Universit`
a degli studi di Pisa
Dipartimento di Matematica
Corso di Laurea Magistrale in Matematica
On the Willmore Functional:
Classical Results and
New Extensions
——
Relatore:
Prof. Matteo Novaga
Controrelatore:
Prof. Giovanni Alberti
Candidato:
Marco Pozzetta
July 2017
De qu´e trata el experimento?, dijo Rosa [...] No es ning´un experimento en el sentido literal de la palabra, dijo Amalfitano [...] la idea es de Duchamp, dejar un libro de geometr´ıa colgado a la intemperie para ver si aprende cuatro cosas de la vida real. Lo vas a destrozar, dijo Rosa. Yo no, dijo Amalfitano, la naturaleza.
Contents
Introduction 7
List of Symbols 11
1 Preliminaries from Geometry and Analysis 13
1.1 Second Fundamental Form and Curvature . . . 13
1.2 Conformal transformations . . . 17
1.3 Spherical Inversions and Curvature . . . 20
1.4 The Stereographic Projection . . . 22
1.5 Rectifiable Sets and Area Formula . . . 23
1.6 Varifolds . . . 26
1.7 Currents . . . 29
1.8 CW-Complexes . . . 33
2 The Willmore Functional: Properties and Existence Results 39 2.1 Conformal Invariance . . . 39
2.2 Lower Bound for Immersed Surfaces . . . 41
2.3 Global Minima . . . 43
2.4 Minima of Prescribed Genus . . . 46
2.5 The Clifford and the Willmore Tori . . . 60
3 The Willmore Conjecture 63 3.1 The Canonical Family . . . 64
3.2 The Min-Max Family . . . 67
3.3 Almgren-Pitts Min-Max Theory . . . 69
3.4 Lower Bound on Width . . . 74
3.5 Theorem B . . . 79
3.6 Theorem A . . . 82
4 Extensions with Area, Volume and Confinement Constraints 85 4.1 Relations between Willmore and Area . . . 87
4.2 Rescaling Invariance . . . 89
4.4 Problems (P 2)B1,Λ and (P 3)B1 . . . 92
4.5 Problems (P 2)Ω,Λ and (P 3)Ω . . . 95
4.6 Problems (Q1)Ω,m, (Q2)Ω,Λ and (Q3)Ω . . . 103
Introduction
Abstract
For an immersed surface Σ in R3 with mean curvature H is defined its Willmore energy as:
W(Σ) = Z
Σ
H2dH2,
and the functional W is then called Willmore functional. It is natural to start a varia-tional study of this funcvaria-tional, that is the study of its minimisation properties among surfaces in R3. This is the problem introduced by Thomas James Willmore in 1965 ([25]). In the first part of this thesis we retrace some of the main results concerning this problem.
Willmore himself found that global minimisers for the functional are only round spheres. So the next step is to introduce some constraints in the minimisation problem. Histor-ically the first requirement has been to fix the genus of the surface. In fact this request started from the so called Willmore Conjecture about the behaviour of the functional among tori.
Conjecture 0.1 (Willmore Conjecture, 1965). If Σ ⊂ R3 is a genus 1 immersed surface, then:
Z
Σ
H2dH2 ≥ 2π2,
so the torus given by the immersion:
[0, 2π]2 3 (u, v) 7−→ ((√2 + cos(u)) cos(v), (√2 + cos(u)) sin(v), sin(u)) ∈ R3, that is called Willmore Torus, is a global minima among genus 1 surfaces.
The proof of the Willmore Conjecture turned out to be very difficult and through the years the study of the functional extended to parallel and related problems. It turns out that W has a number of properties, first of all the important invariance under conformal transformations of the surface and then an estimate that allows to consider embedded surfaces instead of general immersed ones. Moreover the minimisa-tion problem can be appropriately extended for surfaces in arbitrary Rn with the same
Rn minimisers are only round spheres and then we will sum up the great work done by Simon ([23]) and completed by Bauer and Kuwert ([4]) that leads to the existence of a minimiser for all fixed genus g surfaces in arbitrary Rn.
The conjecture has been proved only in 2014 by Fernando Coda Marques and Andr´e Neves ([13]). The proof is extraordinarily ambitious and we retrace its main geometric and variational ideas.
After this bibliographical study, we consider some extensions of the minimisation problem of the Willmore functional involving also the area of the surface (or a volume related to it, when well defined) as a constraint or as a competitor, the whole restricting to surfaces to be contained in some preassigned bounded open set Ω ⊂ R3. We always consider embedded, compact, 2-dimensional submanifold without boundary, denoted by Σ in what follows. In particular we study minimisation properties of the six following problems: (P 1)Ω,Λ: minW(Σ) : Σ ⊂ ¯Ω, |Σ| = Λ , (P 2)Ω,Λ: minWΛ(Σ) := W(Σ) − Λ|Σ| : Σ ⊂ ¯Ω , (P 3)Ω: min W(Σ) := W(Σ)|Σ| : Σ ⊂ ¯Ω ,
(Q1)Ω,m : minW?(E) : E ∈ X, ∂E 2-submanifold, |E| = m ,
(Q2)Ω,Λ: minWΛ?(E) := W ?
(E) − Λ|E| : E ∈ X, ∂E 2-submanifold ,
(Q3)Ω : min
W?(E) :=
W?(E)
|E| : E ∈ X, ∂E 2-submanifold
,
with fixed m, Λ > 0 where they appear, and with X an appropriate metric space of subsets of ¯Ω of finite perimeter (W? in fact denotes the relaxation of the Willmore functional in this space).
We find sufficient conditions under which there exist minimisers in the set of 2 - di-mensional integer rectifiable varifolds and we investigate the existence of a generalized mean curvature in L2. In this way we give a complete description of the existence
(and also non existence) of minima in this weaker sense for all the proposed problems. Moreover we provide several examples devoted to cases in which the minimum does not exist and others devoted to underline the differences that arise when we consider an unbounded domain Ω.
The main results are the following. Problems (P 1) and (Q1) have a solution for every choice of admissible parameters Λ, m. Problem (P 2) presents an unexpected threshold behaviour in function of the parameter Λ, in the sense that the problem passes from having positive minima to have infimum −∞. On the contrary Problem (Q2) is far more regular and we prove the existence of minimum for all the parameters Λ. Prob-lems (P 3) and (Q3) are solvable and they are strictly related to (P 2) and (Q2) for particular values of the parameter Λ.
Organization
This thesis has been thought as a self contained work, and since some prerequisites are necessary to approach the study of the Willmore functional, Chapter 1 is devoted to sum them up. However it is supposed a good knowledge of the basic concepts of Differential Geometry, Real and Functional Analysis and Measure Theory, for which we respectively refer to [2], [5] and [8].
It is crucial the concept of curvature, so this notion will be explained in the context of Riemannian Geometry at the beginning of Chapter 1. We also provide proofs of the most technical facts that will be necessary: they involve the notion of conformal transformation and the change of the curvature under such maps. Then we recall some basic facts in Geometric Measure Theory that will be widely used. At the end of Chapter 1 a section is devoted to CW-complexes, since they have been used in the proof of the Willmore Conjecture.
In Chapter 2 we study the main properties of the functional: conformal invariance, lower bound for immersed surfaces and the explicit characterization of the relation between the Willmore and the Clifford Tori. We also study the proofs leading to the main existence results for surfaces of prescribed genus.
Chapter 3 completes the bibliographical part and it is entirely devoted to the main steps necessary to the proof of the Willmore Conjecture. This involves all the results obtained at that point and it combines ideas from Differential Geometry, Calculus of Variations and even Algebraic Topology in order to complete the proof.
Finally Chapter 4 contains the study of the problems described above. This mainly make use of the basic properties of the Willmore functional and of the results about varifolds recalled in Chapter 1.
List of Symbols
Here we collect the undefined symbols present in the text. h·, ·i = Standard euclidean product in Rm for all m.
BR≡ BR(0) ≡ BRn(0) = Open ball of radius R and centre the origin in Rn.
BR(p) ≡ BRn(0) = Open ball of radius R and centre p in Rn.
dH = Hausdorff distance.
G(k, n) = Grassmannian of k-planes in Rn.
In= n-dimensional cube [0, 1]n.
Λn(V ) = Subspace of alternating tensors of
Nn
i=1V , where V is a finite dimensional
vector space.
N (M ) = Set on vector fields on a manifold ˜M that are normal respect the submanifold M .
Pn(R) = Real projective space of dimension n.
SR ≡ SR(0) ≡ SRn(0) = Sphere of radius R and centre the origin in Rn+1.
SR(p) ≡ SRn(p) = Sphere of radius R and centre p in Rn+1.
T (M ) = Set of vector fields on a manifold M . Th
Chapter 1
Preliminaries from Geometry and
Analysis
In this chapter we recall some classical definitions and results from Differential and Riemannian Geometry and from Geometric Measure Theory, in particular the general definitions and properties of the second fundamental form and of sectional curvature and then some very classical definitions and theorems about compactness, regularity and topology in the theory of rectifiable sets, varifolds and currents. The proofs about these general preliminaries can be found in [2] and [17] for the geometric part, and in [22] and [19] for the measure theoretic part.
Moreover, since the notion of conformal transformation will be fundamental, we also discuss through the whole chapter some well known technical facts about the topic whose proofs are not so easy to be found in the usual textbooks, such as explicit calculations of quantities like the Levi-Civita connection or the curvature tensor when a conformal transformation comes into play. We also give a simple proof of the fact that the stereographic projection is conformal and as last section of this chapter we recall some definitions about CW-complexes that will turn out to be useful in Chapter 3.
