Minmax Methods in the Calculus of Variations of Curves and Surfaces

Tristan Rivière∗

I Lecture 2

Deformations in Infinite Dimensional . The goal of this section is to present the main lines of Palais deformation theory exposed in [3]. We shall start by recalling some notions from infinite dimensional Banach Manifolds theory that will be useful to us. For a more systematic study we suggest the reader to consult [1]. In this lecture as in the previous one N n denotes a closed C∞ sub- of the Euclidian Space Rm. We recall that a is Hausdorff if every pair of points can be included in two disjoint open sets containing each exactly one of the two points. A topological space is called normal if for any two disjoint closed sets have disjoint open neighborhoods.

Definition I.1. A Cp Banach Manifold M for p ∈ N ∪ {∞} is an Hausdorff topological space together with a covering by open sets (Ui)i∈I, a family of Banach vector spaces (Ei)i∈I and a family of continuous mappings (ϕi)i∈I from Ui inton Ei such that i) for every i ∈ I

ϕi Ui −→ ϕi(Ui) is an ii) for every pair of indices i 6= j in I

−1 ϕj ◦ ϕi : ϕi(Ui ∩ Uj) ⊂ Ei −→ ϕj(Ui ∩ Uj) ⊂ Ej is a Cp diffeomorphism

∗Department of , ETH Zentrum, CH-8093 Zürich, Switzerland.

1 2 Example 1. Let Σk be a closed oriented k−dimensional manifold and let l ∈ N and p ≥ 1. Define l,p k n n l,p k m n ko M := W (Σ ,N ) := ~u ∈ W (Σ , R ); ~u(x) ∈ N for a.e. x ∈ Σ where on Σk we can choose any arbitrary reference smooth metric, they are all equivalent since Σk is compact. Assume l p > k then W l,p(Σk,N n) defines a Banach Manifold. This comes mainly from the fact that, under our assumptions,

l,p k m 0 k m W (Σ , R ) ,→ C (Σ , R ) . (I.1) The Banach manifold structure is then defined as follows. Choose δ > 0 N n n such that each geodesic ball Bδ (z) for any z ∈ N is strictly convex and the exponential map

n N n expz : Vz ⊂ TzN −→ Bδ (z) realizes a C∞ diffeomorphism for some open neighborhood of the origin in n N n TzN into the geodesic ball Bδ (z). Because of the (I.1) there exists ε0 > 0 such that l,p k n ∀ ~u, ~v ∈ W (Σ ,N ) k~u − ~vkW l,p < ε0

=⇒ kdistN (~u(x),~v(x))kL∞(Σk) < δ . We equip the space W l,p(Σk,N n) with the W l,p which makes it a and for any ~u ∈ M = W l,p(Σk,N n) we denote by BM(~u) the ε0 open ball in M of center ~u and radius ε0. As a covering of M we take (BM(~u)) . We denote by ε0 ~u∈M ~u  −1  n l,p k m n ko E := ΓW l,p ~u TN := ~w ∈ W (Σ , R ); ~w(x) ∈ T~u(x)N ∀ x ∈ Σ this is the of W l,p−sections of the bundle ~u−1TN and for any ~u ∈ M and ~v ∈ BM(~u) we define ~w ~u(~v) to be the following element ε0 of E~u ~u −1 ∀ x ∈ Σ ~w (~v)(x) := exp~u(x)(~v(x)) It is not difficult to see that ~w ~v ◦ (~w ~u)−1 : ~w u BM(~u) ∩ BM(~v) −→ ~w v BM(~u) ∩ BM(~v) ε0 ε0 ε0 ε0 defines a C∞ diffeomorphism. 2

