salongw. Ifγ
2.2. THE GAUSS MAP AND THE SECOND FUNDAMENTAL FORM 15 is a parametrization by arc-length of this normal section as in the proposition, then the curvature ofγatp=γ(0)is
κw=hγ′′(0), ν(p)i=IIp(w, w),
where we viewγas a plane curve and we view its supporting plane as oriented by{w, ν(p)}.
Since the Weingarten operator−dνpis symmetric, there exists an orthonor- mal basis{e1, e2}ofTpSsuch that
−dνp(e1) =κ1e1, −dνp(e2) =κ2e2.
The eigenvaluesκ1,κ2are calledprincipal curvaturesatp, and the eigenvectors e1,e2are calledprincipal directionsatp. Of course,II(e1, e1) =κ1,II(e2, e2) = κ2,II(e1, e2) = 0. It follows that for a unit vectorw= cosθe1+ sinθe2 ∈TpS we haveEuler’s formula:
κw=Ip(w, w) =κ1cos2θ+κ2sin2θ.
Since this a convex linear combination (sin2θ+ cos2θ = 1), Euler’s formula also shows thatκ1,κ2are the extrema of the curvatures of the normal sections throughp.
In general, a smooth curve γ in S is called aline of curvature if γ(t)˙ is a principal direction atγ(t)for allt. Note that ifκ1(p)6=κ2(p), then the principal directions atpare uniquely defined, but not otherwise. Ifκ1(p) = κ2(p), we say thatpis anumbilic pointofS.
Proposition 2.6 If all the points of a connected surfaceSare umbilic, thenSis con- tained in a plane or a sphere.
Proof. We first prove the result in caseSis the imageV of a parametrization ϕ: U →V. SetN =ν◦ϕ. By assumption,dν =λ·I, whereλ: V →Ris a smooth function. It follows that
Nu=dν(ϕu) = (λ◦ϕ)ϕu, Nv=dν(ϕv) = (λ◦ϕ)ϕv.
We differentiate the first (resp. second) of these equations with respect to v (resp.u) to get
Nuv = (λ◦ϕ)vϕu+ (λ◦ϕ)ϕuv, Nvu = (λ◦ϕ)uϕv+ (λ◦ϕ)ϕvu.
Taking the difference,
0 = (λ◦ϕ)vϕu−(λ◦ϕ)uϕv.
Since{ϕu, ϕv}is linearly independent, the partial derivatives ofλ◦ϕonUare zero. SinceU is connected,λ◦ϕis constant.
Next, we consider two cases. If λ◦ϕ = 0, thenNu = Nv = 0 so N is constant. This implies ∂u∂ hϕ, Ni = ∂v∂ hϕ, Ni= 0and henceV is contained in an affine plane parallel tohNi⊥. On the other hand, ifλ◦ϕ = R 6= 0, then ϕ−R1 Nis a constantq∈R3and henceV is contained in the sphere of center qand radius1/|R|.
In the case of arbitraryS, fix a pointp ∈ Sand a parametrized neighbor- hoodV0ofp. By the previous case,V0is contained in a plane or a sphere. Given x∈S, by connectedness ofS, there exists a continuous curveγ: [0,1]→Sjoin- ingptox(Sis locally arcwise connected, so it is arcwise connected). For any t∈[0,1], there exists a parametrized neighborhood ofγ(t)which is contained in a plane or a sphere. By compactness ofγ([0,1]), it is possible to cover it by open setsV0,V1, . . . , Vnsuch that eachViis contained in a plane or a sphere and Vi∩Vi+1 6=∅(check this!); the latter condition implies thatVi+1is contained in the same plane or sphere that containsVi. The result follows.
2.3 Curvature of surfaces
Let S ⊂ R3 and consider its Weingarten operator −dνp : TpS → TpS. Re- call that−dνpis symmetric and its eigenvaluesκ1(p), κ2(p)are the principal curvatures ofSatp. We define:
Gaussian curvature:K(p) = det(−dνp) =κ1(p)·κ2(p), Mean curvature:H(p) = 1
2trace(−dνp) = 1
2(κ1(p) +κ2(p)). Note thatκ1,κ2 =H±√
H2−K; we will soon see thatH andKare smooth functions onS, and so it follows from this equation thatκ1,κ2are continuous functions onSwhich are smooth away from umbilic points (points character- ized byH2=K).