In this chapter any 2-dimensional manifold will be compact, connected and without boundary.
1.1
Second Fundamental Form and Curvature
In this section M will denote a n-dimensional (n ≥ 2) abstract differentiable Rieman-nian manifold, that is equipped with a RiemanRieman-nian metric tensor g, and let ∇ the induced Levi-Civita connection on M .
First we introduce the concept of sectional curvature of a manifold.
Definition 1.1. Let X, Y ∈ T (M ), k, h ∈ N, we define the curvature endomorphism R(X, Y ) : Tkh(M ) → Tkh(M ) by:
Actually, one can verify that R(X, Y ) is C∞(M )-linear in all its variables, then R : T (M ) × T (M ) × Th
k (M ) → Tkh(M ) defines a tensor field R ∈ T h+k
h+k+2(M ). We are
interested in the case h + k = 1, in which case we define:
Definition 1.2. The curvature tensor is the tensor field R ∈ T1
3 (M ) given by:
R(X, Y, Z) = R(X, Y )Z ∀X, Y, Z ∈ T (M ).
The next proposition allows us to define the first notion of curvature we need. Proposition 1.3. For all p ∈ M , π ⊂ TpM 2-plane and v, w a basis of π, the quantity:
gp(R(w, v)v, w)
gp(v, v)gp(w, w) − gp(v, w)2
(1.1) is independent of the choice of v and w.
So we can give the next:
Definition 1.4. In the notation of Proposition 1.3 we define sectional curvature K(π) of M in p along π the quantity given by equation (1.1) calculated on an arbitrary basis v,w of π.
We observe that if M is 2-dimensional then necessarily π = TpM and the sectional
curvature coincides with the Gaussian curvature of the manifold in p (one can verify it by writing the curvatures in local coordinates, see [2]).
Now we want to describe the relations between the curvature of the manifold M thought as a Riemannian submanifold of ˜M , whose metric tensor and Levi-Civita connection are denoted by ˜g and ˜∇. We call > : T ˜M → T M and ⊥: T ˜M → (T M )⊥ the natural orthogonal projections.
Definition 1.5. Let M a Riemannian submanifold of ˜M . The second fundamental form is the trilinear form:
A : N (M ) × T (M ) × T (M ) → C∞(M ) given by:
A(N, X, Y )(p) = ˜gp( ˜∇XN, Y ).
Making use of the properties of the Levi-Civita connection (symmetry and com-patibility with metric tensor) we immediately see that A is symmetric in the last two arguments and moreover we have:
A(N, X, Y ) = −˜g(N, ˜∇XY ) = −˜g(N, ˜∇YX) = g(>( ˜∇XN ), Y ) = g(X, >( ˜∇YN )).
Definition 1.6. The shape operator of M is S : T (M ) × T (M ) → N (M ) given by: S(X, Y ) = − ⊥ ( ˜∇XY ).
The shape endomorphism of M associated to N ∈ N (M ) is AN : T (M ) → T (M ) given
by:
AN(X) = >( ˜∇XN ).
Hence, using the definitions and the fact that in general ∇XY = >( ˜∇XY ), we have
the important relations:
A(N, X, Y ) = g(S(X, Y ), N ) = g(AN(X), Y ), (1.2)
˜
∇XY = ∇XY − S(X, Y ), (1.3)
and we remind that S is symmetric in its argument and C∞(M )-multilinear, while AN(X) is C∞(M )-multilinear in N and X and it’s symmetric with respect to the
scalar product, that is g(AN(X), Y ) = g(X, AN(Y )).
Here we recall a notion that we will use:
Definition 1.7. The submanifold M is totally geodesic in ˜M if any geodesic on M is also a geodesic on ˜M ; or, equivalently (see [7], page 132,133), if the shape operator vanishes.
So keep in mind that totally geodesic submanifolds of a standard sphere Sn⊂ Rn+1 are just the intersection of it with linear subspaces of Rn+1.
It will be often useful to express the second fundamental form in local coordinates when the manifold is given by an immersion in Rm = ˜M . So suppose that we are given
an immersion of an oriented Riemannian manifold in Rm, that is ϕ : (M, g) → Rm.
Calling N the normal field that orients the manifold, by saying that we express A in local coordinates we mean that we calculate it on the local coordinates given by the immersion ϕ, that is:
Aij = ˜g(N, ∂2ϕ ∂xi∂xj) i, j = 1, . . . , n, thus having: A(N, X, Y ) = n X i,j=1 AijXiYj,
having decomposed the fields X and Y in the same local coordinates. In this way it is explicit the calculation of the norm of A, that is |A| = (˜gikg˜jlA
ijAkl)1/2, where ˜gab are
Remark 1.8. It is important to observe that we have chosen to define the second fundamental form as a scalar quantity, that is as a differentiable function on M to R (once fields X and Y in the argument of A are fixed), but in this last case of an immersed oriented manifold it is sometimes simpler to see it as a family of normal vectors, that are {⊥ (∂x∂i2∂xϕj)}i,j=i,...,n; we will make use of this notation only a few
times and only in sections where specified, otherwise the second fundamental form has to be intended as the scalar quantity of Definition 1.5.
We can now give this last fundamental definition:
Definition 1.9. If M is a n-dimensional Riemannian submanifoldof ˜M whith shape operator S, we define the mean curvature field H ∈ N (M ) as:
H = 1 ntr(S), that is: H(p) = 1 n n X j=1 S(ej, ej),
where p ∈ M and {e1, . . . , en} is an arbitrary orthonormal basis of TpM . And when we
will write |H| we will mean the norm of the vector H calculated in the metric of ˜M . It will be useful to note that if ν ∈ (TpM )⊥ and {e1, . . . , en} is an arbitrary
or-thonormal basis of TpM one has:
˜ gp(H(p), ν) = 1 n n X j=1 A(ν, ej, ej)(p) = 1 n n X j=1 gp(Aν(ej), ej) = 1 ntr(Aν), (1.4) so we will usually write, using N = H/|H| where |H| 6= 0, that:
|H|2 = 1 ntr(AN) 2 = 1 n n X i=1 −ki 2 , (1.5)
being {−k1, . . . , −kn} the eigenvalues of AN.
We conclude this section by giving an important result:
Theorem 1.10 (Gauss equation). Let M be a Riemannian submanifold of ˜M with shape operator S, detoning by ˜R the curvature tensor of ˜M , then for all X, Y, Z, W ∈ T (M ):
g(>( ˜R(X, Y )Z), W ) = g(R(X, Y )Z, W ) + ˜g(S(X, Z), S(Y, W )) − ˜g(S(Y, Z), S(X, W )) (1.6) This theorem gives a relation between sectional curvature of a manifold and its ambient:
Corollary 1.11. In the notation of Theorem 1.10, if ˜K is the sectional curvature of ˜
M and if {e1, e2} is a basis of the 2-plane π ⊂ TpM , then:
˜
K(π) = K(π) + |Sp(e1, e2)|2˜gp− ˜gp(Sp(e1, e1), Sp(e2, e2)). (1.7)
Remark 1.12. It will be useful to observe that, fixed N ∈ N (M ), taking {e1, e2} a
g-orthonormal basis of the 2-plane π ⊂ TpM made of eigenvectors of AN (such a basis
does exist since AN is a symmetric endomorphism) so that AN(ei) = −kiei, we get:
˜
g(S(ei, ej), N ) = g(AN(ei), ej) = −kiδij, S(e1, e2) = 0.
With the above notation Equation (1.7) reduces to: ˜
K(π) = K(π) − k1k2. (1.8)
1.2
Conformal transformations
In this section we briefly introduce the concept of conformal transformation of Rie-mannian manifolds and then we prove some results about the link between Levi-Civita connections and sectional curvatures of two conformal manifolds.
Definition 1.13. Let M, N two Riemannian manifolds with metric tensors gM, gN. A
conformal transformation between M and N is a diffeomorphism F : M → N such that there exists a strictly positive function λ2 ∈ C∞(M ) for which F∗g
N = λ2gM,
where F∗gN is the pull-back metric of gN through F . In this case we say that M and
N (or their metric tensors) are conformal.
Roughly speaking the definition of conformal transformation says that the metrics measure the angles between vectors in tangent spaces in the same way, even if there is a distortion in their length.
Here we have a classical result that classifies the conformal transformations of Euclidean space (an interesting proof that requires less regularity on the map can be found in [18] and makes use of non-standard analysis):
Theorem 1.14 (Liouville’s classification of conformal maps). The conformal trans-formations of the Euclidean space of dimension n > 2, that is Rn with n > 2 with euclidean metric, are only composition of translations, dilatations, orthogonal trans-formations and spherical inversions.
Now we want to prove a useful formula:
Lemma 1.15. Let M a n-dimensional manifold equipped with two conformal metric tensors g and gλ = λ2g and corresponding Levi-Civita connections ∇ and ∇λ. Then
for all X, Y ∈ T (M ):
where grad denotes the gradient operator in the metric g. This means that the shape operator of (M, gλ) in (M, g) is:
S(X, Y ) = X(log λ)Y + Y (log λ)X − g(X, Y )grad(log λ).
Proof. Let {e1, . . . , en} a local frame that is orthonormal with respect to gλ. Then:
∇λ XY − ∇XY = n X i=1 gλ(∇λXY, ei)ei− λ2g(∇XY, ei)ei.