2 The goal of the present section is to construct, for C1 lagrangians on some special Banach manifolds, a substitute to the gradient of such la- grangian in order to be able to deform each level to a lower level set if there is no critical point in between. The strategy for construct- ing such pseudo-gradient will consist in pasting together “pieces” using partitions of unity of suitable regularity. To that aim we introduce the following notion. Definition I.2. A topological Hausdorff space is called paracompact if every open covering admits a locally finite1 open refinement. 2 We have the following result Theorem I.1. [Stones 1948] Every metric space is paracompact. 2 There is a more restrictive “separation axiom” for topological spaces than being Hausdorff which is called normal. Definition I.3. A topological space is called normal if any pair of disjoint closed sets have disjoint open neighborhoods. 2 we have the following proposition Proposition I.1. Every Hausdorff paracompact space is normal. 2 We have the following important lemma (which looks obvious but re- quires a proof in infinite dimension) Lemma I.1. Let M be a normal Banach Manifold and let (U, ϕ) be a chart on M, i.e. U ⊂ M is an open of M and ϕ is an homeomor- phism from U into an ϕ(U) ⊂ E of a Banach Space (E, k · kE). For any x0 ∈ U and for r small enough

−1 E Bϕ(x0, r) := {y ∈ U ; kϕ(x0) − ϕ(y)kE ≤ r} = ϕ (Br (x0)) is closed in M and it’s interior is given by

int −1 E Bϕ (x0, r) := {y ∈ U ; kϕ(x0) − ϕ(y)kE < r} = ϕ (Br (x0)) 2 Proof of lemma I.1. Since the Banach manifold is assumed to be normal there exists two disjoint open sets V1 and V2 such that M\ U ⊂ V1 and −1 x0 ∈ V2. Since ϕ is an homeomorphism, the preimage by ϕ of V2 is open 1locally finite means that any point posses a neighborhood which intersects only finitely many open sets of the subcovering

3 in the Banach space E. Choose now radii r > 0 small enough such that E Br (x0) ⊂ ϕ(V2), hence for such a r we have that −1  E  M\ Bϕ(x0, r) = V1 ∪ φ E \ Br (x0)

So M\Bϕ(x0, r) is the union of two open sets. It is then open and Bϕ(x0, r) is closed in M. 2 Remark I.1. As counter intuitive as it could be at a first glance, there are −1 E counter-examples of the closure of ϕ (Br (x)) when r is not assumed to E be small enough and even with Br (x) ⊂ ϕ(U) ! (see for instance [3]). We shall need the following lemma Lemma I.2. Let M be a normal Banach Manifold and let (U, ϕ) be a chart on M, i.e. U ⊂ M is an open subset of M and ϕ is an homeomor- phism from U into an open set ϕ(U) ⊂ E of a Banach Space (E, k · kE). −1 E For any x0 ∈ U and for r small enough such that ϕ (Br (x0)) is closed and included in U according to lemma I.1 then the function defined by  n −1 E o  ∀ x ∈ U g(x) := inf kϕ(x) − ϕ(y)kE ; y ∈ U \ ϕ (B (x0))  r   ∀ x ∈ M \ U g(x) = 0

−1 E is locally Lipschitz on M and strictly positive exactly on ϕ (Br (x0)). 2 Proof of lemma I.2. First of all we prove that g is globally lipschitz on −1 E U. Let x, y ∈ U and let ε > 0. Choose z ∈ U \ ϕ (Br (x0)) such that

kϕ(x) − ϕ(z)kE < g(x) + ε We have by definition

kϕ(y) − ϕ(z)kE ≥ g(y) Combining the two previous inequalities give

g(y) − g(x) ≤ kϕ(y) − ϕ(z)kE − kϕ(y) − ϕ(z)kE + ε

≤ kϕ(y) − ϕ(x)kE + ε exchanging the role of x and y gives the lipschitzianity of g on U. Take −1 E now y∈ / U since M\ U and ϕ (Br (x0)) are closed and disjoint, since −1 E ϕ (Br (x0)) ⊂ U, the normality of M gives the existence of two dis- −1 E joint open neighborhoods containing respectively M\U and ϕ (Br (x0)).

