Minmax Methods for Geodesics and Minimal Surfaces

Home , Banach manifold, Finsler manifold

Tristan Rivi`ere

ETH Z¨urich Lecture 2 : Palais Deformation Theory

in ∞ Dimensional Spaces. Banach Manifolds Definition A C p Banach Manifold M for p ∈ N ∪ {∞} is an Hausdorff topological space Banach Manifolds Definition A C p Banach Manifold M for p ∈ N ∪ {∞} is an Hausdorff topological space together with a covering by open sets (Ui )i∈I , Banach Manifolds Definition A C p 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 Banach Manifolds Definition A C p 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 Banach Manifolds Definition A C p 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 homeomorphism Banach Manifolds Deﬁnition A C p 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 homeomorphism

ii) for every pair of indices i 6= j inI

−1 ϕj ◦ ϕi : ϕi (Ui ∩ Uj ) ⊂ Ei −→ ϕj (Ui ∩ Uj ) ⊂ Ej is a C p diffeomorphism Banach Manifolds Definition A C p 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 homeomorphism

ii) for every pair of indices i 6= j inI

−1 ϕj ◦ ϕi : ϕi (Ui ∩ Uj ) ⊂ Ei −→ ϕj (Ui ∩ Uj ) ⊂ Ej is a C p diﬀeomorphism Example : Let lp > k

M := W l,p(Σk , Nn) := u~ ∈ W l,p(Σk , Rm) ; u~(x) ∈ Nn a.e. x ∈ Σk n o Paracompact Banach Manifolds Definition A topological Hausdorff space is called paracompact if every open covering admits a locally finite1 open refinement.

1locally finite means that any point posses a neighborhood which intersects only finitely many open sets of the sub-covering Paracompact Banach Manifolds Definition A topological Hausdorff space is called paracompact if every open covering admits a locally finite1 open refinement.

Theorem [Stone 1948] Every metric space is paracompact.

Deﬁnition A topological space is called normal if any pair of disjoint closed sets have disjoint open neighborhoods.

Theorem [Stone 1948] Every metric space is paracompact.

Deﬁnition A topological space is called normal if any pair of disjoint closed sets have disjoint open neighborhoods.

Proposition Every Hausdorﬀ paracompact space is normal.

Theorem [Stone 1948] Every metric space is paracompact.

Deﬁnition A topological space is called normal if any pair of disjoint closed sets have disjoint open neighborhoods.

Proposition Every Hausdorﬀ paracompact space is normal.

Proof : https://topospaces.subwiki.org/wiki/

Theorem [Stone 1948] Every metric space is paracompact.

Deﬁnition A topological space is called normal if any pair of disjoint closed sets have disjoint open neighborhoods.

Proposition Every Hausdorﬀ paracompact space is normal.

Proof : https://topospaces.subwiki.org/wiki/

Warning ! M Banach Paracompact Manifold, (φ, U) a chart s.t.

φ : U −→ φ(U) = (E, k · k) homeomorphism

Theorem [Stone 1948] Every metric space is paracompact.

Deﬁnition A topological space is called normal if any pair of disjoint closed sets have disjoint open neighborhoods.

Proposition Every Hausdorﬀ paracompact space is normal.

Proof : https://topospaces.subwiki.org/wiki/

Warning ! M Banach Paracompact Manifold, (φ, U) a chart s.t.

φ : U −→ φ(U) = (E, k · k) homeomorphism

−1 then φ (Br (x)) might not be closed in M. 1locally ﬁnite means that any point posses a neighborhood which intersects only ﬁnitely many open sets of the sub-covering Partition of Unity on Paracompact Banach Manifolds Partition of Unity on Paracompact Banach Manifolds 1 Proposition Let (Oα)α∈A be an arbitrary covering of a C paracompact Banach manifold M. Partition of Unity on Paracompact Banach Manifolds 1 Proposition Let (Oα)α∈A be an arbitrary covering of a C paracompact Banach manifold M. Then there exists a locally lipschitz partition of unity subordinated to (Oα)α∈A Partition of Unity on Paracompact Banach Manifolds 1 Proposition Let (Oα)α∈A be an arbitrary covering of a C paracompact 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 Partition of Unity on Paracompact Banach Manifolds 1 Proposition Let (Oα)α∈A be an arbitrary covering of a C paracompact 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 Partition of Unity on Paracompact Banach Manifolds 1 Proposition Let (Oα)α∈A be an arbitrary covering of a C paracompact 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 Partition of Unity on Paracompact Banach Manifolds 1 Proposition Let (Oα)α∈A be an arbitrary covering of a C paracompact 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)

