Plateaus problem, isoperimetric inequalities, and asymptotic - PowerPoint PPT Presentation

Plateau’s problem, isoperimetric inequalities, and asymptotic geometry Stefan Wenger University of Fribourg, Switzerland British Mathematical Colloquium – March 2016 Stefan Wenger Plateau’s problem and applications

The Problem Classical Problem of Plateau To find surface of least area with prescribed boundary Stefan Wenger Plateau’s problem and applications

The Problem Classical Problem of Plateau To find surface of least area with prescribed boundary Originally: Surfaces of disc-type with prescribed rectifiable Jordan boundary Γ in X = R n , M n , . . . . D ⊂ R 2 → X with u | S 1 param. Γ and minimal Want u : ¯ � � � ∂ u ∂ x ∧ ∂ u � � Area ( u ) = � dx dy . � � ∂ y � D Stefan Wenger Plateau’s problem and applications

The Problem Classical Problem of Plateau To find surface of least area with prescribed boundary Originally: Surfaces of disc-type with prescribed rectifiable Jordan boundary Γ in X = R n , M n , . . . . D ⊂ R 2 → X with u | S 1 param. Γ and minimal Want u : ¯ � � � ∂ x ∧ ∂ u ∂ u � � Area ( u ) = � dx dy . � � ∂ y � D Solutions: X = R n : Douglas, Radó ’30. X = M n : Morrey ’48. Stefan Wenger Plateau’s problem and applications

Plateau’s Problem Classical proof in R n : 1 Minimize energy among (Sobolev) maps whose trace param. Γ . 2 Show that energy minimizers are conformal and minimize area. Stefan Wenger Plateau’s problem and applications

Plateau’s Problem Classical proof in R n : 1 Minimize energy among (Sobolev) maps whose trace param. Γ . 2 Show that energy minimizers are conformal and minimize area. Variants of Plateau’s problem: Surfaces with fixed genus in Riemannian manifolds (Courant ’37, Jost ’85). Integral currents in R n (Federer-Fleming ’60, . . . ). Chains mod 2 (Fleming ’66, . . . ). Stefan Wenger Plateau’s problem and applications

Plateau’s problem in metric spaces Generalizations: Integral currents in metric spaces: Cpt metric & some Banach spaces (Ambrosio-Kirchheim ’00). Dual Banach and CAT ( 0 ) -spaces (W.’05). Non-compact boundaries (Ambrosio-Schmidt ’13, W. ’14). Stefan Wenger Plateau’s problem and applications

Plateau’s problem in metric spaces Generalizations: Integral currents in metric spaces: Cpt metric & some Banach spaces (Ambrosio-Kirchheim ’00). Dual Banach and CAT ( 0 ) -spaces (W.’05). Non-compact boundaries (Ambrosio-Schmidt ’13, W. ’14). Discs in: CAT ( 0 ) -spaces (Nikolaev ’79). some Alexandrov spaces (Mese-Zulkowski ’10). Finsler 3-space (Overath-von der Mosel ’14). Stefan Wenger Plateau’s problem and applications

Plateau’s problem in metric spaces Generalizations: Integral currents in metric spaces: Cpt metric & some Banach spaces (Ambrosio-Kirchheim ’00). Dual Banach and CAT ( 0 ) -spaces (W.’05). Non-compact boundaries (Ambrosio-Schmidt ’13, W. ’14). Discs in: CAT ( 0 ) -spaces (Nikolaev ’79). some Alexandrov spaces (Mese-Zulkowski ’10). Finsler 3-space (Overath-von der Mosel ’14). Aim: Existence of area min. discs in proper metric spaces. Stefan Wenger Plateau’s problem and applications

Overview Part I Area minimizing discs in metric spaces (existence and regularity) Part II Applications to some problems in geometry and geometric group theory Stefan Wenger Plateau’s problem and applications

Metric space valued Sobolev maps Let ( X , d ) be complete metric space, D open unit disc in R 2 , p > 1. Definition (Reshetnyak ’97, Ambrosio ’90) A map u : D → X is in W 1 , p ( D , X ) if u measurable and essentially separably valued ∃ g ∈ L p ( D ) such that ∀ ϕ ∈ Lip 1 ( X ) have ϕ ◦ u ∈ W 1 , p ( D ) with |∇ ( ϕ ◦ u ) | ≤ g a.e. Stefan Wenger Plateau’s problem and applications

Metric space valued Sobolev maps Let ( X , d ) be complete metric space, D open unit disc in R 2 , p > 1. Definition (Reshetnyak ’97, Ambrosio ’90) A map u : D → X is in W 1 , p ( D , X ) if u measurable and essentially separably valued ∃ g ∈ L p ( D ) such that ∀ ϕ ∈ Lip 1 ( X ) have ϕ ◦ u ∈ W 1 , p ( D ) with |∇ ( ϕ ◦ u ) | ≤ g a.e. Equivalent definitions: Korevaar-Schoen ’93, Jost ’94, Hajłasz ’96 Heinonen-Koskela-Shanmugalingam-Tyson ’01, ’15 Stefan Wenger Plateau’s problem and applications

Metric space valued Sobolev maps Reshetnyak’s energy of u : � � E p � g � p + ( u ) := inf L p ( D ) : g Reshetnyak gradient of u . Stefan Wenger Plateau’s problem and applications

Metric space valued Sobolev maps Reshetnyak’s energy of u : � � E p � g � p + ( u ) := inf L p ( D ) : g Reshetnyak gradient of u . Trace of u : u ( tv ) is abs. cont. for a.e. v ∈ S 1 . ∃ ¯ u rep. such that t �→ ¯ The trace of u is defined by tr ( u )( v ) := lim t ր 1 ¯ u ( tv ) and satisfies tr ( u ) ∈ L p ( S 1 , X ) . Stefan Wenger Plateau’s problem and applications