By the fundamental relations of the Levi-Civita connection with respect to its metric tensor we have: gλ(∇λXY, Z) − λ2g(∇XY, Z) = = 1 2 X(λ 2g(Y, Z)) + Y (λ2g(Z, X)) − Z(λ2g(X, Y ) + λ2g([X, Y ], Z)+ − λ2g([Y, Z], X) + λ2g([Z, X], Y ) − λ21 2 X(g(Y, Z)) + Y (g(Z, X))+ − Z(g(X, Y ) + g([X, Y ], Z) − g([Y, Z], X) + g([Z, X], Y ) = = 1 2 X(λ 2)g(Y, Z) + Y (λ2)g(Z, X) − Z(λ2)g(X, Y ). Then: ∇λ XY −∇XY = 1 2 n X i=1
X(λ2)g(Y, ei) + Y (λ2)g(ei, X) − g(X, Y )ei(λ2)ei =
= λX(λ) n X i=1 λ2 λ2g(Y, ei)ei + λY (λ) n X i=1 λ2 λ2g(X, ei)ei− n X i=1 g(X, Y )λei(λ)ei =
= X(log λ)Y + Y (log λ)X − g(X, Y )grad(log λ),
where the last term is obtained observing that in this local frame gij = λ2δij.
The last claim of the lemma follows by Equation (1.3).
As a first application of the previous lemma, we can calculate the relation between the sectional curvatures of two conformal manifolds:
Proposition 1.16. Let M a n-dimensional manifold equipped with two conformal met-ric tensors g and gλ = λ2g and corresponding sectional curvatures K and Kλ. Then:
K = λ2Kλ+ ∆(log λ), (1.10)
Proof. Substituting formula (1.9) in the definition of the curvature tensor, that is Rλ(X, Y )Z = ∇λ
X∇λYZ − ∇λY∇λXZ − ∇λ[X,Y ]Z, and after a not so short calculation ∗ we
obtain a relation for the curvature tensors:
Rλ(X, Y )Z =R(X, Y )Z + g(∇Xgrad(log λ), Z)Y − g(∇Ygrad(log λ), Z)X+
+ g(X, Y )∇Ygrad(log λ) − g(Y, Z)∇Xgrad(log λ)+
+ (Y log λ)(Z log λ)X − (X log λ)(Z log λ)Y +
− g(grad(log λ), grad(log λ))[g(Y, Z)X − g(X, Z)Y ]+ + [(X log λ)g(Y, Z) − (Y log λ)g(X, Z)]grad(log λ).
(1.11)
Now let π be a 2-plane contained in TpM , we are going to calculate K and Kλ on this
subspace. By the arbitrariness on the choice of the basis generating π, we can choose a couple {E1, E2} that is orthonormal with respect to g (then only orthogonal with
respect to gλ). Hence we obtain:
K(π) = g(R(E1, E2)E2, E1), Kλ(π) = g λ(Rλ(E 1, E2)E2, E1) gλ(E 1, E1)gλ(E2, E2) = 1 λ2g(R λ(E 1, E2)E2, E1).
Thus using Equation (1.11) and the fact that for all f ∈ C∞(M ): grad(f ) = 2 X i=1 g(grad(f ), Ei)Ei = 2 X i=1 (Eif )Ei, and: ∆f = tr(∇grad(f )) = 2 X i=1 g(∇Eigrad(f ), Ei),
∗We made use of properties like the compatibility of the metric tensor and the Levi-Civita
connec-tion and the fact that:
g(W, grad(f )) = W f ∀W ∈ T (M ), f ∈ C∞(M ), g(grad(f ), grad(f )) = (grad(f ))f ∀f ∈ C∞(M ),
we get the claim:
g(Rλ(E1, E2)E2, E1) = K(π) − g(∇E1grad(log λ), E1) − g(∇E2grad(log λ), E2)+
+ (E2log λ)2− g(grad(log λ), grad(log λ))+
+ E1log λg(grad(log λ), E1) = = K(π) − 2 X i=1 g(∇Eigrad(log λ), Ei) + (E1log λ) 2 + (E2log λ)2+ − g(grad(log λ), grad(log λ)) = = K(π) − 2 X i=1 g(∇Eigrad(log λ), Ei) = = K(π) − ∆ log λ.
1.3
Spherical Inversions and Curvature
This section is devoted to the calculation of the change of quantities like the second fundamental form and the sectional curvature under conformal transformations of the Euclidean space, that are completely known for dimension greater than 2 by Liouville’s Theorem 1.14. In particular we concentrate ourselves to the less trivial conformal transformations, that is the spherical inversions.
In this section we shall adopt the so called parametrization point of view, that is we consider immersions ϕ : Σ → Rn with Σ abstract 2-dimensional manifold. This will allow us to express some calculation in a more explicit way. Moreover we are going to use the representation of the second fundamental form as described in Remark 1.8; our treatment of the problem is taken from [4], a work that will be very important later on.
In this context we will denote the standard Euclidean product as h. . . , . . . i and the identity operator by id (both in R2 and in Rn) and we can write the metric tensor G on Σ in term of the differential dϕ : R2 → Rn as:
G = dϕtdϕ : Σ → R2×2, G = (gij). (1.12)
Also we are able to write the projection onto the tangent space Im (dϕ) as:
π>= dϕG−1dϕt: Σ → Rn×n, (1.13)
in fact if w ∈ (Im (dϕ))⊥ we see that:
hdϕG−1dϕtw, dϕvi = hdϕtw, vi = hw, dϕvi = 0 ∀v ∈ R2,
and on the other hand for all w ∈ Rn of the form w = dϕv we have dϕG−1dϕtw =
In particular, we are able to write explicitly the components of the second fundamental form (in the representation chosen before) as:
Aij =⊥ ∂2ϕ ∂xi∂xj = (id − dϕG−1dϕt) ∂ 2ϕ ∂xi∂xj. (1.14)
Let us consider now a spherical inversion I; for simplicity, we assume the inversion is the one given by the standard sphere Sn−1, that is:
I : Rn\ {0} → Rn\ {0}, I(x) = x
|x|2. (1.15)
We immediately note that the differential dI : Rn→ Rn has the properties:
(dIx)ij = 1 |x|2 δij − 2xixj |x|2 , dIxdIxt = dI t xdIx = 1 |x|4id. (1.16)
Finally, we will denote by µ the canonical surfaces measure induced by ϕ on Σ. Assuming 0 6∈ ϕ(Σ), we want now to compare the geometric properties of the im-mersions ϕ and ϕλ = I(ϕ) (whose related quantities will be denoted by a λ at the
exponent). We obtain:
Proposition 1.17. In the notation of Section 1.3, we have: gijλ = 1 |ϕ|4gij, µ λ = 1 |ϕ|4µ, Aλij = dIϕ Aij − 2gij |ϕ|2ϕ ⊥ , |Hλ|2µλ = |H|2µ + ∆ log(|ϕ|2)µ. Proof. We have (dϕλ)
x = dIϕ(x)dϕx (from now on we will lighten the notation by
dropping the dependence of the differential on the point), so: Gλ = (dϕλ)tdϕλ = 1
|ϕ|4G,
then the claim on gλ
ij and µλ follows. Thus we have (Gλ)
−1 = |ϕ|4G−1 and using the
properties of dI we have: Aλij = (id − dϕλ(Gλ)−1(dϕλ)t) ∂ 2ϕλ ∂xi∂xj = = |ϕ|4dIϕ(id − dϕG−1dϕt)dIϕt d2I∂ϕ ∂xj ∂ϕ ∂xi + dIϕ ∂2ϕ ∂xi∂xj = = |ϕ|4dIϕ(id − dϕG−1dϕt) 1 2d(dI t ϕdϕ)∂ϕ ∂xj ∂ϕ ∂xi + 1 |ϕ|4 ∂2ϕ ∂xi∂xj = = |ϕ|4dIϕ(id − dϕG−1dϕt) − 2 |ϕ|6ϕ n X s=1 ∂fs ∂xj ∂fs ∂xi + 1 |ϕ|4 ∂2ϕ ∂xi∂xj = = dIϕ(id − dϕG−1dϕt) ∂2ϕ ∂xi∂xj − 2gij |ϕ|2ϕ ,
that is the claim about Aλij.
For the last claim we first need to see that, writing A as a bilinear form A(v, w) = Pn
i,j=1hAij, νiviwj where ν is fixed in N (ϕ(Σ)) and denoting B(v, v) = A(v, v) −
1/2tr(A)|v|2 its tracefree part, then:
|Bλ|2µλ = |B|2µ, (1.17)
where one has |B|2 = 1/4(k
1− k2)2 if e1, e2 are such that A(ei, ei) = −ki. In fact, if e
is such that A(e, e) = −ke, and |e| = 1, then (with η = dIν):
hAλ, ηi = dIϕ Aij − 2gij |ϕ|2ϕ ⊥ , η = = |ϕ|2 Aij − 2gij |ϕ|2ϕ ⊥ , ν ,
where we used the fact that I is conformal. So one has Aλ(e, e) = −|ϕ|2k
e− 2hϕ⊥, νi =
−kλ
e, thus |Bλ|2 = |ϕ|4|B|2 and (1.17) holds. Now, in the notation of Proposition 1.16
we have λ2 = 1/|ϕ|4 and by the same proposition we get:
Kλµλ = Kµ + ∆ log(|ϕ|2)µ. (1.18)
So, having in general that |H|2 = |B|2+ K by Equation (1.5), the last claim follows
immediately from (1.18).
