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 = Rn, Mn, . . . . Want u : ¯ D ⊂ R2 → X with u|S1 param. Γ and minimal Area(u) =

• D
• ∂u

∂x ∧ ∂u ∂y

• dx dy.

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 = Rn, Mn, . . . . Want u : ¯ D ⊂ R2 → X with u|S1 param. Γ and minimal Area(u) =

• D
• ∂u

∂x ∧ ∂u ∂y

• dx dy.

Solutions: X = Rn: Douglas, Radó ’30. X = Mn: Morrey ’48.

Stefan Wenger Plateau’s problem and applications

Plateau’s Problem

Classical proof in Rn:

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 Rn:

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 Rn (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 R2, 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 ∈ Lp(D) such that ∀ϕ ∈ Lip1(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 R2, 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 ∈ Lp(D) such that ∀ϕ ∈ Lip1(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

+(u) := inf

• gp

Lp(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

+(u) := inf

• gp

Lp(D) : g Reshetnyak gradient of u

• .

Trace of u: ∃ ¯ u rep. such that t → ¯ u(tv) is abs. cont. for a.e. v ∈ S1. The trace of u is defined by tr(u)(v) := lim

tր1 ¯

u(tv) and satisfies tr(u) ∈ Lp(S1, 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 mdz u on R2 with ap − lim

v→0

d(u(z + v), u(z)) − mdz u(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 mdz u on R2 with ap − lim

v→0

d(u(z + v), u(z)) − mdz u(v) |v| = 0. Remarks: If X = (Rn, · ) then mdz u(·) = dzu(·).

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 mdz u on R2 with ap − lim

v→0

d(u(z + v), u(z)) − mdz u(v) |v| = 0. Remarks: If X = (Rn, · ) then mdz u(·) = dzu(·). Reshetnyak’s energy satisfies E p

+(u) =

• D

max

• mdz u(v)p : v ∈ S1

dz.

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 mdz u on R2 with ap − lim

v→0

d(u(z + v), u(z)) − mdz u(v) |v| = 0. Remarks: If X = (Rn, · ) then mdz u(·) = dzu(·). Reshetnyak’s energy satisfies E p

+(u) =

• D

max

• mdz u(v)p : v ∈ S1
• Ip

+(mdz u)

dz.

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 mdz u on R2 with ap − lim

v→0

d(u(z + v), u(z)) − mdz u(v) |v| = 0. Remarks: If X = (Rn, · ) then mdz u(·) = dzu(·). Reshetnyak’s energy satisfies E p

+(u) =

• D

max

• mdz u(v)p : v ∈ S1
• Ip

+(mdz u)

dz. X has property (ET) if mdz 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) =

• D

J2(mdz u) dz, where J2( · ) 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) =

• D

J2(mdz u) dz, where J2( · ) is Hausdorff measure w.r.t. · of Eucl. unit square. Remarks: If u has Lusin’s property (N) then Area(u) =

• X

#u−1(x) dH2

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 mdz u(v) ≤ Q · mdz u(w) for all v, w ∈ S1.

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 mdz u(v) ≤ Q · mdz u(w) for all v, w ∈ S1. Remark: If X Riemannian manifold then u is 1-quasi-conformal ⇔ u weakly conformal, i.e.

• ∂u

∂x

• =
• ∂u

∂y

• and

∂u ∂x , ∂u ∂y

• = 0.

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 mdz u(v) ≤ Q · mdz u(w) for all v, w ∈ S1. Remark: If X Riemannian manifold then u is 1-quasi-conformal ⇔ u weakly conformal, i.e.

• ∂u

∂x

• =
• ∂u

∂y

• and

∂u ∂x , ∂u ∂y

• = 0.

Example: The identity map from D to (R2, · ∞) 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

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}. Theorem (Lytchak-W. ’15) If X is proper and Γ ⊂ X with Λ(Γ) = ∅ then there exists u ∈ Λ(Γ) with Area(u) = inf{Area(v) : v ∈ Λ(Γ)} and which is √ 2-quasi-conformal.

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}. Theorem (Lytchak-W. ’15) If X is proper and Γ ⊂ X with Λ(Γ) = ∅ then there exists u ∈ Λ(Γ) with Area(u) = inf{Area(v) : v ∈ Λ(Γ)} and which is √ 2-quasi-conformal. Remarks: √ 2-qc. is optimal in general. If X has property (ET) then obtain u conformal.

Stefan Wenger Plateau’s problem and applications

Energy minimizers

Theorem (Lytchak-W. ’15) If X is complete and u ∈ W 1,2(D, X) is such that E 2

+(u) ≤ E 2 +(u ◦ ψ)

for every biLipschitz homeomorphism ψ of D then u is √ 2-qc. Remarks: If X has property (ET) then u conformal.