If we changeνto−ν, thenH is changed to−H butKis unchanged. The next example analyses the meaning of the sign ofK.
Examples 2.7 1. Let us compute the Gaussian curvature of the graphS of a smooth function f : U → R, where U ⊂ R2 is open. In general, for a parametrizationϕandN=ν◦ϕ,
II(ϕu, ϕu) =−hdν(ϕu), ϕui=−hNu, ϕui=hN, ϕuui. Similarly,
II(ϕu, ϕv) =−hNu, ϕvi=hN, ϕuvi and
II(ϕv, ϕv) =−hNv, ϕvi=hN, ϕvvi.
2.4. LOCAL EXPRESSIONS FORK,H 17 In our case,
ϕu = (1,0, fu), ϕv = (0,1, fv), ϕuu = (0,0, fuu), ϕuv = (0,0, fuv), ϕvv = (0,0, fvv),
N = (−fu,−fv,1) p1 +fu2+fv2. Hence
(II) = 1
p1 +fu2+fv2
fuu fuv
fvu fvv
= 1
p1 +fu2+fv2 (Hess(f)).
We specialize to the casep= (0,0,0) = f(0,0)andTpS is thexy-plane. Then fu(0,0) =fv(0,0) = 0andIIp = Hess(0,0)(f). In particular, iff(u, v) =au2+ bv2, then
(II) =
2a 0 0 2b
.
It follows thatK(p) = 4abis positive (resp. negative) ifaandbhave the same sign (resp. opposite signs), and it is zero if one ofa,bis zero.
Another interesting case isf(u, v) =u4+v4. We getII = 0.
2. Consider the sphereS2(R)of radiusR >0. We can takeν(p) =−R1p, so
−dνp= R1 idTpSandK(p) = R12 >0,H(p) = R1.
A pointpin a surfaceSis calledelliptic(resp.hyperbolic,parabolic) ifK(p)>
0(resp.K(p)<0,K(p) = 0).
2.4 Local expressions for K , H
Fix a parametrizationϕofS. Then{ϕu, ϕv}is a tangent frame with respect to which we consider the matrices of the fundamental forms and the Weingarten operator and introduce a new (index) notation for the coefficients.
(I) =
hϕu, ϕui hϕu, ϕvi hϕv, ϕui hϕv, ϕvi
=
E F F G
=
g11 g12
g21 g22
,
(II) =
hN, ϕuui hN, ϕuvi hN, ϕvui hN, ϕvvi
=
ℓ m
m n
=
h11 h12
h21 h22
, (−dν) =
h11 h12 h21 h22
.
We have that
h11=II(ϕu, ϕu) =−hdν(ϕu), ϕui=hh11ϕu+h21ϕv, ϕui, so
h11=h11g11+h21g21. Similarly,
h12=h11g12+h21g22, h21=h12g11+h22g21, h22=h12g12+h22g22. In matrix form.
h11 h21 h12 h22
g11 g12
g21 g22
=
h11 h12
h21 h22
. Recall that(I)is invertible since it is positive definite. Thus
h11 h21 h12 h22
= 1
g11g22−g12g21
h11 h12
h21 h22
g22 −g12
−g21 g11
. Back to the classical notation,
(−dν) = 1 EG−F2
ℓ m
m n
G −F
−F E
= 1
EG−F2
ℓG−mF −ℓF+mE mG−nF −mF+nE
. Hence
K= ℓn−m2
EG−F2 = det(II) det(I) and
H = ℓG−2mF +nE 2(EG−F2) .
It follows from these expressions thatK,H are smooth onS.