1.4
The Stereographic Projection
Here we give a simple proof that the stereographic projection is conformal as seen as a map between the Riemannian manifolds Sn\ {p} and Rn with their natural metric
tensors, that is, we see that the abstract differentiable submanifold diffeomorphic to Rn has two conformal metrics, one that makes it isometric to Sn\ {p}, the other that makes it isometric to the Euclidean n-dimensional space. Actually we are interested only in the case n = 3 and we give the result in this case, but this can be improved to arbitrary n.
Proposition 1.18. The stereographic projection π : (S3\ {p}, gS3) → (R3, g
R3), where p = (0, 0, 0, 1), given by: π(x, y, z, t) = x 1 − t, y 1 − t, z 1 − t is conformal.
Proof. We verify that the inverse π−1 preserves angles. Recall that: π−1(X, Y, Z) = 2X 1 + |P |2, 2Y 1 + |P |2, 2Z 1 + |P |2, |P |2− 1 1 + |P |2 ,
with |P |2 = X2+ Y2+ Z2.
Let us consider an angle θ in TPR3, that is, we fix a point P ∈ R3 and two unitary
vectors v, w in TPR3 that give that angle; we identify it with the angle described by
the intersection of the straight lines r = {P + tv}t∈R and s = {P + tw}t∈R. We want to
calculate the angle ϕ given by dπp−1(v) and dπ−1p (w) and check that it coincides with θ.
Modulo rotations and translations of R3, that are clearly conformal transformations, we can suppose that P = (x0, 0, 0) and that the plane generated by the lines r, s is
the one generated by the first two elements of the canonical basis. In this way we can write r = {(x0+ t, 0, 0)}t∈R and s = {(x0+ tcosθ, tsenθ, 0)}t∈R. Now the lines r, s are
mapped by π−1 into S3\ {p} as the curves:
α(t) = 1 1 + (x0 + t)2 2(x0+ t), 0, 0, (x0+ t)2− 1 , β(t) = 1 1 + (x0+ tcosθ)2+ (tsenθ)2 2(x0+ tcosθ), 2tsenθ, 0, (x0+ tcosθ)2+ (tsenθ)2− 1 .
By calculating the derivatives α0(0), β0(0) we obtain precisely dπ−1p (v) and dπp−1(w), that are: α0(0) = 1 (1 + x2 0)2 2(1 − x20), 0, 0, 4x0 , β0(0) = 1 (1 + x2 0)2
2cosθ(1 − x20), 2senθ, 0, 4x0cosθ
. Now a standard calculation gives the claim:
cosϕ = hα
0(0), β0(0)i
|α0(0)||β0(0)| = cosθ.
1.5
Rectifiable Sets and Area Formula
In this section we recall a few basic definitions of Geometric Measure Theory and a very important relation, known as the Area Formula. With x ∈ Rn+k and l > 0, we
shall adopt the notation:
ηx,l : Rn+k → Rn+k, ηx,l(y) =
y − x
l . (1.19)
Definition 1.19. Let M ⊂ Rn+k a set. We say that M is countably n-rectifiable if
M ⊂ M0∪
S∞
j=1Fj(Rn), with Hn(M0) = 0 and Fj : Rn→ Rn+k lipschitz, where Hn is
Remark 1.20. We remind that, as equivalent definitions of n-rectifiable set, one can consider the set M as: M = M0∪S
∞
j=1Fj(Aj) with Fj : Aj ⊂ R
n → Rn+k lipschitz, or
M ⊂ S∞
j=0Nj with Hn(N0) = 0 and {Nj}j≥1 C1 n-dimensional submanifolds of Rn+k,
or again M = t∞j=0Mj with Hn(M0) = 0 and Mj ⊂ Nj ∀j ≥ 1 (with Nj as in the
previous case).
For a rectifiable set it is well defined a sort of tangent space as:
Definition 1.21. Let M ⊂ Rn+k be Hn-measurable and let θ : M → R>0 a Hn-locally
integrable function. Let P be a n-dimensional subspace of Rn+k. We say that P is the approximate tangent space of M in x ∈ M with respect to θ if:
lim
l→0
Z
ηx,l(M )
ϕ(y)θ(x + ly) dHn(y) = θ(x) Z
P
ϕ(y) dHn(y) ∀ϕ ∈ Cc0(Rn+k). (1.20) If such a P exists, it will be denoted as TxM .
In fact it holds the:
Theorem 1.22. Let M a Hn-measurable set. Then: M in countably n-rectifiable if
and only if there exists a Hn-integrable function θ : M → R
>0 such that for Hn-almost
all x ∈ M there exists TxM with respect to θ.
Now, let us consider a lipschitz function f : U → R with U ⊂ Rn+k open and let M ⊂ U n-rectifiable and Hn-measurable. We are able to define weak notions of
gradient and differential as:
Definition 1.23. With the notation above of Section 1.5 and of Remark 1.20, we define the gradient on M of f as:
∇Mf (x) = ∇Njf (x), (1.21)
if x ∈ Nj and f is differentiable at x (we note that this is Hn-ae possible by Rademacher’s
Theorem). Moreover, we define the differential on M of f at x as:
dMfx : TxM → R, dMfx(τ ) = hτ, ∇Mf (x)i, (1.22)
if the above objects exist (again this is clearly possible Hn-ae).
If f is vector-valued, that is f = (f1, . . . , fN) : U → RN, then the differential on M is
defined as: dMfx: TxM → RN, dMfx(τ ) = N X j=1 hτ, ∇Mfj(x)ie j, (1.23)
with {e1, . . . , eN} canonical basis of RN, and for Hn-almost all x ∈ M is defined the
divergence on M as: divMf (x) = n+k X j=1 hej, ∇Mfj(x)i. (1.24)
Finally, if f is vector-valued, that is f : U → RN, with N ≥ n, it is Hn-ae defined the function: JMf : M → R, JMf (x) = p det((dMf x)tdMfx). (1.25)
We will need another important notion:
Definition 1.24. We say that E ⊂ M ⊂ Rn+k+1, with M a (n + 1)-submanifold
of Rn+k+1, is a set of locally finite perimeter in U, with U open in M , if it is Hn+1
-measurable and the characteristic function χE ∈ BVloc(M, Hn+1
¬
M ).
Remark 1.25. If E ⊂ M is a set of locally finite perimeter, then there exist a Radon measure µE on M and a function ν : M → T (M ) that is µE-measurable and |ν(x)| = 1
for µE-ae x ∈ M such that:
Z E div(g) dHn+1 = − Z M hg, νi dµE ∀g ∈ Cc1(U ; T (M )). (1.26)
And we can define:
Definition 1.26. Let E be a set of locally finite perimeter. Then its reduced boundary is the set: ∂∗E = x ∈ U : ∃ lim ρ→0 R Bρ(x)ν dµE µE(Bρ(x))
:= νE(x) and it has lenght 1
, (1.27)
with the notation of Remark 1.25. We have the important result:
Theorem 1.27 (De Giorgi). Let E ⊂ M ⊂ Rn+k+1 a locally finite perimeter set
contained in a (n + 1)-dimensional manifold M . Then ∂∗E is countably n-rectifiable and µE = Hn
¬
∂∗E. Moreover for each x ∈ ∂∗E the approximate tangent space Tx
exists with multiplicity 1 and:
Tx = {y ∈ TxM : hy, νE(x)i = 0}.
We are ready to give the announced result, that will turn out to be extremely important:
Theorem 1.28 (Area Formula). If f : U → RN is lipschitz, with U ⊂ Rn+k open, and
if M ⊂ U n-rectifiable and Hn-measurable with N ≥ n, if g : M → R is Hn-measurable then: Z RN Z f−1(y) g dH0dHn(y) = Z M gJMf dHn. (1.28)
1.6
Varifolds
Here we recall some basic definitions and results about the theory of varifolds, such as compactness and regularity theorems.
In this section the letter π will denote the projection:
π : U × G(n, n + k) 3 (x, S) 7−→ x ∈ U, U ⊂ Rn+k. (1.29)
Definition 1.29. A n-varifold V in U ⊂ Rn+k is a Radon measure on the
Grassman-nian Gn(U ) = U × G(n, n + k). We call weight of V the Radon measure µV on U
defined by:
µV(A) = V (π−1(A)), A ⊂ U. (1.30)
And we call mass of V the number M(V ) = µV(U ) = V (Gn(U )).
Moreover we say that a sequence of varifolds Vk converges to a varifold V if there is
convergence Vk→ V in the sense of Radon measures.
We also define, if U, ˜U are open sets, if f : U → ˜U is C1 and f |
suppµV∩U is proper, the
image varifold f#V on ˜U as:
f#V (A) = Z F−1(A) JSf (x) dV (x, S), (1.31) with A ⊂ Gn( ˜U ), F : {(x, S) ∈ Gn(U )|JSf (x) 6= 0} → Gn( ˜U ) defined by F (x, S) = (f (x), dfx(S)) and JSf (x) = (det(dfx|∗Sdfx|S))1/2.
Definition 1.30. Let M, ˜M countably n-rectifiable sets and Hn-measurable, θ, ˜θ :
M → R>0 a Hn-locally integrable functions. Let us define an equivalence relation as
(M, θ) ∼ ( ˜M , ˜θ) if and only if Hn(M \ ˜M ∪ ˜M \ M ) = 0 and θ = ˜θ Hn-ae on M ∩ ˜M . We call the equivalence class v(M, θ) a n-rectifiable varifold ; θ is called multiplicity and v(M, θ) is said to be integer if θ(x) ∈ N>0 Hn-ae. The associated measure is the
Radon measure µ = Hn ¬θ.
Moreover we say that a sequence of rectifiable varifolds v(Mk, θk) converges to a
recti-fiable varifold v(M, θ) if there is convergence µk → µ in the sense of Radon masures.