4 Hence there exists an open neighborhood of y which does not intersect −1 E ϕ (Br (x0)) and on which g is identically zero. This implies the local lipschitzianity of g. 2 One of the reasons why we care about paracompactness in our context comes from the following property. 1 Proposition I.2. Let (Oα)α∈A be an arbitrary covering of a C paracom- pact Banach manifold M. Then there exists a locally lipschitz partition of unity subordinated to (Oα)α∈A, i.e. there exists (φα)α∈A where φα is locally lipschitz in M and such that i) Supp(φα) ⊂ Oα ii) φα ≥ 0 iii) X φα ≡ 1 α∈A where the sum is locally finite. 2

Proof of proposition I.2. To each point x in Oα we assign an open neigh- −1 Ei borhood of the form ϕi (Br (ϕ(x))) included in Oα for r small enough given by lemma I.1. From the total union of all the families

−1 Ei (ϕi (Br (ϕ(x))))x∈Oα where α ∈ A we extract a locally finite sub covering that we denote −1 Ei (ϕi (Br (ϕ(xi))))i∈Iα and α ∈ A (we can have possibly Iα = ∅). To each −1 Ei i open set ϕi (Br (ϕ(xi))) we assign the function gα given by lemma I.2 −1 Ei which happens to be strictly positive on ϕi (Br (ϕ(xi))) and zero outside. i −1 Ei The functions gα are locally lipschitz and since the family (ϕi (Br (ϕ(xi))))i∈Iα P P i is locally finite, α i∈Iα gα is locally lipschitz too. we take X i gα i∈Iα φα := X X i gα α∈A i∈Iα with the convention that φα ≡ 0 on M if Iα = ∅. The family (φα)α∈A solves i), ii) and iii) and proposition I.2. 2

5 We introduce more structures in order to be able to perform deforma- tions in Banach Manifolds. Definition I.4. A Banach manifold V is called Cp− Banach Space Bun- dle over another Banach manifold M if there exists a Banach Space E, a submersion π from V into M, a covering (Ui)i∈I of M and a family of −1 homeomorphism from π Ui into Ui × E such that the following diagram commutes −1 τi π Ui −→ Ui × E π& ↓ ρ Ui where ρ is the canonical projection from Ui × E onto Ui. The restric- −1 tion of τi on each fiber Vx := π ({x}) for x ∈ Ui realizes a continuous onto Ei. Moreover the map

x ∈ U ∩ U −→ τ ◦ τ −1 ∈ L(E,E) i j i j π−1(x) is Cp. 2 Definition I.5. Let M be a normal Banach manifold and let V be a Ba- nach Space Bundle over M.A Finsler structure on V is a k · k : V −→ R such that for any x ∈ M

k · kx := k · k|π−1({x}) is a norm on Vx .

Moreover for any local trivialization τi over Ui and for any x0 ∈ Ui we define on Vx the following norm −1 −1 ∀ ~w ∈ π ({x}) k~wkx0 := kτi (x0, ρ(τi(~w))) kx0 and there exists Cx0 > 1 such that ∀ x ∈ U C−1 k · k ≤ k · k ≤ C k · k . i x0 x x0 x0 x 2 Definition I.6. Let M be a normal Cp Banach manifold. T M equipped with a Finsler structure is called a Finsler Manifold. 2 Remark I.2. A Finsler structure on T M defines in a canonical way a dual Finsler structure on T ∗M. 2

6 Example. Let Σ2 be a closed oriented 2−dimensional manifold and N n be a closed sub-manifold of Rm. For q > 2 we define   2,q 2 n ~ 2,q 2 n 2 M := Wimm(Σ ,N ) := Φ ∈ W (Σ ,N ); rank (dΦx) = 2 ∀x ∈ Σ

2,q 2 n The set Wimm(Σ ,N ) as an open subset of the normal Banach Manifold W 2,q(Σ2,N n) inherits a Banach Manifold structure. The tangent space ~ ~ −1 n 2,q to M at a point Φ is the space ΓW 2,q (Φ TN ) of W −sections of the bundle Φ~ −1TN n, i.e.   2,q 2 m n 2 TΦ~ M = ~w ∈ W (Σ , R ); ~w(x) ∈ TΦ(~ x)N ∀ x ∈ Σ .