Stefan Wenger Plateau’s problem and applications

Energy minimizers

Theorem (Lytchak-W. ’15) If X is complete and u ∈ W 1,2(D, X) is such that E 2

+(u) ≤ E 2 +(u ◦ ψ)

for every biLipschitz homeomorphism ψ of D then u is √ 2-qc. Remarks: If X has property (ET) then u conformal. The classical proof: {ψt} diffeos, ψ0 = id, (ξ, η) := ∂ψt

∂t

|t=0 ⇒ 0 = d dt E2(u ◦ ψt) =

• D

(|ux |2 − |uy |2) · (ξx − ηy ) dx dy +

• D

ux , uy · (ξy + ηx ) dx dy

can be adapted when X has (ET) but breaks down in general.

Stefan Wenger Plateau’s problem and applications

Proof that energy minimizers are quasi-conformal

Suppose √ 2-qc. fails at some point, say z = 0. Set s := mdz u.

Stefan Wenger Plateau’s problem and applications

Proof that energy minimizers are quasi-conformal

Suppose √ 2-qc. fails at some point, say z = 0. Set s := mdz u. Then: ∃ T ∈ SL2(R) such that I2

+(s ◦ T) < I2 +(s).

Stefan Wenger Plateau’s problem and applications

Proof that energy minimizers are quasi-conformal

Suppose √ 2-qc. fails at some point, say z = 0. Set s := mdz u. Then: ∃ T ∈ SL2(R) such that I2

+(s ◦ T) < I2 +(s).

r > 0 ⇒ ∃ ρ biLip. homeo. of R2 s.th.

ρ = T −1 on B := B(0, r). ρ conformal on R2 \ B.

Stefan Wenger Plateau’s problem and applications

Proof that energy minimizers are quasi-conformal

Suppose √ 2-qc. fails at some point, say z = 0. Set s := mdz u. Then: ∃ T ∈ SL2(R) such that I2

+(s ◦ T) < I2 +(s).

r > 0 ⇒ ∃ ρ biLip. homeo. of R2 s.th.

ρ = T −1 on B := B(0, r). ρ conformal on R2 \ B.

Consequently, E 2

+(u ◦ ρ−1|ρ(D\B)) = E 2 +(u|D\B),

E 2

+(u ◦ ρ−1|ρ(B)) =

• B

I2

+(mdw u ◦ T) dw

≃ |B| · I2

+(s ◦ T) < |B| · I2 +(s) ≃ E 2 +(u|B).

Stefan Wenger Plateau’s problem and applications

Back to Plateau’s problem

The classical proof in Rn relies on:

1 Existence of energy minimizers in Λ(Γ). 2 Energy minimizers are conformal and minimize area.

Problem: Step 2 breaks down in metric spaces.

Stefan Wenger Plateau’s problem and applications

Back to Plateau’s problem

The classical proof in Rn relies on:

1 Existence of energy minimizers in Λ(Γ). 2 Energy minimizers are conformal and minimize area.

Problem: Step 2 breaks down in metric spaces. Proposition There exists X biLipschitz homeo. to S2 and Γ ⊂ X biLipschitz such that energy minimizers in Λ(Γ) are not area minimizers. However: If X has (ET) then energy minimizers are area minimizers.

Stefan Wenger Plateau’s problem and applications

Back to Plateau’s problem

The classical proof in Rn relies on:

1 Existence of energy minimizers in Λ(Γ). 2 Energy minimizers are conformal and minimize area.

Problem: Step 2 breaks down in metric spaces. Proposition There exists X biLipschitz homeo. to S2 and Γ ⊂ X biLipschitz such that energy minimizers in Λ(Γ) are not area minimizers. However: If X has (ET) then energy minimizers are area minimizers. General X: prove and use lower semi-continuity of Area.

Stefan Wenger Plateau’s problem and applications

Local quadratic isoperimetric inequality

Definition A metric space X admits (C, r0)-isoperimetric inequality if every Lipschitz curve c : S1 → X with ℓ(c) ≤ r0 is the trace of some u ∈ W 1,2(D, X) with Area(u) ≤ C · ℓ(c)2. Examples: Homogeneously regular Riemannian manifolds (Morrey). Compact Lipschitz manifolds. CAT(κ)-spaces and compact Alexandrov spaces. Banach spaces. Some Carnot-Carathéodory spaces (e.g. Heisenberg Hn≥2). ˜ M, where M closed Riem. s.th. π1(M) has quadratic Dehn fct.

Stefan Wenger Plateau’s problem and applications

Regularity of area minimizers

Theorem (Lytchak-W. ’15) Let X be complete metric space, Γ ⊂ X Jordan curve, u ∈ Λ(Γ) such that Area(u) = inf{Area(v) : v ∈ Λ(Γ)} and u is Q-qc. If X admits (C, r0)-isop. then:

1 u is loc. W 1,p>2, thus has Lusin (N). 2 u is locally α-Hölder and extends cont. to D, α =

1 4πQ2C .