φα ≡ 1 αX∈A Partition of Unity on Paracompact Banach Manifolds 1 Proposition Let (Oα)α∈A be an arbitrary covering of a C paracompact 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)

φα ≡ 1 αX∈A where the sum is locally finite. Banach Space Bundles Banach Space Bundles Definition A Banach manifold V is called C p− Banach Space Bundle over another Banach manifold M Banach Space Bundles Definition A Banach manifold V is called C p− Banach Space Bundle over another Banach manifold M if there exists a Banach Space E, Banach Space Bundles Definition A Banach manifold V is called C p− Banach Space Bundle over another Banach manifold M if there exists a Banach Space E, a submersion π from V into M, Banach Space Bundles Definition A Banach manifold V is called C p− Banach Space Bundle 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 Banach Space Bundles Definition A Banach manifold V is called C p− Banach Space Bundle 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 −1 and a family of homeomorphism from π Ui into Ui × E Banach Space Bundles Definition A Banach manifold V is called C p− Banach Space Bundle 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 −1 and a family of homeomorphism from π Ui into Ui × Esuch that the following diagram commutes

τ −1 i π Ui −→ Ui × E πց ↓ σ Ui Banach Space Bundles Deﬁnition A Banach manifold V is called C p− Banach Space Bundle 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 −1 and a family of homeomorphism from π Ui into Ui × Esuch that the following diagram commutes

τ −1 i π Ui −→ Ui × E πց ↓ σ Ui

where σ is the canonical projection from Ui × E onto Ui . Banach Space Bundles Deﬁnition A Banach manifold V is called C p− Banach Space Bundle 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 −1 and a family of homeomorphism from π Ui into Ui × Esuch that the following diagram commutes

τ −1 i π Ui −→ Ui × E πց ↓ σ Ui

where σ is the canonical projection from Ui × E onto Ui . The −1 restriction of τi on each ﬁber Vx := π ({x}) for x ∈ Ui realizes a continuous isomorphism onto E. Banach Space Bundles Deﬁnition A Banach manifold V is called C p− Banach Space Bundle 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 −1 and a family of homeomorphism from π Ui into Ui × Esuch that the following diagram commutes

τ −1 i π Ui −→ Ui × E πց ↓ σ Ui

where σ is the canonical projection from Ui × E onto Ui . The −1 restriction of τi on each ﬁber Vx := π ({x}) for x ∈ Ui realizes a continuous isomorphism onto E. Moreover the map

−1 x ∈ Ui ∩ Uj −→ τi ◦ τj ∈L(E, E) π−1(x)

is C p. Finsler Structures on Banach Bundles.

Deﬁnition Let M be a normal Banach manifold and let V be a Banach Space Bundle over M. Finsler Structures on Banach Bundles.

Deﬁnition Let M be a normal Banach manifold and let V be a Banach Space Bundle over M.A Finsler structure on V is a continuous function k · k : V −→ R Finsler Structures on Banach Bundles.

Deﬁnition Let M be a normal Banach manifold and let V be a Banach Space Bundle over M.A Finsler structure on V is a continuous function k · k : V −→ R such that for any x ∈ M

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

Deﬁnition Let M be a normal Banach manifold and let V be a Banach Space Bundle over M.A Finsler structure on V is a continuous function k · k : V −→ R such that for any x ∈ M

k · kx := k · k|π−1({x}) is a norm on Vx . and the norms are locally uniformly comparable using any trivialization. Finsler Structures on Banach Bundles.

Deﬁnition Let M be a normal Banach manifold and let V be a Banach Space Bundle over M.A Finsler structure on V is a continuous function k · k : V −→ R such that for any x ∈ M

k · kx := k · k|π−1({x}) is a norm on Vx . and the norms are locally uniformly comparable using any trivialization.