We equip TΦ~ M with the following norm

"  q/2 #1/q Z 2 2 2 2 ∞ k~vk~ := |∇ ~v|g + |∇~v|g + |~v| dvolg~ + k |∇~v|g~ kL (Σ) Φ Σ Φ~ Φ~ Φ Φ where we keep denoting, for any j ∈ N, ∇ to be the connection on ∗ ⊗j −1 g ∗ h g (T Σ) ⊗Φ~ TN over Σ defined by ∇ := ∇ Φ~ ⊗Φ~ ∇ and ∇ Φ~ is the Levi h n Civita connection on (Σ, gΦ~ ) and ∇ is the Levi-Civita connection on N . We check for instance that ∇2~v defines a C0 section of (T ∗Σ)2 ⊗ Φ~ −1TN. Observe that, using Sobolev embedding and in particular due to the fact 2,q m 1 m W (Σ, R ) ,→ C (Σ, R ) for q > 2, the norm k · kΦ~ as a function on the Banach tangent bundle T M is obviously continuous. 2 Proposition I.3. The norms k · kΦ~ defines a C −Finsler structure on the space M. 2 Proof of proposition I.3. We introduce the following trivialization of ~ the Banach bundle. For any Φ ∈ M we denote PΦ(~ x) the orthonormal projection in Rm onto the n−dimensional vector subspace of Rm given by T N n and for any ξ~ in the ball BM(Φ)~ for some ε > 0 and any Φ(~ x) ε1 1 ~−1 ~v ∈ Tξ~M = ΓW 2,q (ξ TN) we assign the map ~w(x) := PΦ(~ x)~v(x). It is straightforward to check that for ε1 > 0 chosen small enough the map which to ~v assigns ~w is an isomorphism from Tξ~M into TΦ~ M and that there exists k > 1 such that ∀~v ∈ TBM(Φ)~ Φ~ ε1 k−1 k~vk ≤ k~wk ≤ k k~vk Φ~ ξ~ Φ~ Φ~ ξ~ This concludes the proof of proposition I.3. 2

7 Theorem I.2. [Palais 1970] Let (M, k·k) be a Finsler Manifold. Define on M × M

Z 1 dω

d(p, q) := inf dt ω∈Ωp,q 0 dt ω(t) where n 1 o Ωp,q := ω ∈ C ([0, 1], M); ω(0) = p ω(1) = q . Then d defines a distance on M and (M, d) defines the same as the one of the Banach Manifold. d is called Palais distance of the Finsler manifold (M, k · k). 2 Contrary to the first appearance the non degeneracy of d is not straight- forward and requires a proof (see [3]). This last result combined with theorem I.1 gives the following corollary. Corollary I.1. Let (M, k·k) be a Finsler Manifold then M is paracompact. 2 The following result is going to play a central role in this course Proposition I.4. Let M be the space   2,q 2 n ~ 2,q 2 n 2 Wimm(Σ ,N ) := Φ ∈ W (Σ ,N ); rank (dΦx) = 2 ∀x ∈ Σ where Σ2 is a closed oriented surface and N n a closed sub-manifold of Rm. The Finsler Manifold given by the structure

"  q/2 #1/q Z 2 2 2 2 ∞ k~vk~ := |∇ ~v|g + |∇~v|g + |~v| dvolg~ + k |∇~v|g~ kL (Σ) Φ Σ Φ~ Φ~ Φ Φ is complete for the Palais distance. 2 We have also.

Proposition I.5. For N n a closed sub-manifold of Rm and p > 1 we define on

2,p 1 n n 2,p 1 n 1o M := Wimm(S ,N ) := ~γ ∈ W (S ,N ); rank (dγx) = 1 ∀x ∈ S the following Finsler structure

"  p/2 #1/p Z 2 2 2 2 k~vk~γ := |∇ ~v| + |∇~v| + |~v| dvolg S1 g~γ g~γ ~γ Then (M, k · k) is complete for the Palais distance . 2

8 We shall present only the proof of proposition I.4. The proof of proposi- tion th-complete-S1 is very similar and can be found in [2]. ~ Proof of proposition I.4. For any Φ ∈ M and ~v ∈ TΦ~ M we introduce the in (T ∗Σ)⊗2 given in coordinates by

2   ∇~v ⊗˙ dΦ~ + dΦ~ ⊗˙ ∇~v = X ∇ ~v · ∂ Φ~ + ∂ Φ~ · ∇ ~v dx ⊗ dx ∂xi xj xi ∂xj i j i,j=1 2 " # X h ~ ~ h = ∇∂ Φ~ ~v · ∂xj Φ + ∂xi Φ · ∇∂ Φ~ ~v dxi ⊗ dxj i,j=1 xi xj where · denotes the scalar product in Rm. Observe that we have