2.5 Surfaces of revolution
Consider the parametrized surface of revolution
ϕ(u, v) = (f(u) cosv, f(u) sinv, g(u))
where(u, v) ∈ (a, b)×(v0, v0+ 2π)and the geratrixγ(s) = (f(s),0, g(s))is parametrized by arc length. Then
ϕu= (f′cosv, f′sinv, g′), ϕv= (−fsinv, fcosv,0),
2.5. SURFACES OF REVOLUTION 19 so
E=f′2+g′2= 1, F = 0, G=f2. Moreover,
ϕuu = (f′′cosv, f′′sinv, g′′), ϕuv = (−f′sinv, f′cosv,0), ϕvv = (−fcosv,−fsinv,0).
We compute the coefficients ofII.
ℓ = hN, ϕuui= hϕu×ϕv, ϕuui
||ϕu×ϕv|| = 1 EG−F2
f′cosv f′sinv g
−fsinv fcosv 0 f′′cosv f′′sinv g′′
= f′g′′−f′′g′. Similarly,
m=hN, ϕuvi= 1 f
f′cosv f′sinv g
−fsinv fcosv 0
−f′sinv f′cosv 0
= 0
and
n=hN, ϕvvi= 1 f
f′cosv f′sinv g
−fsinv fcosv 0
−fcosv −fsinv 0
=f g′.
Note thatf′g′′−f′′g′is the signed curvatureκγofγ, forκγ =hγ′′,(−g′,0, f′)i. Now
K= ℓn−m2
EG−F2 = (f′g′′−f′′g′)g′
f =κγg′
f (2.8)
and
H= ℓG−2mF+nE 2(EG−F2) =1
2
κγ+g′ f
. (2.9)
It follows that the principal curvatures
κ1=κγ, κ2= g′ f.
The identitiesF =m = 0mean that the fundamental forms are diagonalized in the frame{ϕu, ϕv}. In particular,ϕu,ϕvare always principal directions and thus the curvesu-constant (parallels) andv-constant (meridians) are lines of curvature.
We can also derive some other useful formulas forK,H. Differentiation of f′2+g′2= 1givesf′f′′+g′g′′= 0. Substituting this identity into (2.8) yields
K= −f′′
f .
The identity also givesκ1=κγ = gf′′′ iff′6= 0, so H =1
2 g′′
f′ +g′ f
=(f g′)′
(f2)′ . (2.10)
We use this formula to prove the following theorem. A surface satisfyingH ≡0 is calledminimal. This terminology will be explained in section??.
Theorem 2.11 The only minimal surfaces of revolution are the plane and the catenoid (the surface of revolution generated by a catenary, which is the graph of hyperbolic cosine).
Proof. We use the above notation. Iff′ = 0on an interval, then eqn. (2.9) givesg′ = 0, which is a contradiction to the fact thatγis regular. Therefore we can assume thatf′ is never zero. By formula (2.10), we need to solve the equationf g′ =k, wherekis a constant. Usingg′=±p
1−f′2, we get f′=±p
1−(k/f)2.
Note that|f| ≥ |k|is a necessary condition. This equation can be easily inte- grated by rewriting it as
f df
pf2−k2 =±ds.
We get
f(s) =±p
k2+ (s+c1)2.
The constantc1can be chosen to be zero by redefining the instants = 0, and we recall thatf >0, so we have
f(s) =p k2+s2.
Ifk= 0thenf(s) =±sandgis constant, which corresponds to the case of the plane. Supposek6= 0and integrateg′=k/fto get
g(s) =klog(s+p
k2+s2) +c2.
We choose the constant c2 = −klog|k| so thatγ(0) = (|k|,0,0). Changing the sign ofkis equivalent to changing the sign ofg, which corresponds to a reflection on the planez = 0, so we may assumek > 0. Finally, we make the change of variable
t=g(s) =klog(p
1 + (s/k)2+s/k) to get
γ(t) = (kcosh(t/k),0, t)
which is a catenary.
2.6. RULED SURFACES 21
2.6 Ruled surfaces
A ruled surface is a surface generated by a smooth one-parameter family of lines. More precisely, a (nonnecessarily regular) parametrized surfaceϕ:U ⊂ R2 →Sis called aruled surfaceif there exist a smooth curvesγ :I →R3and w:I→S2such that
ϕ(u, v) =γ(u) +v w(u)
where(u, v)∈I×R=U. The curveγis called adirectrixand the linesRw(u) are called therulings.