Deﬁnition Let M be a normal C p Banach manifold. T M equipped with a Finsler structure is called a Finsler Manifold. A Finsler Structure on Sobolev Immersions. A Finsler Structure on Sobolev Immersions. Let Σ2 be a closed oriented 2−dim manifold and Nn be a closed sub-manifold of Rm. A Finsler Structure on Sobolev Immersions. Let Σ2 be a closed oriented 2−dim manifold and Nn be a closed sub-manifold of Rm. Let q > 2

2,q 2 n M := Wimm(Σ , N )

2,q 2 n 2 := Φ~ ∈ W (Σ , N ) ; rank (dΦx ) = 2 ∀x ∈ Σ n o A Finsler Structure on Sobolev Immersions. Let Σ2 be a closed oriented 2−dim manifold and Nn be a closed sub-manifold of Rm. Let q > 2

2,q 2 n M := Wimm(Σ , N )

2,q 2 n 2 := Φ~ ∈ W (Σ , N ) ; rank (dΦx ) = 2 ∀x ∈ Σ n o The tangent space to M at a point Φ~ is

2,q 2 Rm n 2 TΦ~ M = w~ ∈ W (Σ , ) ; w~ (x) ∈ TΦ(~ x)N ∀ x ∈ Σ . n o A Finsler Structure on Sobolev Immersions. Let Σ2 be a closed oriented 2−dim manifold and Nn be a closed sub-manifold of Rm. Let q > 2

2,q 2 n M := Wimm(Σ , N )

2,q 2 n 2 := Φ~ ∈ W (Σ , N ) ; rank (dΦx ) = 2 ∀x ∈ Σ n o The tangent space to M at a point Φ~ is

2,q 2 Rm n 2 TΦ~ M = w~ ∈ W (Σ , ) ; w~ (x) ∈ TΦ(~ x)N ∀ x ∈ Σ . n o

We equip TΦ~ M with the following norm

q/2 1/q 2 2 2 2 k~vk := |∇ ~v| + |∇~v| + |~v| dvol +k |∇~v| k ∞ Φ~ g~ g~ g~Φ g~Φ L (Σ) ZΣ h Φ Φ i A Finsler Structure on Sobolev Immersions. Let Σ2 be a closed oriented 2−dim manifold and Nn be a closed sub-manifold of Rm. Let q > 2

2,q 2 n M := Wimm(Σ , N )

2,q 2 n 2 := Φ~ ∈ W (Σ , N ) ; rank (dΦx ) = 2 ∀x ∈ Σ n o The tangent space to M at a point Φ~ is

2,q 2 Rm n 2 TΦ~ M = w~ ∈ W (Σ , ) ; w~ (x) ∈ TΦ(~ x)N ∀ x ∈ Σ . n o

We equip TΦ~ M with the following norm

q/2 1/q 2 2 2 2 k~vk := |∇ ~v| + |∇~v| + |~v| dvol +k |∇~v| k ∞ Φ~ g~ g~ g~Φ g~Φ L (Σ) ZΣ h Φ Φ i

2 Proposition k · kΦ~ deﬁne a C −Finsler struct. on M. The Palais Distance. The Palais Distance.

Theorem [Palais 1970] Let (M, k · k) bea Finsler Manifold. Deﬁne on M × M

1 dω d(p, q):= inf dt ω∈Ωp,q Z dt 0 ω(t)

The Palais Distance.

Theorem [Palais 1970] Let (M, k · k) bea Finsler Manifold. Deﬁne on M × M

1 dω d(p, q):= inf dt ω∈Ωp,q Z dt 0 ω(t)

where

1 Ωp,q := ω ∈ C ([0, 1], M) ; ω(0) = p ω(1) = q . The Palais Distance.

Theorem [Palais 1970] Let (M, k · k) bea Finsler Manifold. Deﬁne on M × M

1 dω d(p, q):= inf dt ω∈Ωp,q Z dt 0 ω(t)

where

1 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 topology as the one of the Banach Manifold. The Palais Distance.

Theorem [Palais 1970] Let (M, k · k) bea Finsler Manifold. Deﬁne on M × M

1 dω d(p, q):= inf dt ω∈Ωp,q Z dt 0 ω(t)

where

1 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 topology as the one of the Banach Manifold.

d is called Palais distance of the Finsler manifold (M, k · k). The Palais Distance.

Theorem [Palais 1970] Let (M, k · k) bea Finsler Manifold. Deﬁne on M × M

1 dω d(p, q):= inf dt ω∈Ωp,q Z dt 0 ω(t)

where

1 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 topology as the one of the Banach Manifold.

d is called Palais distance of the Finsler manifold (M, k · k).