~ ~ ∇~v ⊗˙ dΦ + dΦ ⊗˙ ∇~v ≤ 2 |∇~v|g Φ~ gΦ~ 1 ~ ~ Hence, taking a C path Φs in M one has for ~v := ∂sΦ

2 ~ ~ 2 X ij kl k|d~v⊗˙ dΦ + dΦ⊗˙ d~v| kL∞(Σ) = g g~ ∂s(g~ )ik ∂s(g~ )jl gΦ~ Φ~ Φ Φ Φ i,j,k,l=1 L∞(Σ)

2 2 = |∂s(gijdxi ⊗ dxj)|g = |∂sg~ |g Φ~ L∞(Σ) Φ Φ~ L∞(Σ) (I.2) Hence Z 1 Z 1 2 ~ |∂sg~ |g ds ≤ 2 k∂sΦk~ ds (I.3) 0 Φ Φ~ L∞(Σ) 0 Φs We now use the following lemma 1 Lemma I.3. Let Ms be a C path into the space of positive n by n sym- metric matrix then the following inequality holds

−2 2 2 2 Tr (M (∂sM) ) ≥ k∂s log Mk = Tr ((∂s log M) ) Proof of lemma I.3. We write M = exp A and we observe that

2 2 Tr (exp(−2A)(∂s exp A) ) = Tr (∂sA) Then the lemma follows. 2 Combining the previous lemma with (I.2) and (I.3) we obtain in a given chart Z 1 Z 1 Z 1 q 2 ~ k∂s log(gij)k ds ≤ Tr ((∂s log gij) ) ds ≤ 2 k∂sΦk~ ds (I.4) 0 0 0 Φs

This implies that in the given chart the log of the matrix (gij(s)) is uni- ~ formly bounded for s ∈ [0, 1] and hence Φ1 is an immersion. It remains

9 to show that it has a controlled W 2,q norm. We introduce p = q/2 and denote ~ Z ~ 2 p Hessp(Φ) := [1 + |∇dΦ|g ] dvolg~ Σ Φ~ Φ and we compute

d Z (Hess (Φ))~ = p ∂ |∇dΦ~ |2 [1 + |∇dΦ~ |2 ]p−1 dvol p s g~ g~ g~ ds Σ Φ Φ Φ (I.5) Z ~ 2 p + [1 + |∇dΦ|g ] ∂s(dvolg~ ) Σ Φ~ Φ Classical computations give   ∂ (dvol ) = ∇∂ Φ~ , dΦ~ dvol s gΦ~ s gΦ~ gΦ~ So we have Z Z 2 p 2 p ~ ~ ∞ ~ [1 + |∇dΦ|g ] ∂s(dvolg~ ) ≤ k |∇∂sΦ|g~ kL (Σ) [1 + |∇dΦ|g ] dvolg~ Σ Φ~ Φ Φ Σ Φ~ Φ ~ Z ~ 2 p ≤ k∂sΦk~ [1 + |∇dΦ|g ] dvolg~ Φ Σ Φ~ Φ (I.6) In local charts we have 2 * + ~ 2 X ij kl h ~ h ~ |∇dΦ| = g g~ ∇ ~ ∂x Φ, ∇ ~ ∂x Φ gΦ~ Φ~ Φ ∂x Φ k ∂x Φ l i,j,k,l=1 i j h

R ~ 2 ~ 2 p−1 Thus in bounding Σ ∂s|∇dΦ| [1 + |∇dΦ| ] dvolg we first have to gΦ~ gΦ~ Φ~ control terms of the form

Z 2 * + X ij kl h ~ h ~ ~ 2 p−1 ∂sg g~ ∇ ~ ∂xk Φ, ∇ ~ ∂xl Φ [1 + |∇dΦ|g ] dvolg~ Σ Φ~ Φ ∂x Φ ∂x Φ Φ~ Φ i,j,k,l=1 i j h (I.7) We write 2 * + X ij kl h ~ h ~ ∂sg g~ ∇ ~ ∂x Φ, ∇ ~ ∂x Φ Φ~ Φ ∂x Φ k ∂x Φ l i,j,k,l=1 i j h 2 * + X ij tr kl h ~ h ~ = ∂sg gjt g g~ ∇ ~ ∂x Φ, ∇ ~ ∂x Φ Φ~ Φ ∂x Φ k ∂x Φ l i,j,k,l,t,r=1 i j h 2  2  * + X X tr ij kl h ~ h ~ = −  ∂sgjt g  g g~ ∇ ~ ∂x Φ, ∇ ~ ∂x Φ Φ~ Φ ∂x Φ k ∂x Φ l i,j,k,l,=1 t,r=1 i j h