Obvious examples of ruled surfaces are planes, cilinders and cones. Other examples are the helicoid, the one-sheeted hyperboloid and the hyperbolic paraboloid (given by the equationz=xyinR3).
We make some local considerations. Assume thatw′(u) 6= 0for allu, in other words,wis regular; this condition is sometimes expressed by saying that the ruled surface isnoncylindrical. Then it is possible to introduce the so called standard parametersonS.
Proposition 2.12 There exists a unique reparametrization
˜
ϕ(˜u,v) = ˜˜ γ(˜u) + ˜vw(˜˜ u) such that||w˜′||= 1andhγ˜′,w˜′i= 0.
Proof. Sincewis regular, we can introduce arc-length parameteru˜ so that
˜
w(˜u) = w(u(˜u)), and then||w˜′|| = 1. Next, we writeγ(˜˜ u) = γ(˜u)−˜v(˜u) ˜w(˜u) and impose the conditionhγ˜′,w˜′i= 0to get˜v(˜u) =−hd˜duγ,w˜′(u)i.
The curveγ˜is called thestriction lineof the surface and its points are called central pointsof the surface; note that˜γis not necessarily regular.
Using the standard parametrization, we can compute the curvature of a ruled surface
ϕ(u, v) =γ(u) +v w(u) where||w||=||w′||= 1,hγ′, w′i= 0. We have
ϕu=γ′+v w′, ϕv=w, so
E=||γ′||2+v2, F =hγ′, wi, G= 1.
Sincew′ is orthogonal tow andγ′, there exists a smooth functionλ = λ(u), called thedistribution parameter, such that
γ′×w=λ w′. (2.13)
It follows that
||ϕu×ϕv||=p
EG−F2=||γ′||2− hγ′, wi2+v2=λ2+v2.
In particular, the singular points of ϕ occur along the striction line (v = 0) precisely whenλ(u) = 0.
Next, we compute the coefficients ofII. We have ϕuu=γ′′+v w′, ϕuv=w′, ϕvv= 0.
This implies
m= hϕu×ϕv, ϕuvi
||ϕu×ϕv|| = hλw′+v w×w′, w′i
√λ2+v2 = λ
√λ2+v2 and
n= 0,
what is sufficient to get the formula for the Gaussian curvature:
K= ℓn−m2
EG−F2 =− λ(u)2 (λ(u)2+v2)2.
Note thatK ≤0, andK = 0precisely along the rulings that meet the striction line at a singular point (λ(u) = 0), except of course the singular point itself (v 6= 0). Ifλ(u)6= 0, this formula also shows the striction line is characterized by the property that the maximum ofKalong each ruling occurs exactly at the central point.
The computation ofℓis more involved. We have
hϕu×ϕv, ϕuui (2.14)
= λhw′, γ′′i+λvhw′, w′′i+vhw′×w, γ′′i+v2hw′×w, w′′i. We analyse separately the four terms on the right-hand side. Introduce the parameter
J =hw×w′, w′′i. Since{w, w′, w×w′}is an orthonormal frame,
γ′ =hγ′, wiw+hγ′, w×w′iw×w′=F w+λ w×w′ and
hw′, γ′′i=−hw′′, γ′i=−Fhw′′, wi −λhw′′, w×w′i=F−λJ,
where we have usedhw′′, wi=−hw′, w′i=−1. Equation||w′||= 1also implies hw′, w′′i = 0. In order to analyse the third term in eqn. (2.14), differentiate eqn. (2.13) to get
γ′′×w+γ′×w′=λ′w′+λw′′, and multiply through byw′to write
hγ′′×w, w′i=λ′. Now eqn. (2.14) is
hϕu×ϕv, ϕuui=−Jv2−λ′v+λ(F−λJ)
2.7. MINIMAL SURFACES 23 and
ℓ= −λ2J+λ′v+λ(F−λJ)
√λ2+v2 . Hence
H =ℓG−2mF+nE
2(EG−F2) =−Jv2+λ′v+λ(λJ+F) 2(λ2+v2)3/2 .