With this notion of convergence we will denote with Vn(M ) the closure of the space of
n-rectifiable varifolds with support contained in M .
Of course a n-dimensional submanifold Σ of Rn+k defines a rectifiable varifold de-noted by v(Σ) = v(Σ, 1). Also a rectifiable varifold v(M, θ) defines a varifold V taking V (A) = µ(π(T M ∩ A)) with T M = {(x, TxM )|x ∈ M : ∃TxM } (so note that in this
case µV = µ). We will usually denote the varifold corresponding to a rectifiable one
with the same symbol v(M, θ). It is an important issue to find out under which con-ditions the contrary holds, that is when a general varifold is a rectifiable one. In order to give a first answer we need a new definition:
Definition 1.31. If S ∈ G(n, n + k) and V is a varifold on U , we say that S is the tangent space of V at x with multiplicity θ(x) if:
Vx,l −→ θ(x)v(S) := θ(x)v(S, 1) as l → 0 as Radon measures, (1.32)
where v(S, 1) denotes the corresponding general varifold given by the rectifiable vari-fold, while Vx,l is the varifold on Rn+k defined by:
Vx,l(A) =
1
lnV ({(x + ly, R)|(y, R) ∈ A} ∩ Gn(U )) A ⊂ Gn(R
n+k). (1.33)
So we can now state the:
Theorem 1.32 (First Theorem of Rectifiability). Let V a n-varifold on U such that for µV-ae x ∈ U there exists the tangent space Tx with multiplicity θ(x), then V is
rectifiable. This means that M = {x ∈ U |∃Tx with multiplicity θ(x)} is countably
n-rectifiable and Hn-measurable, θ is Hn locally integrable on M , and V = v(M, θ). Now we recall the notion of first variation of varifolds. Let us consider U ⊂ Rn+k
open and a family of diffeomorphisms {Φt : U → U }−1<t<1 such that:
Φ(t, x) := Φt(x) : (−1, 1) × U → U is C2,
Φ0(x) = x ∀x ∈ U,
Φt(x) = x ∀t ∀x ∈ U \ K with K ⊂ U compact,
(1.34)
and call X(x) = ∂Φ(t,x)∂t |t=0 that is a vector field with compact support contained in K.
If K(U, Rn+k) is the space of vector fields with compact support contained in U , we
define:
Definition 1.33. If V is a varifold we define its first variation the linear functional δV : K(U, Rn+k) → R defined by:
δV (X) = d dtM Φt#(V ¬ Gn(K)) t=0 , (1.35)
where {Φt} is a family of diffeomorphisms originated by X as in Equation (1.34). It
turns out that:
δV (X) = Z
Gn(U )
divS(X) dV (x, S), (1.36)
that in case of n-rectifiable varifold V = v(M, θ) it simplifies to: δV (X) =
Z
U
divM(X) θ dHn. (1.37)
A µV locally integrable function H : U → Rn+k is called generalized mean curvature in
U if:
δV (X) = −n Z
U
where the presence of the constant n is due to the fact that in the case of a n-manifold then H would be the classical mean curvature vector. We say that V is stationary in U if δV = 0, that is if H = 0 (in case it exists).
Moreover we will denote by ||δV || the total variation measure of the functional δV and we say that V has locally bounded first variation if:
∀W ⊂⊂ U ∃C : |δV (X)| ≤ C sup
x∈U
|X(x)| ∀X ∈ K(U, Rn+k) : supp(X) ⊂ W. (1.39)
It is useful to remind that the first variation as seen as a functional:
δ : Vn(M ) × K(U, Rn+k) → R (1.40)
is continuous with respect to the natural product topology of the space Vn(M ) ×
K(U, Rn+k). With these notions we can give a new result of rectifiability:
Theorem 1.34 (Second Theorem of Rectifiability). Let V a n-varifold on U with locally bounded first variation such that for µV-ae the function Θn(µV, x) exists and
it’s > 0, where Θn(µV, x) = limρ→0µV(U ∩ Bρ(x))/(ωnρn), then V is rectifiable (in the
same sense of Theorem 1.32).
Having the rectifiability of a varifold is the first step toward the study of its regu-larity, as stated for example in next theorem.
Theorem 1.35 (Allard’s Regularity Theorem). Let δ ∈ (0, 1/2) and p > n integer, let v(M, θ) a rectifiable n-varifold on U with generalized mean curvature H and with:
θ ≥ 1 µV − ae in U, 0 ∈ suppµV, Bρ(0) ⊂ U, µV(Bρ(0)) ωnρn ≤ 1 + δ, Z Bρ(0) |H|p 1/p ρ1−np ≤ δ,
Then there exists γ ∈ (0, 1), an isometry Q of Rn+k and a C1,1−np
function u : ¯Bn
γρ(0) →
Rk such that:
u(0) = 0 supp(V ) ∩ Bγρn (0) = Q(graph(u)) ∩ Bγρn (0).
The last tools we will need concern convergence of varifolds. First we state a classical result.
Theorem 1.36 (Compactness of Varifolds). Let Vn = v(Mn, θn) be a sequence of
rectifiable varifolds in U such that:
(1) sup
n
(2) ∃Θ(Vn, x) ≥ 1 on U \ An : µVn(An∩ W ) → 0 ∀W ⊂⊂ U.
Then there exists a subsequence Vnk converging to a rectifiable varifold V with locally
bounded first variation with the properties that:
∃Θ(µV, x) ≥ 1 µV-ae in U,
lim inf
n ||δVn||(W ) ≥ ||δV ||(W ) ∀W ⊂⊂ U.
Moreover if each Vnk is integer, then V is integer too.
Remark 1.37. It is very important to observe that if in Theorem 1.36 the varifolds Vn are integer, then the hypothesis (2) is automatically satisfied (with sets such that
µVn(An) = 0).
Finally, it will be useful to remind the concept of F-metric (see [19], page 66), defined as follows.
Definition 1.38. The F-metric on Vn(M ) is defined as:
F(V, W ) = sup{V (f ) − W (f ) : f ∈ Cc(Gn(Rn+k)), |f | ≤ 1, Lip(f ) ≤ 1}. (1.41)
And we have the useful:
Lemma 1.39 ([19], page 66). In sets Vn(U ) ∩ {V : M(V ) ≤ C < +∞} with U ⊂ Rn+k
open, the convergence of varifolds is equivalent to the convergence in the F-metric.
1.7
Currents
This section contains some basic definitions and results about convergence in the theory of currents.
Let us consider U ⊂ Rn+k open. We denote by An(U ) the space on n-differential forms
on U and by Dn(U ) the subspace of n-forms with compact support. So an element
ω ∈ Dn(U ) is written as: ω = X i1,...,in ai1,...,indx i1 ∧ · · · ∧ dxin :=X α aαdxα, with ai1,...,in ∈ C ∞ c (U ).
Definition 1.40. If ωj = Pαajαdxα is sequence in Dn(U ), we say that ωj converges
in Dn(U ) to ω = P
αaαdx
α ∈ Dn(U ) if there exists a compact K ⊂ U such that
supp(aj
α) ⊂ K ∀α ∀j and Dβajα→ Dβaα uniformly ∀α, β multiindexes.
The space Dn(U ) of the n-dimensional currents on U is the space of continuous linear
functionals on Dn(U ). If T ∈ D
n(U ) the boundary ∂T ∈ Dn−1(U ) is the current defined
by:
and if W ⊂ U open, then:
MW(T ) = sup
|ω|≤1, suppω⊂W
T (ω), (1.43)
and the mass of T is M(T ) := MU(T ).
We say that a sequence of n-currents Tj converges in Dn(U ) (or in the sense of currents)
to a n-current T if for each ω ∈ Dn(U ) it occurs that T
j(ω) → T (ω).
Remark 1.41. If Tj → T is a convergence in the sense of currents, then the mass is
lower semicontinuous in the sense that: lim inf
j MW(Tj) ≥ MW(T ) ∀W ⊂ U open. (1.44)
Moreover we observe that if a sequence of currents Tj is such that for each W ⊂ U
open there exists a constant C such that MW(Tj) ≤ C for all j, then applying
Banach-Alaoglu’s Theorem there exists a subsequence convergent in the sense of currents to a current T .
Remark 1.42. Note that if T ∈ Dn(U ) is such that MW(T ) < +∞ for all W ⊂⊂ U ,
then by Riesz representation theorem there exist a Radon measure µT on U and a
function ¯T : U → Λn(Rn+k) µT-measurable with | ¯T | = 1 µT-ae such that:
T (ω) = Z
U
hω(x), ¯T (x)i dµT ∀ω ∈ Dn(U ). (1.45)
In such a case we will always denote by µT the Radon measure given by applying Riesz
theorem.