Corollary Let (M, k · k) be a Finsler Manifold then M is paracompact. Completeness of the Palais Distance.

2Recall that every metric space is normal. Completeness of the Palais Distance.

Proposition Let q > 2 and let M be the normal2 Banach manifold

2,q 2 n ~ 2,q 2 n 2 Wimm(Σ , N ) := Φ ∈ W (Σ , N ) ; rank (dΦx ) = 2 ∀x ∈ Σ n o

2Recall that every metric space is normal. Completeness of the Palais Distance.

Proposition Let q > 2 and let M be the normal2 Banach manifold

2,q 2 n ~ 2,q 2 n 2 Wimm(Σ , N ) := Φ ∈ W (Σ , N ) ; rank (dΦx ) = 2 ∀x ∈ Σ n o The Finsler Manifold given by

q/2 1/q 2 2 2 2 k~vk := |∇ ~v| + |∇~v| + |~v| dvol +k |∇~v| k ∞ Φ~ g~ g~ g~Φ g~Φ L (Σ) ZΣ h Φ Φ i is complete for the Palais distance.

2Recall that every metric space is normal. Pseudo-gradients Pseudo-gradients Definition Let M bea C 2 Finsler Manifold and E be a C 1 function on M. Pseudo-gradients Definition Let M bea C 2 Finsler Manifold and E be a C 1 function on M. Denote ∗ M := {u ∈ M ; DEu 6= 0} . Pseudo-gradients Definition Let M bea C 2 Finsler Manifold and E be a C 1 function on M. Denote ∗ M := {u ∈ M ; DEu 6= 0} . A pseudo-gradient is a Lipschitz continuous section X : M∗ → T M∗ such that Pseudo-gradients Definition Let M bea C 2 Finsler Manifold and E be a C 1 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 Pseudo-gradients Definition Let M bea C 2 Finsler Manifold and E be a C 1 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 Pseudo-gradients Deﬁnition Let M bea C 2 Finsler Manifold and E be a C 1 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

Proposition Every C 1 function on a Finsler Manifold admits a pseudo-gradient. Pseudo-gradients Deﬁnition Let M bea C 2 Finsler Manifold and E be a C 1 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

Proposition Every C 1 function on a Finsler Manifold admits a pseudo-gradient.

“Proof” Use that Finsler Manifolds are Paracompact and “glue together” local pseudo-gradients constructed by local trivializations with an ad-hoc partition of unity. The Palais-Smale condition : (PS) The Palais-Smale condition : (PS) Definition Let E be a C 1 function on a Finsler manifold (M, k · k) and β ∈ E(M). The Palais-Smale condition : (PS) Definition Let E be a C 1 function on a Finsler manifold (M, k · k) and β ∈ E(M). One says that E fulfills the Palais-Smale condition at the level β if for any sequence un satisfying

E(un) −→ β and kDEun kun −→ 0 , The Palais-Smale condition : (PS) Deﬁnition Let E be a C 1 function on a Finsler manifold (M, k · k) and β ∈ E(M). One says that E fulﬁlls the Palais-Smale condition at the level β if for any sequence un satisfying

E(un) −→ β and kDEun kun −→ 0 ,

then there exists a subsequence un′ and u∞ ∈ M such that

dP(un′ , u∞) −→ 0 . The Palais-Smale condition : (PS) Deﬁnition Let E be a C 1 function on a Finsler manifold (M, k · k) and β ∈ E(M). One says that E fulﬁlls the Palais-Smale condition at the level β if for any sequence un satisfying

E(un) −→ β and kDEun kun −→ 0 ,

then there exists a subsequence un′ and u∞ ∈ M such that

dP(un′ , u∞) −→ 0 . and hence E(u∞)= β and DEu∞ = 0. The Palais-Smale condition : (PS) Deﬁnition Let E be a C 1 function on a Finsler manifold (M, k · k) and β ∈ E(M). One says that E fulﬁlls the Palais-Smale condition at the level β if for any sequence un satisfying

E(un) −→ β and kDEun kun −→ 0 ,

then there exists a subsequence un′ and u∞ ∈ M such that

dP(un′ , u∞) −→ 0 . and hence E(u∞)= β and DEu∞ = 0.