10 Hence

Z 2 * + X ij kl h ~ h ~ ~ 2 p−1 ∂sg g~ ∇ ~ ∂x Φ, ∇ ~ ∂x Φ [1 + |∇dΦ| ] dvolg Φ~ Φ ∂x Φ k ∂x Φ l gΦ~ Φ~ Σ i,j,k,l=1 i j h Z 2 p ∞ ~ ≤ k |∂sg~ |g~ kL (Σ) [1 + |∇dΦ|g ] dvolg~ Φ Φ Σ Φ~ Φ ~ Z ~ 2 p ≤ k∂sΦk~ [1 + |∇dΦ|g ] dvolg~ Φs Σ Φ~ Φ (I.8) We have also * + h ~ h ~ ∂s ∇∂ Φ~ ∂xk Φ, ∇∂ Φ~ ∂xl Φ xi xj h *   + * !+ h h ~ h ~ h ~ h h ~ = ∇∂ Φ~ ∇∂ Φ~ ∂xk Φ , ∇∂ Φ~ ∂xl Φ + ∇∂ Φ~ ∂xk Φ, ∇∂ Φ~ ∇∂ Φ~ ∂xl Φ s xi xj h xi s xj h By definition we have     h h ~ h h ~ h ~ ~ ~ ∇ ~ ∇ ~ ∂xk Φ = ∇ ~ ∇ ~ ∂xk Φ + R (∂xi Φ, ∂sΦ)∂xk Φ ∂sΦ ∂xi Φ ∂xi Φ ∂sΦ ~ ~ ~ where we have used the fact that [∂sΦ, ∂xi Φ] = Φ∗[∂s, ∂xi ] = 0. Using also ~ ~ h that [∂sΦ, ∂xk Φ] = 0, since ∇ is torsion free, we have finally     h h ~ h h ~ h ~ ~ ~ ∇ ~ ∇ ~ ∂xk Φ = ∇ ~ ∇ ~ ∂sΦ + R (∂xi Φ, ∂sΦ)∂xk Φ (I.9) ∂sΦ ∂xi Φ ∂xi Φ ∂xk Φ where Rh is the Riemann tensor associated to the Levi-Civita connection ∇h. We have   h h ~ h 2 ~ h ~ ∇ ~ ∇ ~ ∂sΦ = (∇ ) ~ ~ ∂sΦ + ∇ h ~ ∂sΦ (I.10) ∂xi Φ ∂xk Φ ∂xi Φ∂xk Φ ∇ ~ ∂xk Φ ∂xi Φ Hence * ! + * + ∇h ∇h ∂ Φ~ , ∇h ∂ Φ~ = (∇h)2 ∂ Φ~ , ∇h ∂ Φ~ ∂ Φ~ ∂ Φ~ xk ∂ Φ~ xl ∂ Φ~ ∂ Φ~ s ∂ Φ~ xl s xi xj h xi xk xj h * + * + h ~ h ~ h ~ ~ ~ h ~ + ∇ h ~ ∂sΦ, ∇ ~ ∂xl Φ + R (∂xi Φ, ∂sΦ)∂xk Φ, ∇ ~ ∂xl Φ ∇ ~ ∂xk Φ ∂xj Φ ∂xj Φ ∂xi Φ h h (I.11)

11 Combining all the previous gives then

Z 2 * + X ij kl h ~ h ~ g g~ ∂s ∇ ~ ∂x Φ, ∇ ~ ∂x Φ dvolg Φ~ Φ ∂x Φ k ∂x Φ l Φ~ Σ i,j,k,l=1 i j h