Definition 1.43. A current T ∈ Dn(U ) is a n-rectifiable integer current (or simply
integer current ) if there exist M ⊂ U countably n-rectifiable and Hn-measurable,
θ : M → N>0a Hn-locally integrable function and ξ : M → Λn(Rn+k) a Hn-measurable
function such that ξ(x) = τ1∧· · ·∧τnfor Hn-ae x ∈ M with {τ1, . . . , τn} an orthonormal
basis of TxM (if it exists) and it holds:
T (ω) = Z
M
hω(x), ξ(x)iθ(x) dHn(x) ∀ω ∈ Dn(U ). (1.46)
The function ξ is called orientation and θ multiplicity. In such a case we write T = t(M, θ, ξ) = [|M |] and we are able to associate an integer varifold, that is denoted |T | = v(M, θ). The space of n-dimensional integer currents in Rn+k with support
contained in M , with M that is (n + k1)-rectifiable and H(n+k1)-measurable, k1 ≤ k,
will be denoted by In(M ). The space of T ∈ In(M ) such that ∂T = 0 will be denoted
If M is an oriented n-dimesional manifold in Rn+k, it defines a n-rectifiable integer current (with multiplicity 1) denoted by [|M |] as:
[|M |](ω) = Z
M
hω(x), ξ(x)i dHn(x) ∀ω ∈ Dn(U ), (1.47)
where ξ(x) = ±τ1 ∧ · · · ∧ τn ∈ Λn(TxM ) is a field inducing the given orientation on
M and {τ1, . . . , τn} is an orthonormal basis of TxM , and h·, ·i denotes here the natural
duality.
As in the case of varifolds, it is often useful to know when a general current T is rectifiable, that is when it can be expressed as T = t(M, θ, ξ) for some M, θ, ξ. Classical answers are next theorems.
Theorem 1.44 (Rectifiability of Currents). Let T ∈ Dn(U ) such that:
∃C : MW(T ) + MW(∂T ) < +∞ ∀W ⊂⊂ U, ∃Θ∗n(µT, U, x) := lim sup ρ→0 µT(U ∩ Bρ(x)) ωnρn > 0 for µT-ae x ∈ U.
Then T is rectifiable, that is T = t(M, θ, ξ) for some M, θ, ξ as in Definition 1.43. Moreover one has µT = Hn
¬
θ.
Theorem 1.45 (Rectifiability of Boundaries). Let T ∈ In(U ) such that MW(T ) < +∞
for all W ⊂⊂ U , with U open. Then ∂T ∈ Zn(U ), in particular ∂T is integer rectifiable.
Now we describe the possible senses of convergence of currents. We begin from the next two results.
Theorem 1.46 (Compactness of Integer Currents). Let Tj be a sequence of integer
currents in Dn(U ) such that:
sup
j
MW(Tj) + MW(∂Tj) < +∞ ∀W ⊂⊂ U.
Then there exist an integer current T ∈ Dn(U ) and a subsequence Tjk such that Tjk → T
in the sense of currents as k → ∞.
Theorem 1.47 (Strong Completeness of Integer Currents). The space In(U ) is
com-plete with respect to the topology of the family of seminorms {MW}W ⊂⊂U. This means
that if a sequence {Tj} ⊂ In(U ) is such that limj,kMW(Tj − Tk) = 0 ∀W ⊂⊂ U , then
there exists T ∈ In(U ) such that limjMW(Tj − T ) = 0 ∀W ⊂⊂ U .
These notions of convergence are related to some definitions of appropriate metrics as follows.
Definition 1.48. The flat metric on In(U ) is defined for all S, T ∈ In(U ) as:
F (S, T ) = inf{M(P ) + M(Q) : S − T = P − ∂Q, P ∈ In(U ), Q ∈ In+1(U )}. (1.48)
Moreover we define the F-metric on In(U ) for all S, T ∈ In(U ) as:
F(S, T ) = F (S, T ) + F(|S|, |T |), (1.49)
where the second symbol F is the F-metric on Vn(U ) and |S|, |T | are the varifolds
associated to the integer currents S, T .
These metrics are related to the convergence in the sense of currents as stated in the next results.
Lemma 1.49. Let Tj be a sequence in In(U ) such that supjMW(Tj)+MW(∂Tj) < +∞
for all W ⊂⊂ U and let T ∈ In(U ). Then Tj → T in the sense of currents if and only
if Tj → T in the topology induced by the flat topology, that is F (Tj, T ) → 0.
Remark 1.50. Given all the definitions of possible metrics on currents and varifolds, we collect here some simple results and comparisons between them. In particular it holds:
F (T, 0) ≤ M(T ) ∀T ∈ In(U ), (1.50)
F(|S|, |T |) ≤ M(S − T ) ∀S, T ∈ In(U ), (1.51)
F (S, T ) ≤ F(S, T ) ≤ 2M(S − T ) ∀S, T ∈ In(U ). (1.52)
The mass of currents M is continuous in the F-metric topology, while, as we already said, it is lower semicontinuous in the flat metric topology.
We conclude the section proving a tool:
Lemma 1.51. Let K ⊂ Zk(M ) be compact in the F-metric topology. Then for every
there is δ such that:
∀S ∈ K ∀T ∈ Zk(M ) : M(T ) < M(S) + δ, F (S, T ) ≤ δ ⇒ F(S, T ) ≤ .
Proof. Note that in general limi→∞F(S, Ti) = 0 if and only if limi→∞M(Ti) = M(S)
and limi→∞F (S, Ti) = 0 for Ti, S ∈ Zk(M ) (see [19], page 63 and 68). Now suppose
by contradiction that there exist sequences {Si} ⊂ K and Ti ⊂ Z2(M ) such that:
M(Ti) < M(Si) + δi, F (Si, Ti) ≤ δi, δi → 0, and F(Si, Ti) > C0 > 0 ∀i.
By compactness Si → S ∈ K in the F-metric, so M(Si) → M(S) and F (Si, S) → 0.
Then F (Ti, S) ≤ F (Ti, Si) + F (Si, S) → 0, thus by lower semicontinuity of the mass:
M(S) ≤ lim inf
i M(Ti) ≤ lim infi M(Si) + δi = M(S).
Then (after passing to subsequence if needed) we have F(Si, Ti) ≤ F(Si, S)+F(S, Ti) →
1.8
CW-Complexes
This last section mainly contains definitions and basic results about CW-complexes and cubical singular homology. We will also provide a specific terminology that we will use later, in fact the setting of CW-complexes will be used to simplify some statements of Chapter 3. So the exposed concepts will be adapted to what we will need later. The references for this section are [9] and [14].
Let us start by recalling the notion of cubical singular homology. The symbol In
will denote the n-dimensional cube [0, 1]n and X will be a topological space that is
Hausdorff and with a topology that admits a countable basis.
Definition 1.52. A singular n-cube is a continuous map ϕ : In → X. Qn(X) will
denote the free Z-module generated by all singular n-cubes in X. A singular n-cube ϕ is degenerate if the function ϕ(x1, . . . , xn) does not depend on xi for at least an index
i ∈ {1, . . . , n}. Dn(X) ⊂ Qn(X) is the subgroup of degenerate singular n-cubes and
the quotient group Cn(X) = Qn(X)/Dn(X) is the group of cubical singular n-chains
in X (or just n-chains).
The group {Cn(X)}n are free Z-modules and we now define on them a structure of
algebraic complex. If ϕ is a singular n-cube, we get the singular (n-1)-cubes:
Aiϕ : In−1 → X Aiϕ(x1, . . . , xn−1) = ϕ(x1, . . . , xi−1, 0, xi, . . . , xn−1), (1.53)
Biϕ : In−1 → X Biϕ(x1, . . . , xn−1) = ϕ(x1, . . . , xi−1, 1, xi, . . . , xn−1). (1.54)
And we define:
Definition 1.53. If ϕ is a singular n-cube with n > 0, then: ∂nϕ =
n
X
i=1
(−1)i Aiϕ − Biϕ. (1.55)
It easily turns out that:
∂n−1∂nϕ = 0, ∂n(Dn(X)) ⊂ Dn−1(X),
so the maps ∂n pass to the quotient and define the boundary of the complex made of
the groups Cn(X).
Definition 1.54. The complex {(Cn(X), ∂n)} is the complex of chains of X and we
denote Zn(X) = ker(∂n) ⊂ Cn(X) the group of n-dimensional cycles and Bn(X) =
Im (∂n+1) ⊂ Zn(X) the group of n-dimensional boundaries. And the n-dimensional
cubical singular homology group of X (or simply n-dimensional homology group of X ) is the quotient Hn(X) = Zn(X)/Bn(X) (with the 0-dimensional group defined as
H0 = C0(X)).
Definition 1.55. If f : X → Y is a continuous function between two topological spaces, it induces a homomorphism f# : Qn(X) → Qn(Y ) as:
f#(ϕ) = f ϕ. (1.56)
Since f#(Dn(X)) ⊂ Dn(Y ), the map f# induces a homomorphism between the group
of chains (denoted with the same symbol) as:
f#: Cn(X) → Cn(Y ) f#( ¯ϕ) = f ϕ, (1.57)
where the bars denote the class both in the quotient Cn(X) and in Cn(Y ). Since
f# : {(Cn(X), ∂nX)} → {(Cn(Y ), ∂nY)} is a homomorphism of complexes, it induces a
homomorphism between the homology groups as:
f∗ : Hn(X) → Hn(Y ) f∗([ϕ]) = [f ϕ], (1.58)
where the square parentheses denote the class both in the quotient Hn(X) and in
Hn(Y ).
Since the homology group Hn(M ) of a compact connected orientable n-dimensional
manifold M without boundary is isomorphic to Z (see [9], Theorem 3.26), we can define:
Definition 1.56. If f : M → N is continuous between two compact connected ori-entable n-dimensional manifolds M, N without boundary, then we call fundamental classes the elements [M ], [N ] that alone generate the groups Hn(M ) and Hn(N ) and
the degree of f is the number deg(f ) ∈ N>0 such that:
f∗([M ]) = deg(f )[N ].
From these definitions we get some results of isomorphisms adding hypotheses to the function f .
Lemma 1.57. Let f, g : X → Y two continuous maps that are homotopic, then the homomorphisms induced in homology coincide, that is f∗ = g∗.