Example Let M be W 1,2(S1, Nn) for the Finsler structure given by

−1 n ∀ w~ ∈ ΓW 1,2 (u~ TN ) kw~ ku~ := kw~ kW 1,2(S1) The Palais-Smale condition : (PS) Deﬁnition Let E be a C 1 function on a Finsler manifold (M, k · k) and β ∈ E(M). One says that E fulﬁlls the Palais-Smale condition at the level β if for any sequence un satisfying

E(un) −→ β and kDEun kun −→ 0 ,

then there exists a subsequence un′ and u∞ ∈ M such that

dP(un′ , u∞) −→ 0 . and hence E(u∞)= β and DEu∞ = 0.

Example Let M be W 1,2(S1, Nn) for the Finsler structure given by

−1 n ∀ w~ ∈ ΓW 1,2 (u~ TN ) kw~ ku~ := kw~ kW 1,2(S1)

Then the Dirichlet Energy satisﬁes the Palais Smale condition for every level set. Admissible families

Deﬁnition A family of closed subsets A ⊂ P(M) of a Banach manifold M is called admissible family Admissible families

Deﬁnition A family of closed subsets 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 Admissible families

Deﬁnition A family of closed subsets 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,q 2 R3 Example M := Wimm(S , ). Admissible families

Deﬁnition A family of closed subsets 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,q 2 R3 2 R3 Z Z Example M := Wimm(S , ). Let c ∈ π1(Imm(S , )) = 2 × Admissible families

Deﬁnition A family of closed subsets 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,q 2 R3 2 R3 Z Z Example M := Wimm(S , ). Let c ∈ π1(Imm(S , )) = 2 × then

~ 0 2,q 2 R3 ~ ~ ~ A := Φ ∈ C ([0, 1], Wimm(S , )) ; Φ(0, ·)= Φ(1, ·) and[Φ] = c n o is admissible Palais Min-Max Principle Palais Min-Max Principle

Theorem[Palais 1970] Let (M, k · k) bea C 1,1−Finsler manifold. Palais Min-Max Principle

Theorem[Palais 1970] Let (M, k · k) bea C 1,1−Finsler manifold. Assume M is complete for dP Palais Min-Max Principle

Theorem[Palais 1970] Let (M, k · k) bea C 1,1−Finsler manifold. 1 Assume M is complete for dP and let E ∈ C (M). Palais Min-Max Principle

Theorem[Palais 1970] Let (M, k · k) bea C 1,1−Finsler manifold. 1 Assume M is complete for dP and let E ∈ C (M). Let A admissible. Palais Min-Max Principle

Theorem[Palais 1970] Let (M, k · k) bea C 1,1−Finsler manifold. 1 Assume M is complete for dP and let E ∈ C (M). Let A admissible. Let β := inf sup E(u) A∈A u∈A Palais Min-Max Principle

Theorem[Palais 1970] Let (M, k · k) bea C 1,1−Finsler manifold. 1 Assume M is complete for dP and let E ∈ C (M). Let A admissible. Let β := inf sup E(u) A∈A u∈A

Assume (PS)β for the level set β. Then there exists u ∈ M s.t.

DEu = 0   E(u)= β  Proof Proof By contradiction. Proof By contradiction.(PS)β ⇒ Proof By contradiction.(PS)β ⇒

∃ δ > 0 , ∃ ǫ> 0 β − ε< E(u) < β + ε =⇒ kDEuku ≥ δ . Proof By contradiction.(PS)β ⇒

∃ δ > 0 , ∃ ǫ> 0 β − ε< E(u) < β + ε =⇒ kDEuku ≥ δ .

∗ Let u ∈ M and φt

dφt (u) u = − X (φt (u)) η(E(φt (u))) in [0, t )  dt max  φ0(u)= u  Proof By contradiction.(PS)β ⇒

∃ δ > 0 , ∃ ǫ> 0 β − ε< E(u) < β + ε =⇒ kDEuku ≥ δ .

∗ Let u ∈ M and φt

dφt (u) u = − X (φt (u)) η(E(φt (u))) in [0, t )  dt max  φ0(u)= u  where supp(η) ⊂ [β − ε0, β + ε] and η ≡ 1 on [β − ε0/2, β + ε0/2]. Proof By contradiction.(PS)β ⇒

∃ δ > 0 , ∃ ǫ> 0 β − ε< E(u) < β + ε =⇒ kDEuku ≥ δ .