3 If Γ is chord-arc then u is Hölder on D. Stefan Wenger Plateau’s problem and applications

Regularity of area minimizers

Theorem (Lytchak-W. ’15) Let X be complete metric space, Γ ⊂ X Jordan curve, u ∈ Λ(Γ) such that Area(u) = inf{Area(v) : v ∈ Λ(Γ)} and u is Q-qc. If X admits (C, r0)-isop. then:

1 u is loc. W 1,p>2, thus has Lusin (N). 2 u is locally α-Hölder and extends cont. to D, α =

1 4πQ2C .

3 If Γ is chord-arc then u is Hölder on D.

Remarks: Proof along lines of classical proofs. Hölder exponent α is optimal (when Q = 1).

Stefan Wenger Plateau’s problem and applications

Applications

Applications:

1 Regularity of quasi-harmonic maps. 2 Geometry of spaces with quadratic isoperimetric inequality:

Non-positive curvature Negative curvature Asymptotic cones

3 Nilpotent groups Stefan Wenger Plateau’s problem and applications

Regularity of quasi-harmonic (q.h.) maps

Let X complete metric space, Ω ⊂ R2 bounded Lipschitz domain. Definition A map u ∈ W 1,2(Ω, X) is called M-q.h. if for every Lip. domain Ω′ ⊂ Ω and v ∈ W 1,2(Ω′, X) with tr(v) = tr(u|Ω′) have E 2

+(u|Ω′) ≤ M · E 2 +(v).

Stefan Wenger Plateau’s problem and applications

Regularity of quasi-harmonic (q.h.) maps

Let X complete metric space, Ω ⊂ R2 bounded Lipschitz domain. Definition A map u ∈ W 1,2(Ω, X) is called M-q.h. if for every Lip. domain Ω′ ⊂ Ω and v ∈ W 1,2(Ω′, X) with tr(v) = tr(u|Ω′) have E 2

+(u|Ω′) ≤ M · E 2 +(v).

Examples: harmonic maps, solutions of Plateau’s problem. Family of q.h. maps invariant under:

restriction to Lip. domain

• biLip. change of metrics.

Stefan Wenger Plateau’s problem and applications

Regularity of quasi-harmonic maps

Theorem (Lytchak-W. ’15) If X proper with (C, r0)-isop. and u ∈ W 1,2(Ω, X) M-q.h. then

1 u is locally α-Hölder with α = α(C, M). 2 if tr(u) continuous then u extends cont. to Ω. 3 if tr(u) Lipschitz then u globally Hölder. Stefan Wenger Plateau’s problem and applications

Regularity of quasi-harmonic maps

Theorem (Lytchak-W. ’15) If X proper with (C, r0)-isop. and u ∈ W 1,2(Ω, X) M-q.h. then

1 u is locally α-Hölder with α = α(C, M). 2 if tr(u) continuous then u extends cont. to Ω. 3 if tr(u) Lipschitz then u globally Hölder.

For u harmonic and X CAT(0), Heisenberg group, or cpt. Alexandrov: Korevaar-Schoen ’93, Capogna-Lin ’01, Mese-Zulkowski ’10.

Stefan Wenger Plateau’s problem and applications

Regularity of quasi-harmonic maps

Theorem (Lytchak-W. ’15) If X proper with (C, r0)-isop. and u ∈ W 1,2(Ω, X) M-q.h. then

1 u is locally α-Hölder with α = α(C, M). 2 if tr(u) continuous then u extends cont. to Ω. 3 if tr(u) Lipschitz then u globally Hölder.

For u harmonic and X CAT(0), Heisenberg group, or cpt. Alexandrov: Korevaar-Schoen ’93, Capogna-Lin ’01, Mese-Zulkowski ’10. Proof. Every short Lipschitz curve c : S1 → X is trace of u ∈ W 1,p(D, X) with

• E p

+(u)

1

p ≤ C ′ · Lip(c)

for some p > 2 and C ′ depending on C. This + Morrey args.

Stefan Wenger Plateau’s problem and applications

(Asymptotic) geometry

Problem: Study (asymptotic) properties of spaces/groups with quadratic isoperimetric inequality.

Stefan Wenger Plateau’s problem and applications

(Asymptotic) geometry

Problem: Study (asymptotic) properties of spaces/groups with quadratic isoperimetric inequality. Motivation from Geometric Group Theory: Gromov’s program to classify groups up to quasi-isometry:

Finitely generated groups as metric spaces. Quasi-isometry: large scale analog of biLipschitz homeo.