From which we get:
Theorem 1.58 (Homotopic Invariance). If two topological space X, Y are homotopi-cally equivalent, then Hn(X) ' Hn(Y ) for all n.
Now we need the concept of relative homology, in order to define cell complexes. From now on let A ⊂ X be a topological subspace, thus the inclusion defines a homo-morphism i# : Cn(A) → Cn(X) and we have the:
Definition 1.59. The quotient group Cn(X, A) = Cn(X)/i#(Cn(A)) is called group
of n-dimensional chains of the pair (X,A). Since the boundary ∂X
n(i#(Cn(A))) ⊂ i#(Cn−1(A)), then it induces a boundary:
∂n0 : Cn(X, A) → Cn−1(X, A). (1.59)
Then the n-dimensional cubical singular relative homology group of the pair (X,A) (or simply n-dimensional relative homology group of the pair (X,A)) is the group Hn(X, A) = Zn(X, A)/Bn(X, A) := ker(∂n0)/ Im (∂
0
n+1). And we denote the
homol-ogy complex as {(Hn(X, A), ∂∗)}.
Now we can move to the notion of cell and cell complex.
Definition 1.60. Let E ⊂ X be a closed subset such that X \ E is a disjoint union of open subsets en
l, l ∈ Λ. If enl is homeomorphic to the open n-cube ˚In for each l ∈ Λ,
then each enl is called an open n-cell (or simply n-cell ). It is also assumed that for each l ∈ Λ there exists a continuous map fln: In → en
l such that f n
l |˚In is a homeomorphism
with en
l and fln(∂In) ⊂ E. We define this procedure of pasting cells as adjoining cells.
A structure of CW-complex is defined on X by an ascending sequence of closed subsets: X0 ⊂ X1 ⊂ X2 ⊂ . . .
with the properties that:
(i) X0 has the discrete topology,
(ii) for n > 0, Xn is obtained from Xn−1 by adjoining n-cells as defined above,
(iii) X = ∪i≥0Xi,
(iv) a subset E is closed if and only if E ∩ en is closed in en for all n for all n-cells en.
Each Xn is called n-skeleton and the points of X0 are called vertices. A CW-complex
is finite (or infinite) if the number of its cells is finite (or infinite). A subset A of a CW-complex X is a subcomplex if A is union of cells of X and if for any cell en ⊂ A then en⊂ A.
From now on we consider X a CW-complex; we are going to define a particular homology on it. Definition 1.61. We define: ˜ Cn(X) = Hn(Xn, Xn−1), dn: ˜Cn(X) → ˜Cn−1(X), (1.60) as: dn = ∂∗ with Hn(Xn, Xn−1) ∂∗ −→ Hn−1(Xn−1, Xn−2). (1.61)
This defines a complex {( ˜Cn(X), dn)} with homology ˜Hn(X).
And we have the important result that:
Remark 1.63. It is useful to observe that if X is a CW-complex which is a n-dimensional manifold, then Hk(X) = 0 for all k > n. Moreover if X has no m-cells for
some m, then Hm(X) = 0.
Finally, now having all the results and definitions we need, we just introduce some notation and terminology about specific CW-complexes we are going to built and that we will use in Chapter 3.
For each j ∈ N the symbol I(1, j) denotes the CW-complex defined on I whose 1-cells and 0-cells are:
[(n − 1)3−j
, n3−j] : n ∈ {1, . . . , 3j} = [0, 3−j], [3−j, 2 · 3−j], . . . , [1 − 3−j, 1] , [n3−j] : n ∈ {0, . . . , 3j} = [0], [3−j], . . . , [1 − 3−j], [1] ,
where we denoted with the square parentheses the image of an adjoint cell as described in Definition 1.60 (we will usually identify a map that defines a cell with its image). Moreover we define the CW-complex I(n, j) on In as:
I(n, j) = I(1, j) ⊗ · · · ⊗ I(1, j) n times, (1.62)
in the sense that the cells of I(n, j) are given by functions having as image products of images of functions defining cells of I(1, j). We will denoted such a function as α = α1⊗ · · · ⊗ αn and we note that α is a p-cell if Pni=1dimαi = p. The subcomplex
generated by a single cell α is denoted hαi.
Remark 1.64. Note that the boundary homomorphism is now very intuitive:
∂(α1⊗ · · · ⊗ αn) = n X i=1 (−1)σ(i)α1⊗ . . . ⊗ ∂αj⊗ · · · ⊗ αn, with: σ(i) = i−1 X r=1
dimαr, ∂([a, b]) = [b] − [a], ∂([a]) = 0.
Moreover, with α a p-cell of I(n, j), we shall use the following notation: (o) In
0 = ∂In,
(i) I(n, j)p is the set of all p-cells in I(n, j),
(ii) I0(n, j)p is the set of p-cells of I(n, j) contained in I0n,
(iii) I0(n, j) is the subcomplex of I(n, j) generated by all cells contained in I0n,
(iv) α(k) is the p-dimensional subcomplex of I(n, j + k) whose cells are those with support contained in α,
(v) α(k)q with q ≤ p is the set of q-cells of α(k),
boundary of α,
(vii) and the following subcomplexes of I(n, j):
(top) T (n, j) = I(n − 1, j) ⊗ h[1]i, (side) S(n, j) = I0(n − 1, j) ⊗ I(1, j),
Chapter 2
The Willmore Functional:
Properties and Existence Results
This chapter is devoted to the study of the main properties of the Willmore functional. We will discuss first the conformal invariance of the functional. Then, since it is defined for immersed surfaces in Rn, we will first show a lower bound that will usually allow us
to restrict to embedded surfaces. Finally we will discuss the existence of global minima in Rn, and of minima in a family of surfaces of prescribed genus. The last section of
the chapter will deal with the geometric characterization of the so called Clifford torus and Willmore torus that we will consider in Chapter 3.
In this chapter any 2-dimensional manifold will be compact, connected and without boundary.
2.1
Conformal Invariance
In this section we are going to prove one of the most important properties of the Willmore functional, that is its conformal invariance under. Let us consider Σ a 2-dimensional Riemannian submanifold of M and let g and gλ = λ2g two conformal
metric tensors on M . Under this notation we prove that: Theorem 2.1 (Conformal Invariance). It holds:
Z Σ |H|2+ K dµ = Z Σ |Hλ|2+ Kλdµλ, (2.1)
where K (or Kλ) is the sectional curvature of M calculated on T Σ with respect to g
(or gλ).
Proof. Denote by ∇ and ∇λ the Levi-Civita connections corresponding to the given
metric tensors. Let N = H/|H|g ∈ N (Σ) such that g(N, N ) = 1 and Nλ = H/|H|gλ ∈
Lemma 1.15 that:
g(N, ∇λXY ) − g(N, ∇XY ) = −g(X, Y )g(N, grad(log λ)),
that is:
g(AλN(X), Y ) = g(AN(X), Y ) + g(X, Y )g(N, grad(log λ)). (2.2)
Let us denote the eigenvectors of AN as AN(ei) = −ki(N )ei, for i = 1, 2. Then we
observe that: AλN ei λ = −kiλ(N )ei λ, (2.3)
in fact using Equation (2.2) for example with X = e1/λ and Y = ej/λ for j = 1, 2, one
has: gλ AλN e1 λ ,e2 λ = 0, gλ AλN e1 λ ,e1 λ
= g(AN(e1), e1) + g(e1, e1)g(N, grad(log λ)) = −kλ1(N ).
Moreover, by the definition of the shape endomorphism and using what we already found, we get: gλ AλNλ e1 λ ,e2 λ = gλ Nλ, ∇λe1 λ e2 λ = 1 λg λ N, ∇λe1 λ e2 λ = 0 gλ AλNλ e1 λ ,e1 λ = gλ Nλ, ∇λe1 λ e1 λ = 1 λg λ N, ∇λe1 λ e1 λ = −k λ 1(N ) λ ,
and the same using e2. Hence even the operator AλNλ has eigenvectors ei/λ as AλN,
with: AλNλ ei λ = 1 λA λ N ei λ = 1 λ(−k λ i(N )) ei λ = −k λ i(N λ)ei λ. (2.4)
Now putting X = Y = ei in Equation (2.2) and using the obtained relations between
the eigenvalues, we get: −λkλ i(N λ) = −k i(N ) + g(N, grad(log λ)), so that: λ2 k λ 1(Nλ) − k2λ(Nλ) 2 2 = k1(N ) − k2(N ) 2 2 . (2.5)
Now we observe that in general: k1(N ) − k2(N ) 2 2 = k1(N ) + k2(N ) 2 2 − k1(N )k2(N ) = |H|2− KΣ+ KM, (2.6)
where the last inequality follows by Remark 1.12 (KΣ, or KM, is the sectional curvature
of Σ, or KM, calculated on T Σ). So, having µλ = λ2µ, we obtain:
Z Σ |H|2+ K M dµ − Z Σ KΣdµ = Z Σ |Hλ|2+ Kλ Mdµ λ− Z Σ KΣλdµλ.
By Gauss-Bonnet theorem the last term on the left is equal to the last term on the right (they are equal to −2πχ(Σ), with χ(Σ) the Euler characteristic of Σ), so they cancel out and we get the claim.
Corollary 2.2. If Σ ⊂ Rn is an immersed 2-dimensional Riemannian submanifold and π : Sn\ {p} → Rn is the stereographic projection, then:
W(Σ) = Z Rn |HΣ|2dH2 = Z Sn 1 + |Hπ−1(Σ)|2dH2. (2.7)
Proof. The claim follows immediately by Theorem 2.1 identifying Rn with Sn \ {p}
with different metric tensors, reminding that the sectional curvature of Rn is zero,
while the one of Snis constantly one, and by the fact that the stereographic projection is conformal (see Section 1.4).