∗ Let u ∈ M and φt

dφt (u) u = − X (φt (u)) η(E(φt (u))) in [0, t )  dt max  φ0(u)= u  where supp(η) ⊂ [β − ε0, β + ε] and η ≡ 1 on [β − ε0/2, β + ε0/2].

1/2 1/2 d(φt1 (u), φt2 (u)) ≤ 2 |t2 − t1| [E(φt1 (u)) − E(φt2 (u))] Proof By contradiction.(PS)β ⇒

∃ δ > 0 , ∃ ǫ> 0 β − ε< E(u) < β + ε =⇒ kDEuku ≥ δ .

∗ Let u ∈ M and φt

dφt (u) u = − X (φt (u)) η(E(φt (u))) in [0, t )  dt max  φ0(u)= u  where supp(η) ⊂ [β − ε0, β + ε] and η ≡ 1 on [β − ε0/2, β + ε0/2].

1/2 1/2 d(φt1 (u), φt2 (u)) ≤ 2 |t2 − t1| [E(φt1 (u)) − E(φt2 (u))]

u If tmax < +∞ Proof By contradiction.(PS)β ⇒

∃ δ > 0 , ∃ ǫ> 0 β − ε< E(u) < β + ε =⇒ kDEuku ≥ δ .

∗ Let u ∈ M and φt

dφt (u) u = − X (φt (u)) η(E(φt (u))) in [0, t )  dt max  φ0(u)= u  where supp(η) ⊂ [β − ε0, β + ε] and η ≡ 1 on [β − ε0/2, β + ε0/2].

1/2 1/2 d(φt1 (u), φt2 (u)) ≤ 2 |t2 − t1| [E(φt1 (u)) − E(φt2 (u))]

u If tmax < +∞ then Completeness of (M, d) Proof By contradiction.(PS)β ⇒

∃ δ > 0 , ∃ ǫ> 0 β − ε< E(u) < β + ε =⇒ kDEuku ≥ δ .

∗ Let u ∈ M and φt

dφt (u) u = − X (φt (u)) η(E(φt (u))) in [0, t )  dt max  φ0(u)= u  where supp(η) ⊂ [β − ε0, β + ε] and η ≡ 1 on [β − ε0/2, β + ε0/2].

1/2 1/2 d(φt1 (u), φt2 (u)) ≤ 2 |t2 − t1| [E(φt1 (u)) − E(φt2 (u))]

u If tmax < +∞ then Completeness of (M, d) ⇒

u ∗ limu φt ( ) ∈ M t→tmax Proof By contradiction.(PS)β ⇒

∃ δ > 0 , ∃ ǫ> 0 β − ε< E(u) < β + ε =⇒ kDEuku ≥ δ .

∗ Let u ∈ M and φt

dφt (u) u = − X (φt (u)) η(E(φt (u))) in [0, t )  dt max  φ0(u)= u  where supp(η) ⊂ [β − ε0, β + ε] and η ≡ 1 on [β − ε0/2, β + ε0/2].

1/2 1/2 d(φt1 (u), φt2 (u)) ≤ 2 |t2 − t1| [E(φt1 (u)) − E(φt2 (u))]

u If tmax < +∞ then Completeness of (M, d) ⇒

u ∗ limu φt ( ) ∈ M Impossible ! t→tmax Proof By contradiction.(PS)β ⇒

∃ δ > 0 , ∃ ǫ> 0 β − ε< E(u) < β + ε =⇒ kDEuku ≥ δ .

∗ Let u ∈ M and φt

dφt (u) u = − X (φt (u)) η(E(φt (u))) in [0, t )  dt max  φ0(u)= u  where supp(η) ⊂ [β − ε0, β + ε] and η ≡ 1 on [β − ε0/2, β + ε0/2].

1/2 1/2 d(φt1 (u), φt2 (u)) ≤ 2 |t2 − t1| [E(φt1 (u)) − E(φt2 (u))]

u If tmax < +∞ then Completeness of (M, d) ⇒

u ∗ t R A A limu φt ( ) ∈ M Impossible ! ⇒∀ ∈ + ∀ ∈A φt ( ) ∈A t→tmax Proof By contradiction.(PS)β ⇒

∃ δ > 0 , ∃ ǫ> 0 β − ε< E(u) < β + ε =⇒ kDEuku ≥ δ .