Z   2 ~ ~ ~ 2 p−1 ≤ C ∇ ∂sΦ, ∇dΦ [1 + |∇dΦ|g ] dvolg~ Σ Φ~ Φ gΦ~ (I.12) Z ~ ~ 2 ~ 2 p−1 + C |∇∂sΦ|g~ |∇dΦ|g [1 + |∇dΦ|g ] dvolg~ Σ Φ Φ~ Φ~ Φ

h Z 2 p−1 ∞ n ~ ~ ~ + C kR kL (N ) |∂sΦ|h |∇dΦ|g~ [1 + |∇dΦ|g ] dvolg~ Σ Φ Φ~ Φ Combining all the above we finally obtain that   1−1/2p ∂ Hess (Φ)~ ≤ C k∂ Φ~ k Hess (Φ)~ + Hess (Φ)~ (I.13) s p s Φ~ p p Combining (I.4) and (I.13) we deduce using Gromwall lemma that if we take a C1 path from [0, 1) into M with finite length for the Palais distance ~ 2,q n d, the limiting map Φ1 is still a W −immersion of Σ into N , which proves the completeness of (M, d). 2

Definition I.7. Let M be a C2 Finsler Manifold and E be a C1 function on M. Denote

∗ M := {u ∈ M ; DEu 6= 0} . A pseudo-gradient is a Lipschitz continuous section X : M∗ → T M∗ such that i) ∗ ∀ u ∈ M kX(u)ku < 2 kDEuku ii) ∗ 2 ∀ u ∈ M kDEuk < hX(u),DEui ∗ ∗ ∗ u TuM ,Tu M 2 The following result is mostly using the existence of a Lipschitz partition of unity for any covering of a Finsler Manifold (combine proposition I.2 and corollary I.1). Proposition I.6. Every C1 function on a Finsler Manifold admits a pseudo- gradient. 2 The following definition is central in Palais deformation theory.

12 Definition I.8. Let E be a C1 function on a Finsler manifold (M, k · k) and β ∈ E(M). On says that E fulfills the Palais-Smale condition at the level β if for any sequence un staisfying

E(un) −→ β and kDEun kun −→ 0 , then there exists a subsequence un0 and u∞ ∈ M such that

d(un0 , u∞) −→ 0 .

and hence E(u∞) = β and DEu∞ = 0. 2 Example. Let M be W 1,2(S1,N n) for the Finsler structure given by

−1 n ∀ ~w ∈ ΓW 1,2 (~u TN ) k~wk~u := k~wkW 1,2(S1) Then the Dirichlet Energy satisfies the Palais Smale condition for every level set. 2

Definition I.9. A family of A ⊂ P(M) of a Banach manifold M is called admissible family if for every homeomorphism Ξ of M isotopic to the identity we have ∀ A ∈ A Ξ(A) ∈ A 2

Example 1. A closed 2 dimensional sub-manifold N 2 of Rm being given 2 1,2 1 2 and α ∈ π2(N ) 6= 0, considering the Banach Manifold M := W (S ,N ) we can take  0 1,2 1 2   u ∈ C ([0, 1],W (S ,N )) ; u(0, ·) and u(1, ·) are constant  A :=   1 2  and u(t, θ) [0, 1] × S −→ N realizes α  (I.14) 2 2 i.e. for N ' S A corresponds to a class of sweep-outs of the form Ωσ0 . 2 2,q 2 3 2 3 Example 2. Consider M := Wimm(S , R ) and take c ∈ π1(Imm(S , R )) = Z2 × Z then the following family is admissible   ~ 0 2,q 2 3 ~ ~ ~ A := Φ ∈ C ([0, 1],Wimm(S , R )) ; Φ(0, ·) = Φ(1, ·) and [Φ] = c 2 We can now state the main theorem in this section.