Example 2.15 The standard parametrization of the helicoid hasγ(u) = (0,0, bu) andw(u) = (cosu,sinu,0). Sinceγ′×w=bw′, the distribution parameterλ=b is constant andK =−b2/(b2+v2)2. Note thatF = 0and, sincew′′=−w, we also haveJ = 0. HenceH = 0.
Proposition 2.16 The only minimal ruled surfaces are the plane and the helicoid.
Proof. We have just seen thatH = 0says that Jv2+λ′v+λ(λJ+F) = 0.
This is a quadratic polynomial invwhose coefficients, being functions depend- ing only onu, must vanish. It follows thatλis constant andJ =λF = 0.
SinceJ = 0,w′′ is a linear combination ofw, w′. But hw′′, w′i = 0 and hw′′, wi=−1, sow′′=−wandwis a circle.
Ifλ= 0thenII = 0, which corresponds to the case of the plane. Suppose λ6= 0. ThenF = 0implying thatγ′ =λ w×w′. Differentiation of this equation yieldsγ′′ = 0. It follows thatγis a line perpendicular to the circle defined by
w. Hence the surface is the helicoid.
2.7 Minimal surfaces
LetS ⊂ R3 be a surface. We say thatS is aminimal surfaceif the mean cur- vatureH ≡0. Historically speaking, this concept is related to the problem of characterizing the surface with smallest area spanned by a given boundary, a problem raised by Lagrange in 1760. The question of showing the existence of such a surface is called thePlateau problem, in honor of the Belgian physicist who performed experiments with soap films around 1850, and it was solved completely only in 1930, independently by Jesse Douglas and Tibor Rad ´o.
In order to explain the relation between mean curvature and minimization of surface area, consider a parametrization ϕ : U ⊂ R2 → S, N = ν ◦ϕ the induced unit normal, and a smooth functionf : U → R. Then we can introduce thenormal variationofϕalongf:
ϕǫ=ϕ+ǫ f N.
Let us compute the first fundamental form ofϕǫ:
ϕǫu =ϕu+ǫ(fuN+f Nu), ϕǫv=ϕv+ǫ(fvN+f Nv).
Sincehϕu, Ni =hϕv, Ni= 0andhϕu, Nui= −ℓ,hϕu, Nvi =hϕv, Nui =−m, hϕv, Nvi=−n, we obtain
Eǫ = E−2ǫf ℓ+O(ǫ2), Fǫ = F−2ǫf m+O(ǫ2), Gǫ = G−2ǫf n+O(ǫ2),
whereO(ǫ2)denotes a continuous function satisfying limǫ→0O(ǫ2)/ǫ = 0. It follows that
EǫGǫ−(Fǫ)2 = EG−F2−2ǫf(ℓG+nE−2mF) +O(ǫ2)
= (EG−F2)(1−4ǫf H) +O(ǫ2).
Let nowD⊂Ube a compact domain and introduce A(ǫ) = area(ϕǫ(D)) =
Z Z
D
pEǫGǫ−(Fǫ)2dudv.
We have
A′(0) = ∂
∂ǫ ǫ=0
Z Z
D
EǫGǫ−(Fǫ)21/2
dudv
= Z Z
D
∂
∂ǫ
ǫ=0EǫGǫ−(Fǫ)2 2(EG−F2)1/2 dudv.
Hence
A′(0) =−2 Z Z
D
f Hp
EG−F2dudv.
This formula is calledfirst variation of surface area. As a corollary, we obtain the following characterization of minimal surfaces as critical points of the area functional.
Proposition 2.17 A surfaceSis minimal if and only ifA′(0) = 0for every parametriza- tionϕ:U →S, every normal variation ofϕ, and every compact domainD⊂U.