∗ Let u ∈ M and φt

dφt (u) u = − X (φt (u)) η(E(φt (u))) in [0, t )  dt max  φ0(u)= u  where supp(η) ⊂ [β − ε0, β + ε] and η ≡ 1 on [β − ε0/2, β + ε0/2].

1/2 1/2 d(φt1 (u), φt2 (u)) ≤ 2 |t2 − t1| [E(φt1 (u)) − E(φt2 (u))]

u If tmax < +∞ then Completeness of (M, d) ⇒

u ∗ t R A A limu φt ( ) ∈ M Impossible ! ⇒∀ ∈ + ∀ ∈A φt ( ) ∈A t→tmax

Take A ∈A s.t. maxu∈A E(u) < β + ε0/2. Proof By contradiction.(PS)β ⇒

∃ δ > 0 , ∃ ǫ> 0 β − ε< E(u) < β + ε =⇒ kDEuku ≥ δ .

∗ Let u ∈ M and φt

dφt (u) u = − X (φt (u)) η(E(φt (u))) in [0, t )  dt max  φ0(u)= u  where supp(η) ⊂ [β − ε0, β + ε] and η ≡ 1 on [β − ε0/2, β + ε0/2].

1/2 1/2 d(φt1 (u), φt2 (u)) ≤ 2 |t2 − t1| [E(φt1 (u)) − E(φt2 (u))]

u If tmax < +∞ then Completeness of (M, d) ⇒

u ∗ t R A A limu φt ( ) ∈ M Impossible ! ⇒∀ ∈ + ∀ ∈A φt ( ) ∈A t→tmax

Take A ∈A s.t. maxu∈A E(u) < β + ε0/2. Apply φt ...cont. ! Birkhoﬀ Existence Result Revisited. Birkhoﬀ Existence Result Revisited.

M := W 1,2(S1, N2) where N2 ≃ S2 Birkhoﬀ Existence Result Revisited.

M := W 1,2(S1, N2) where N2 ≃ S2deﬁnes a complete Finsler manifold. Birkhoﬀ Existence Result Revisited.

M := W 1,2(S1, N2) where N2 ≃ S2deﬁnes a complete Finsler manifold.

E is (PS) on M. Birkhoﬀ Existence Result Revisited.

M := W 1,2(S1, N2) where N2 ≃ S2deﬁnes a complete Finsler manifold.

E is (PS) on M.

2 Let any sweep-out ~σ0 of N corresponding to a non zero element 2 of π2(N ). Birkhoﬀ Existence Result Revisited.

M := W 1,2(S1, N2) where N2 ≃ S2deﬁnes a complete Finsler manifold.

E is (PS) on M.

2 Let any sweep-out ~σ0 of N corresponding to a non zero element 2 of π2(N ).A := Ω~σ0 is admissible. Birkhoﬀ Existence Result Revisited.

M := W 1,2(S1, N2) where N2 ≃ S2deﬁnes a complete Finsler manifold.

E is (PS) on M.

2 Let any sweep-out ~σ0 of N corresponding to a non zero element 2 of π2(N ).A := Ω~σ0 is admissible.

Palais Theorem Birkhoﬀ Existence Result Revisited.

M := W 1,2(S1, N2) where N2 ≃ S2deﬁnes a complete Finsler manifold.

E is (PS) on M.

2 Let any sweep-out ~σ0 of N corresponding to a non zero element 2 of π2(N ).A := Ω~σ0 is admissible.

Palais Theorem ⇒

W~σ0 = inf max E(~σ(t, ·)) > 0 ~σ∈Ω~σ0 ∩Λ t∈[0,1]

is achieved by a closed geodesic. Birkhoﬀ Existence Result Revisited.

M := W 1,2(S1, N2) where N2 ≃ S2deﬁnes a complete Finsler manifold.

E is (PS) on M.

2 Let any sweep-out ~σ0 of N corresponding to a non zero element 2 of π2(N ).A := Ω~σ0 is admissible.

Palais Theorem ⇒

W~σ0 = inf max E(~σ(t, ·)) > 0 ~σ∈Ω~σ0 ∩Λ t∈[0,1]

is achieved by a closed geodesic.

This gives a new proof of Birkhoﬀ existence result.