13 Theorem I.3. [Palais 1970] Let (M, k·k) be a Banach manifold together with a C1,1−Finsler structure. Assume M is complete for the induced Palais distance d and let E ∈ C1(M) satisfying the Palais-Smale condition (PS)β for the level set β. Let A be an admissible family in P(M) such that inf sup E(u) = β A∈A u∈A then there exists u ∈ M satisfying   DEu = 0  (I.15)   E(u) = β 2 Proof of theorem I.3. We argue by contradiction. Assuming there is no u satisfying (I.1) then Palais Smale condition (PS)β implies

∃ δ0 > 0 , ∃ 0 > 0 β −ε < E(u) < β +ε =⇒ kDEuku ≥ δ . (I.16) Let u ∈ M∗. Because of the Local lipschitz nature of a fixed pseudo- u gradient given by proposition I.6 there exists a maximal time tmax ∈ (0, +∞] such that   dφt(u) u  = − X(φ (u)) η(E(φ (u))) in [0, t )  dt t t max    φ0(u) = u where 1 ≥ η(t) ≥ 0 is supported in [β − ε0, β + ε] and is equal to one on [β − ε0/2, β + ε0/2]. u We have for any 0 ≤ t1 < t2 < tmax we have

Z t 2 dφt(u) d(φt1 (u), φt2 (u)) ≤ dt t1 dt φt(u) Z t2 ≤ 2 η(E(φt(u))) kDEφt(u)kφt(u) dt t1

"Z t #1/2 ≤ |t − t |1/2 2 η(E(φ (u))) kDE k2 dt 2 1 t φt(u) φt(u) t1 and Z t 2 η(E(φ (u))) kDE k2 dt t φt(u) φt(u) t1

Z t2 D E ≤ − η(E(φt(u))) X(φt(u)),DEφt(u) dt t1

≤ E(φt1 (u)) − E(φt2 (u))

14 Hence

1/2 1/2 d(φt1 (u), φt2 (u)) ≤ 2 |t2 − t1| [E(φt1 (u)) − E(φt2 (u))] u u Hence, assuming tmax < +∞, φt(u) realizes a Cauchy sequence as t → tmax. Since M is complete, the only possibility for the extinction of the flow is ∗ u that limt→tmax φt(u) belongs to M . But the flow is constant in time outside −1 u E ([β − ε0, β + ε]) hence tmax = +∞.

Hence for any t ∈ R+ φt is an homeomorphism of M isotopic to the identity and, since A is admissible

∀ A ∈ A ∀ t ∈ [0, +∞) φt(A) ∈ A .

Let u now such that β ≤ E(u) ≤ β + ε0/2. For any τ > 0 such that E(φt(u)) ≥ β − ε0/2 we have (taking δ0 < 1)

Z τ dφt(u) 2 − τ δ0 ≤ E(φt(u)) − E(u) = dt ≤ − 2 τ δ 0 dt 0 2 Hence for any τ δ0 ≤ ε0/2 we have 2 E(φτ (u)) ≤ E(u) − 2 τ δ0 . In particular

E(φε0/2δ0 (u)) ≤ E(u) − δ0 ε0 Choose A ∈ A such that

sup E(u) < β + δ0 ε0 u∈A

Hence we have for t0 = ε0/2δ0

sup E(φt0 (u)) < β φt0 (u)∈φt0 (A) which is a contradiction. 2 Application. We take M := W 1,2(S1,N 2) where N 2 ' S2. Let any 2 2 sweep-out ~σ0 of N corresponding to a non zero element of π2(N ). Then

W~σ0 = inf max E(~σ(t, ·)) ~σ∈Ω~σ0 ∩Λ t∈[0,1] is achieved by a closed geodesic. This gives a new proof of Birkhoff exis- tence result. 2 2Observe that this kind of inequality is reminiscent to the condition v) of the definition of Birkhoff curve shortening process.

15 Now, what about surfaces ? The Dirichlet energy of maps into a sub- manifold of is not satisfying the Palais Smale anymore in 2 dimension. So Palais Deformation theory does not apply directly to the construction of minimal surfaces by working with the Dirichlet energy. We would also like to go beyond the Colding-Minicozzi framework which is restricted to spheres.

References

[1] Lang, Serge Fundamentals of differential geometry. Graduate Texts in Mathematics, 191. Springer-Verlag, New York, 1999. [2] Alexis Michelat, Tristan Rivière “A Viscosity Method for the Min-Max Construction of Closed Geodesics” arXiv:1511.04545 (2015). [3] Palais, Richard S. Critical point theory and the minimax principle. 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif, 1968) pp. 185-212 Amer. Math. Soc., Providence, R.I. [4] Stone, A. H. Paracompactness and product spaces. Bull. Amer. Math. Soc. 54 (1948), 977-982.

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