Proof. IfH(p)6= 0for somep∈S, we choose a compact neighborhoodD˜ of pinSsuch thatH does not vanish onD˜ andD˜ is contained in the image of a parametrizationϕ:U →S, setD=ϕ−1( ˜D), and takef =H|U. We get
A′(0) =−2 Z Z
D
H2p
EG−F2dudv <0,
so the given condition is suficient for the minimality ofS. That it is also neces-
sary is obvious.
2.7. MINIMAL SURFACES 25
2.7.1 Isothermal parameters
When studying minimal surfaces, it is useful to introduce special parameters.
A parametrized surfaceϕ:U →Sis calledisothermalif E=G=λ2, F = 0,
whereλ≥0is a smooth function onU; in this case, the parameters(u, v)∈U are also calledisothermal. Note thatϕis regular if and only ifλ >0. An isother- mal parametrizationϕis also calledconformalorangle preservingbecause angles between curves in the surface are equal to the angles between the correspond- ing curves in the parameter plane.
Note that the mean curvature expressed in terms of isothermal parameters becomes
H =ℓG−2mF+nE
2(EG−F2) = ℓ+n
2λ2 . (2.18)
Proposition 2.19 Ifϕis isothermal, then∆ϕ= 2λ2HN(hereN=ν◦ϕis the unit normal alongϕ).
Proof. Here∆denotes the Laplacian operator and∆ϕ=ϕuu+ϕvv. Con- sider the equations hϕu, ϕui = hϕv, ϕviandhϕu, ϕvi = 0; differentiating the first one with respect touand the second one with resptec tov, we obtain
hϕuu, ϕui=hϕvu, ϕui and hϕuv, ϕvi+hϕu, ϕvvi= 0.
Putting these together yieldsh∆ϕ, ϕui = 0. Similarly, differentiating the first equation with respect tov and the second one with resptec tou, we get that h∆ϕ, ϕvi= 0. This shows that∆ϕis a normal vector. Finally,
h∆ϕ, Ni=hϕuu, Ni+hϕvv, Ni=ℓ+n= 2λ2H
by eqn. (2.18).
Corollary 2.20 An isothermal regular parametrized surfaceϕ:U →Sis minimal if and only if the coordinate functions ofϕare harmonic functions onU.
Isothermal parameters exist around any point in a surface. In the next sec- tion, we present a proof of their existence in the case of minimal surfaces.
Theorem 2.21 There exist no compact minimal surfaces inR3.
Proof. Suppose, to the contrary, thatSis a compact minimal surface. With- out loss of generality, we may assumeSis connected. Consider the coordinate functionx: R3 →R. There exists a pointp∈Swhere the restrictionx|S at- tains its maximum. Letϕ:U →ϕ(U)be an isothermal parametrization around pwithU connected. Thenx◦ϕis a harmonic function onU which attains its maximum at an interior point p ∈ U. By the maximum principle, x◦ϕis a constant function onU, or,x≡x(p)onϕ(U).
Next, let q ∈ S be arbitrary and choose a continuous curve γ : [0,1] → S joiningpto q. Cover γ([0,1])by finitely many connected open setsV0 = ϕ(U),V1, . . . , Vn, eachVi equal to the image of an isothermal parametrization ϕi, such thatVi∩Vi+1 6=∅for alli= 0,1, . . . , nandq∈Vn. Sincex|S attains its maximum value along V0 ∩V1 6= ∅, the maximum principle applied to x◦ϕ1 yields that this function is constant alongV1, namely,x ≡ x(p)onV1. Proceeding by induction, we get thatx≡ x(p)onVn and hencex(q) =x(p).
Sinceqis arbitrary, this argument proves thatx|S is a constant function. The same argument applied to the other coordinate functionsy,z:R3→Rfinally
shows thatSmust be a point, a contradiction.
2.7.2 The Enneper-Weierstrass representation
We discuss now an unexpected connection beween minimal surfaces and the- ory of functions of one complex variable. Letϕ : U → S be a parametrized surface. Denote by x1, x2, x3 : U → R the coordinate functions ofϕ. We introduce the complex functions (j= 1,2,3):
fj(ζ) =∂xj
∂u −∂xj
∂v , whereζ=u+iv. (2.22) The functionfjis smooth as a real functionU ⊂R2→R2, so a necessary and suffcient condition forfjto be holomorphic is given by the Cauchy-Riemann equations ∂u∂ ℜfj= ∂v∂ ℑfj,∂v∂ ℜfj=−∂u∂ ℑfj. We deduce that
(a) fjis holomorphic inζif and only ifxjis harmonic inu,v.