Stefan Wenger Plateau’s problem and applications

(Asymptotic) geometry

Problem: Study (asymptotic) properties of spaces/groups with quadratic isoperimetric inequality. Motivation from Geometric Group Theory: Gromov’s program to classify groups up to quasi-isometry:

Finitely generated groups as metric spaces. Quasi-isometry: large scale analog of biLipschitz homeo.

Isoperimetric function FA0(X, r) of a space X:

Measures difficulty to fill curves of length ≤ r by Lip. discs. If G M geometrically then Dehn fct. δG(n) ≃ FA0(M, n). Asymptotic growth of FA0(·, r) is quasi-isometry invariant.

Stefan Wenger Plateau’s problem and applications

(Asymptotic) geometry

Problem: Study (asymptotic) properties of spaces/groups with quadratic isoperimetric inequality. Motivation from Geometric Group Theory: Gromov’s program to classify groups up to quasi-isometry:

Finitely generated groups as metric spaces. Quasi-isometry: large scale analog of biLipschitz homeo.

Isoperimetric function FA0(X, r) of a space X:

Measures difficulty to fill curves of length ≤ r by Lip. discs. If G M geometrically then Dehn fct. δG(n) ≃ FA0(M, n). Asymptotic growth of FA0(·, r) is quasi-isometry invariant.

Spaces with quadratic isoperimetric inequality are not well understood.

Stefan Wenger Plateau’s problem and applications

Non-positive curvature

Theorem (Lytchak-W. ’16) Let X be proper geodesic metric space. Then TFAE:

1 X is CAT(0). 2 X admits quadratic isoperimetric inequality with constant

1 4π.

Stefan Wenger Plateau’s problem and applications

Non-positive curvature

Theorem (Lytchak-W. ’16) Let X be proper geodesic metric space. Then TFAE:

1 X is CAT(0). 2 X admits quadratic isoperimetric inequality with constant

1 4π.

Remarks: "⇒": due to Reshetnyak ’68. "⇐": Idea:

1

Fill a geodesic Jordan triangle in X with a minimal disc Σ.

2

Consider Σ with intrinsic metric; try to show is CAT(0)-space.

3

Problem: The "differential geometric"pullback metric has non-positive curvature; how to link this almost-everywhere

• bject to the intrinsic metric?

Stefan Wenger Plateau’s problem and applications

Gromov hyperbolicity

Theorem (W. ’08) If X is geodesic metric space and ε, r > 0 such that every Lip. loop c in X of length ≥ r is trace of some u ∈ W 1,2(D, X) with Area(u) ≤ 1 − ε 4π · ℓ(c)2 then X is Gromov hyperbolic. Remarks: Gromov ’87:

1 16π for Riem mfds, 1 4000 for metric spaces.

Proof uses currents in metric spaces. If X proper then can use existence/regularity results above.

Stefan Wenger Plateau’s problem and applications

Asymptotic properties

Observation: If X has non-positive curvature (e.g. CAT(0) or convex bicombing) then X has Lipschitz extension property: A X λ-Lipschitz ∩ R2 ∃ Cλ-Lipschitz extension Consequence: X and its asymptotic cones have Lipschitz extension property. are simply connected. have quadratic isoperimetric inequality.

Stefan Wenger Plateau’s problem and applications

Asymptotic properties

Theorem (Lytchak-W.-Young ’16) For proper geodesic metric spaces, a quadratic isoperimetric inequality passes to: pointed Gromov-Hausdorff limits. ultralimits and asymptotic cones.

Stefan Wenger Plateau’s problem and applications

Asymptotic properties

Theorem (Lytchak-W.-Young ’16) For proper geodesic metric spaces, a quadratic isoperimetric inequality passes to: pointed Gromov-Hausdorff limits. ultralimits and asymptotic cones. Theorem (Lytchak-W.-Young ’16) Let X be (asymptotic cone of) a proper geodesic metric space admitting a quadratic isoperimetric inequality. Then X has the α-Hölder extension property for all α ∈ (0, 1): A X α-Hölder ∩ R2 ∃ α-Hölder extension In particular, X is simply connected.

Stefan Wenger Plateau’s problem and applications

Nilpotent groups

Question: Does every finitely generated nilpotent G have δG(n) ≃ nα for some α?

Stefan Wenger Plateau’s problem and applications

Nilpotent groups

Question: Does every finitely generated nilpotent G have δG(n) ≃ nα for some α? Theorem (W. ’11) There exists G nilpotent of step 2 such that n2̺(n) δG(n) n2 log(n) for some function ̺(n) → ∞ as n → ∞. Remarks: Proof uses Ambrosio-Kirchheim currents and compactness. May use results from Part I instead.

Stefan Wenger Plateau’s problem and applications