2.2
Lower Bound for Immersed Surfaces
In this section we discuss a lower bound for the Willmore energy of an immersed surface in Rn that is not embedded, that is, there are points on the surface with multiplicity
greater than one. This result was first given in 1982 by Li and Yau in [12]; the same conclusion, but applicable only for surfaces with multiplicity at most two, can be found in Lemma 1.4 of [23], an article that we will discuss later. Here we present an even simpler proof from [10] by Kusner.
Let us consider an immersion ϕ : Σ → Rn, with Σ a 2-dimensional abstract manifold.
We shall adopt for this section the parametrization point of view, then the definition of second fundamental form is as in Remark 1.8 and we will make use of results and the notation of Section 1.3. In this frame we shall write W(ϕ) for the Willmore functional instead of W(ϕ(Σ)). We are going to prove the following:
Theorem 2.3. If ϕ : Σ → Rn is an immersion of a 2-dimensional manifold, Ia,R is
the spherical inversion given by the sphere of radius R centred in a, ϕλ = I(ϕ), then:
W(ϕλ) = W(ϕ) − 4π#(ϕ−1{a}). (2.8)
Proof. If a 6∈ ϕ(Σ), knowing by Liouville’s Theorem 1.14 that Ia,R is conformal, the
claim follows by the conformal invariance of the functional together with the fact that #(ϕ−1{a}) = 0.
Now if a ∈ ϕ(Σ) we can assume that a = 0 and that the surfaces is locally given by a graph as:
ϕ : DR(0) = {z ∈ R2 : |z| < R} → Rn, ϕ(z) = (z, u(z)),
with u(0) = 0, du0 = 0. Let η ∈ T (DR(0) \ {0}) be the outward unit normal to the
circles ∂Dr(0) ∀0 < r < R. We have: η = G −1e r hG−1e r, eri1/2 , with er(z) = z r(z), r(z) = |z|. (2.9)
In fact by definition grad(r) = G−1er and g(G−1er, G−1er) = hGG−1er, G−1eri =
hG−1e
r, eri. Now applying the operator grad = G−1gradR
2 on |ϕ|2 = |z|2+ |u(z)|2 we get: grad(log |ϕ|2) = G−1gradR2 log |z|2+ log 1 + |u| 2 r2 = = 2 r 1 1 + |u|2/r2G −1 er+ 1 r(du) tu (2.10)
Now with eθ(z) = (−y, x)/|z| if z = (x, y) let us integrate:
I(r) = Z Dr(0) ∆ log |ϕ|2dµ = Z ∂Dr(0) g(grad(ln|ϕ|2), η) dH1 = = Z S1 hGgrad(log |ϕ|2), ηihGe θ, eθi1/2dH1 = = Z 2π 0 hGgrad(log |ϕ|2), ηihGeθ, eθi1/2r dθ = = 2 Z 2π 0 1 1 + |u|2/r2 er+ 1 r(du) tu , G −1e r hG−1e r, eri1/2 hGeθ, eθi1/2dθ.
Since u(0) = 0, du0 = 0 and Gz → id uniformly as r → 0, the previous integrand tend
to 1 as r → 0. So I(r) → 4π as r → 0.
Now integrating the last equation from Proposition 1.17 over the surface where around each point in ϕ−1{0} the neighbourhood corresponding to the disk Dr(0) is deleted
and letting r → 0 we obtain the claim.
Corollary 2.4. If ϕ : Σ → Rn is an immersion of a 2-dimensional manifold, if ϕ(Σ)
has a point with multiplicity k, then:
2.3
Global Minima
The aim of this section is to provide a proof of the existence and the characterization of the global minima of the Willmore functional both in R3 and in Rn. This result was proved for the first time by Willmore himself in his first works on this subject starting from 1965. For example, his proof in R3 can be found in [26] but we will recall it here
in order to extend the strategy to arbitrary codimension, that is in Rn generally. Let us start with the observation that:
Remark 2.5. Both in R3 and in Rn there is not uniqueness of the minimum, in
fact, given a minimiser Σ, the surface F (Σ) is again a minimiser for all F conformal transformation of the euclidean space, because the functional is conformal invariant.
Now we prove the following:
Theorem 2.6 (Global Minima in R3). The Willmore functional has a minimum among all the immersed 2-dimensional Riemannian submanifold of R3, its value is 4π and
W(Σ) = 4π if and only if Σ is a round sphere.
Proof. Let us first consider Σ an embedded surface in R3. Let ν ∈ S2 and Tν,t a 2-plane with normal ν and distance t from Σ (this is possible being Σ compact). If we let t → 0 we see that the plane Tν,t tends to the (translated) tangent space TpνΣ at a
point pν ∈ Σ such that the Gaussian curvature K(pν) ≥ 0 and a normal to Σ in pν is
ν (one has in general two choice in this case). Using this procedure we actually define a map:
N : {p ∈ Σ : K(p) ≥ 0} → S2,
such that N is surjective and differentiable. This is actually a particular restriction of the classical Gauss map of the surface Σ, thus it holds:
H2(p) = k1(p) + k2(p) 2
2
, K(p) = k1(p)k2(p) = det(dNp).
Then using the area formula and the fact that in general H2 ≥ K we have:
W(Σ) = Z Σ H2dH2 ≥ Z {p∈Σ:K(p)≥0} H2dH2 ≥ Z {p∈Σ:K(p)≥0} K dH2 = = Z {p∈Σ:K(p)≥0} det(dN ) dH2 = Z {p∈Σ:K(p)≥0} |det(dN )| dH2 = = Z N ({p∈Σ:K(p)≥0}) Z N−1(ν) dH0dH2(ν) ≥ Z S2 dH2 = 4π = W(S2).
Moreover the equalities above hold if and only if H2(p) = K(p) for all p if and only
if k1(p) = k2(p) for all p if and only if Σ is a round sphere.This concludes that round
spheres are global minima among embedded surfaces.
Now by Corollary 2.4 we see that if a surface Σ is not embedded, that is there exists a point in Σ with multiplicity at least two, then W(Σ) ≥ 8π. Thus this concludes the proof.
Now we want to extend this method to an immersed 2-manifold Σ in Rn. We notice that there is a difficulty in applying the strategy of the proof of Theorem 2.6, in fact since (TpΣ)⊥ has now dimension greater than one it is not possible the same
construction of the map N : two 2-planes with different normals ν1, ν2 could touch the
surface at the same point (in the notation of Theorem 2.6 we could have pν1 = pν2).
So we first need to bypass this problem, by means of the next lemma.
Lemma 2.7. Let Σ ⊂ Rn an immersed 2-submanifold, then there exists a (n −
3)-dimensional subspace P of Rn such that T
pΣ ∩ P 6= {0} for at most a finite number of
points p ∈ Σ.
Proof. Let us consider T ∈ G(2, n) and let:
A(T ) = {A ∈ G(n − 3, n) : A ∩ T = {tv : v ∈ Rn}t∈R},
B(T ) = {B ∈ G(n − 3, n) : T ⊂ B}. (2.11)
We recall that in general G(k, n) is a differentiable manifold of dimension k(n − k), than G(n − 3, n) is a (3n − 9)-manifold.
Now an element A ∈ A(T ) is uniquely determined by a couple ({tv : v ∈ Rn} t∈R, R)
where {tv : v ∈ Rn}
t∈Rsimply denotes a straight line and R ∈ G(n − 4, n − 1) is (n −
4)-plane contained in (P ∩ T )⊥. Then A(T ) is diffeomorphic to P1(R) × G(n − 4, n − 1), so dim(A(T )) = 3n − 11. We also observe that A(T ) is compact.
By the same token we see that if n = 4 then B(T ) = ∅, otherwise an element B ∈ B(T ) is uniquely determined by a (n − 5)-subspace contained in T⊥. Then B(T ) is diffeomorphic to G(n − 5, n − 2), so dim(B(T )) = 3n − 15.
Now we define A(Σ), B(Σ) subsets of Rn× G(n − 3, n) as:
A(Σ) = {(p, A) : p ∈ Σ, A ∈ A(TpΣ)},
B(Σ) = {(p, B) : p ∈ Σ, B ∈ B(TpΣ)}.
(2.12) Clearly A(Σ) and B(Σ) are differentiable manifolds of dimension: dim(A(Σ)) = 2 + 3n − 11 = 3n − 9 and dim(B(Σ)) = 2 + 3n − 15 = 3n − 13. We also observe that A(Σ) is compact.
Now let:
fA: A(Σ) → G(n − 3, n) fA(p, A) = A,
fB: B(Σ) → G(n − 3, n) fB(p, B) = B,
(2.13) that are the projections on the second term, so they are differentiable maps. By Sard’s Lemma we have that there exists an element P ∈ G(n − 3, n) that is a regular value both for fA and for fB. We now show that this is the P that we need in the claim.
Since dim(B(Σ)) < dimG(n−3, n), there are not regular values for fB, so P 6∈ Im (fB).
This means that for all p ∈ Σ the tangent space TpΣ is never contained in P . Moreover
dim(A(Σ)) = dimG(n − 3, n), so fA−1({P }) is discrete (being appropriate restrictions of fA local diffeomorphisms with a neighbourhood of P ) and hence it is finite by
compactness of A(Σ). This means that for at most a finite number of points p in Σ it occurs that Tp∩ P is a straight line. So this completes the claim.