Note also the identities:
f12+f22+f32 = X3
j=1
∂xj
∂u 2
− X3
j=1
∂xj
∂v 2
−2i X3
j=1
∂xj
∂u ·∂xj
∂v
= E−G−2iF and
|f1|2+|f2|2+|f3|2= X3
j=1
∂xj
∂u 2
+ X3
j=1
∂xj
∂v 2
=E+G. (2.23)
It follows from these identities that (b) ϕis isothermal if and only if
f12+f22+f32= 0. (2.24) (c) Ifϕis isothermal, thenϕis regular if and only if
|f1|2+|f2|2+|f3|26= 0. (2.25)
2.7. MINIMAL SURFACES 27 Proposition 2.26 Let ϕ : U → S be an isothermal regular parametrized minimal surface. Then the functionsfj defined by (2.22) are holomorphic and satisfy (2.24) and (2.25). Conversely, iff1,f2,f3are holomorphic functions defined on an simply- connected domainU which satisfy (2.24) and (2.25), then (j= 1,2,3)
xj(ζ) =ℜ Z ζ
ζ0
fj(z)dz, ζ∈U,
(for fixedζ0 ∈U) are the coordinates of an isothermal regular parametrized minimal surfaceϕ:U →Ssuch that eqns. (2.22) are valid.
Proof. One direction follows from assertions (a), (b), (c) above and Corol- lary 2.20. For the converse, note thatζ7→Rζ
ζ0fj(z)dzis well defined becauseU is simply connected andfjis holomorphic, and yields a holomorphic function onUfor which we can apply the Cauchy-Riemann equations:
d dζ
Z ζ
fj = ∂
∂uℜ Z ζ
fj+i ∂
∂uℑ Z ζ
fj
= ∂
∂uℜ Z ζ
fj−i ∂
∂vℑ Z ζ
fj,
so eqns. (2.22) are valid; the rest now follows from (a), (b), (c) and Corollary 2.20
applied in the opposite direction.
Note that the functionsxjin the preceding proposition are defined up to an additive constant so that the surface is defined up to a translation.
Thus we see that the local study of minimal surfaces inR3 is reduced to solving equations (2.24) and (2.25) for triples of holomorphic functions. We next explain how this can be done. Rewrite (2.24) as
(f1+if2)(f1−if2) =−f32. (2.27) Except in casef1 ≡if2andf3 ≡ 0(which is easily seen to correspond to the case of a plane), the functions
f =f1−if2, g= f3
f1−if2
are such thatf is holomorphic andgis meromorphic. Clearly,f3 =f g, and it follows from eqn. (2.27) that
f1+if2= −f32
f1−if2 =−f g2. (2.28) Hence
f1=1
2f(1−g2), and f2= i
2f(1 +g2).
By (2.28),f g2is homolorphic and this says that at every pole ofg,f has a zero of order at least twice the order of the pole. Further, eqn. (2.25) says thatf1,f2,
f3cannot vanish simultanelously, and this means thatf can only have a zero at a pole ofg, and then the order of its zero must be exactly twice the order of the pole ofg. We summarize this dicussion as follows.
Theorem 2.29 (The Enneper-Weierstrass representation) Every minimal surface which is not a plane can be locally represented as
x1 = ℜ Z 1
2f(ζ)(1−g2(ζ)dζ x2 = ℜ
Z i
2f(ζ)(1 +g2(ζ))dζ x3 = ℜ
Z
f(ζ)g(ζ)dζ,
where:fis a holomorphic function on a simply-connected domainU,gis meromorphic onU,fvanishes only at the poles ofg, and the order of its zero at such a point is exactly twice the order of the pole ofg.