Approximate metric differentiability Proposition (Kirchheim ’94, Karmanova ’07) If u ∈ W 1 , p ( D , X ) then for a.e. z ∈ D there exists a unique seminorm md z u on R 2 with d ( u ( z + v ) , u ( z )) − md z u ( v ) ap − lim = 0 . | v | v → 0 Stefan Wenger Plateau’s problem and applications

Approximate metric differentiability Proposition (Kirchheim ’94, Karmanova ’07) If u ∈ W 1 , p ( D , X ) then for a.e. z ∈ D there exists a unique seminorm md z u on R 2 with d ( u ( z + v ) , u ( z )) − md z u ( v ) ap − lim = 0 . | v | v → 0 Remarks: If X = ( R n , � · � ) then md z u ( · ) = � d z u ( · ) � . Stefan Wenger Plateau’s problem and applications

Approximate metric differentiability Proposition (Kirchheim ’94, Karmanova ’07) If u ∈ W 1 , p ( D , X ) then for a.e. z ∈ D there exists a unique seminorm md z u on R 2 with d ( u ( z + v ) , u ( z )) − md z u ( v ) ap − lim = 0 . | v | v → 0 Remarks: If X = ( R n , � · � ) then md z u ( · ) = � d z u ( · ) � . Reshetnyak’s energy satisfies � � md z u ( v ) p : v ∈ S 1 � E p + ( u ) = max dz . D Stefan Wenger Plateau’s problem and applications

Approximate metric differentiability Proposition (Kirchheim ’94, Karmanova ’07) If u ∈ W 1 , p ( D , X ) then for a.e. z ∈ D there exists a unique seminorm md z u on R 2 with d ( u ( z + v ) , u ( z )) − md z u ( v ) ap − lim = 0 . | v | v → 0 Remarks: If X = ( R n , � · � ) then md z u ( · ) = � d z u ( · ) � . Reshetnyak’s energy satisfies � � md z u ( v ) p : v ∈ S 1 � E p + ( u ) = max dz . D � �� � I p + ( md z u ) Stefan Wenger Plateau’s problem and applications

Approximate metric differentiability Proposition (Kirchheim ’94, Karmanova ’07) If u ∈ W 1 , p ( D , X ) then for a.e. z ∈ D there exists a unique seminorm md z u on R 2 with d ( u ( z + v ) , u ( z )) − md z u ( v ) ap − lim = 0 . | v | v → 0 Remarks: If X = ( R n , � · � ) then md z u ( · ) = � d z u ( · ) � . Reshetnyak’s energy satisfies � � md z u ( v ) p : v ∈ S 1 � E p + ( u ) = max dz . D � �� � I p + ( md z u ) X has property (ET) if md z u is (deg.) inner product for all u . Ex: (Sub-)Riem. mfds, spaces of bounded curvature, etc. Stefan Wenger Plateau’s problem and applications

Area of Sobolev maps Definition The parametrized Hausdorff area of u ∈ W 1 , 2 ( D , X ) is � Area ( u ) = J 2 ( md z u ) dz , D where J 2 ( � · � ) is Hausdorff measure w.r.t. � · � of Eucl. unit square. Stefan Wenger Plateau’s problem and applications

Area of Sobolev maps Definition The parametrized Hausdorff area of u ∈ W 1 , 2 ( D , X ) is � Area ( u ) = J 2 ( md z u ) dz , D where J 2 ( � · � ) is Hausdorff measure w.r.t. � · � of Eucl. unit square. Remarks: If u has Lusin’s property (N) then � # u − 1 ( x ) d H 2 Area ( u ) = X ( x ) . X There exist other natural choices of parametrized area. Stefan Wenger Plateau’s problem and applications

An infinitesimal notion of quasi-conformality Definition A map u ∈ W 1 , 2 ( D , X ) is Q -quasi-conformal if for a.e. z ∈ D md z u ( v ) ≤ Q · md z u ( w ) for all v , w ∈ S 1 . Stefan Wenger Plateau’s problem and applications

An infinitesimal notion of quasi-conformality Definition A map u ∈ W 1 , 2 ( D , X ) is Q -quasi-conformal if for a.e. z ∈ D md z u ( v ) ≤ Q · md z u ( w ) for all v , w ∈ S 1 . Remark: If X Riemannian manifold then u is 1-quasi-conformal ⇔ u weakly conformal, i.e. � � � � � ∂ u � ∂ u ∂ u ∂ x , ∂ u � � � � � = and = 0 . � � � � ∂ x ∂ y ∂ y � � � Stefan Wenger Plateau’s problem and applications

An infinitesimal notion of quasi-conformality Definition A map u ∈ W 1 , 2 ( D , X ) is Q -quasi-conformal if for a.e. z ∈ D md z u ( v ) ≤ Q · md z u ( w ) for all v , w ∈ S 1 . Remark: If X Riemannian manifold then u is 1-quasi-conformal ⇔ u weakly conformal, i.e. � � � � � ∂ u � ∂ u ∂ u ∂ x , ∂ u � � � � � = and = 0 . � � � � ∂ x ∂ y ∂ y � � � √ Example: The identity map from D to ( R 2 , � · � ∞ ) is 2-qc. Stefan Wenger Plateau’s problem and applications

Solution to Plateau’s problem Given Γ ⊂ X Jordan curve let Λ(Γ) = { v ∈ W 1 , 2 ( D , X ) : tr ( v ) weakly mon. param. of Γ } . Stefan Wenger Plateau’s problem and applications