Conversely, every pair functionsf,gsatisfying these conditions define an isother- mal regular parametrized minimal surface via the above equations.
Examples 2.30 1. The catenoid is given byf(z) =−e−z,g(z) =−ez. 2. The helicoid is given byf(z) =−ie−z,g(z) =−ez.
3. The minimal surface of Enneper (discovered in 1863) is given byf(z) = 1, g(z) = z. Solving for the parametrization, we obtainx1 = u− 13u3 +uv2, x2=−v−u2v+13v3,x3=u2−v2.
4. The minimal surface of Scherk (discovered in 1834) is given byf(z) = 4/(1−z4), g(z) = iz. It can also be parametrized as the graph of(x, y) 7→
logcoscosxy.
The Enneper-Weierstrass representation not only allows us to construct a great variety of minimal surfaces having interesting properties, but also serve to prove general theorems about minimal surfaces by translating the state- ments into corresponding statements about holomorphic functions. Unfortu- nately, developing this philosophy would take us beyond the scope of these notes, so we content ourselves with a small remark. Let us express the basic geometric quantities of an isothermal regular parametrized minimal surface ϕ:U →Sin terms off,g. We have
E=G=λ2, F = 0, where
λ2 = 1 2
X3
j=1
|fj|2 by(2.23)
= 1
4|f|2|1 +g|2+1
4|f|2|1 +g|2+|f g|2
=
|f|(1 +|g|2) 2
2
.
2.7. MINIMAL SURFACES 29 Moreover,
ϕu×ϕv = (ℑ{f2f¯3},ℑ{f3f¯1},ℑ{f1f¯2})
= |f|2(1 +|g|2)
4 ( 2ℜg,2ℑg,|g|2−1 ), and
||ϕu×ϕv||=p
EG−F2=λ2, so
N =
2ℜg
|g|2+ 1, 2ℑg
|g|2+ 1,|g|2−1
|g|2+ 1
.
Recall thatstereographic projectionπ:S2\ {(0,0,1)} →Cis the map π(x1, x2, x3) = x1+ix2
1−x3
, and its inverse is
π−1(z) =
2ℜz
|z|2+ 1, 2ℑz
|z|2+ 1,|z|2−1
|z|2+ 1
. Hence
N =π−1◦g. (2.31)
Proposition 2.32 Let ϕ : U → S be an isothermal regular parametrized minimal surface, whereU is the entireζ-plane. Then eitherSlies in a plane, or the image of the Gauss map takes on all values with at most two exceptions.
Proof. IfSdoes not lie in a plane, we can construct the functiong(ζ)which is meromorphic on the entireζ-plane; by Picard’s theorem, it either takes all values with at most two exceptions, or else it is constant. Eqn. (2.31) shows that the same alternative applies toN, and in the latter caseSlies in a plane.
2.7.3 Local existence of isothermal parameters for minimal sur- faces
Lemma 2.33 LetSbe a minimal surface. Then every point ofS lies in the image of an isothermal parametrization ofS.
Proof. Letp∈S. First all, we can find a neighborhood ofpin whichSis the graph of a smooth function which, by relabeling coordinates, can be assumed in the form z = h(x, y) for (x, y) ∈ U (Check!). The minimal equation for graphs is easily computed to be
(1 +h2y)hxx−2hxhyhxy+ (1 +h2x)hyy= 0.
We then have equation
∂
∂x 1 +h2y
W = ∂
∂y hxhy
W .
satisfied onU, whereW = q
1 +h2x+h2y. Taking U simply-connected, this implies that we can find a smooth functionΦonU with
∂Φ
∂x = hxhy
W , ∂Φ
∂y = 1 +h2y W . Introduce new coordinates
¯
x=x, y¯= Φ(x, y).
One checks
∂x
∂x¯ = 1, ∂x∂y¯= 0, ∂y∂x¯ =− hxhy
1 +h2y, ∂y∂¯y = W 1 +h2y,
and the coeffcients of the second fundamental form with respect tox,¯ y¯are E¯ = ¯G= W2
1 +h2y, F¯ = 0,
as desired.
