Min-Max methods for surfaces with boundary
Camillo De Lellis
Institut für Mathematik
University of Zürich
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
1 / 30
Introduction
A classical fact (cf. mountain pass, Morse theory etc.):
Theorem
A function with two local minima has always a third critical point
(provided some reasonable assumptions are fulfilled).
Problem
Let Γ be an n − 1-dim. closed surface in Rn+1 for which there exist two
distinct strictly stable embedded minimal hypersurfaces Σ1 and Σ2 with
∂Σi = Γ .
Is there a third embedded minimal surface Σ3 with the same
boundary?
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
2 / 30
Introduction
A classical fact (cf. mountain pass, Morse theory etc.):
Theorem
A function with two local minima has always a third critical point
(provided some reasonable assumptions are fulfilled).
Problem
Let Γ be an n − 1-dim. closed surface in Rn+1 for which there exist two
distinct strictly stable embedded minimal hypersurfaces Σ1 and Σ2 with
∂Σi = Γ .
Is there a third embedded minimal surface Σ3 with the same
boundary?
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
2 / 30
Introduction
Minimality: zero first variation for any normal perturbation which fixes
the boundary.
d δΣ(χ) =
Voln (Φt (Σ))
{z
}
dt t=0 |
=:V (t)
dΦt
= χ(Φt )
dt
χ = fν
χ = 0 on ∂Σ .
Stability: nonnegative second variation, δ 2 Σ(χ) = V 00 (0) ≥ 0.
Strict stability: V 00 (0)>0 when f 6= 0.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
3 / 30
Introduction
Minimality: zero first variation for any normal perturbation which fixes
the boundary.
d δΣ(χ) =
Voln (Φt (Σ))
{z
}
dt t=0 |
=:V (t)
dΦt
= χ(Φt )
dt
χ = fν
χ = 0 on ∂Σ .
Stability: nonnegative second variation, δ 2 Σ(χ) = V 00 (0) ≥ 0.
Strict stability: V 00 (0)>0 when f 6= 0.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
3 / 30
Introduction
Minimality: zero first variation for any normal perturbation which fixes
the boundary.
d δΣ(χ) =
Voln (Φt (Σ))
{z
}
dt t=0 |
=:V (t)
dΦt
= χ(Φt )
dt
χ = fν
χ = 0 on ∂Σ .
Stability: nonnegative second variation, δ 2 Σ(χ) = V 00 (0) ≥ 0.
Strict stability: V 00 (0)>0 when f 6= 0.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
3 / 30
Introduction
Minimality: zero first variation for any normal perturbation which fixes
the boundary.
d δΣ(χ) =
Voln (Φt (Σ))
{z
}
dt t=0 |
=:V (t)
dΦt
= χ(Φt )
dt
χ = fν
χ = 0 on ∂Σ .
Stability: nonnegative second variation, δ 2 Σ(χ) = V 00 (0) ≥ 0.
Strict stability: V 00 (0)>0 when f 6= 0.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
3 / 30
Main theorems
Yes for n = 2 (without embeddedness but control on topology):
I
Douglas-Rado approach, Tomi-Tromba, Tromba, etc.;
I
Harmonic map flow, Struwe.
n = 2 special: existence of conformal parametrization, criticality of the
Dirichlet energy.
No result in higher dimension.
Theorem (De Lellis - Ramic, 2016)
YES for n ≤ 6 if
(C) Γ lies on the boundary of a bounded, uniformly convex open set.
(N) Σ1 and Σ2 do not intersect in the interior.
NB: smoothness up to the boundary. Hence no multiplicity.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
4 / 30
Main theorems
Yes for n = 2 (without embeddedness but control on topology):
I
Douglas-Rado approach, Tomi-Tromba, Tromba, etc.;
I
Harmonic map flow, Struwe.
n = 2 special: existence of conformal parametrization, criticality of the
Dirichlet energy.
No result in higher dimension.
Theorem (De Lellis - Ramic, 2016)
YES for n ≤ 6 if
(C) Γ lies on the boundary of a bounded, uniformly convex open set.
(N) Σ1 and Σ2 do not intersect in the interior.
NB: smoothness up to the boundary. Hence no multiplicity.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
4 / 30
Main theorems
Yes for n = 2 (without embeddedness but control on topology):
I
Douglas-Rado approach, Tomi-Tromba, Tromba, etc.;
I
Harmonic map flow, Struwe.
n = 2 special: existence of conformal parametrization, criticality of the
Dirichlet energy.
No result in higher dimension.
Theorem (De Lellis - Ramic, 2016)
YES for n ≤ 6 if
(C) Γ lies on the boundary of a bounded, uniformly convex open set.
(N) Σ1 and Σ2 do not intersect in the interior.
NB: smoothness up to the boundary. Hence no multiplicity.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
4 / 30
Main theorems
Yes for n = 2 (without embeddedness but control on topology):
I
Douglas-Rado approach, Tomi-Tromba, Tromba, etc.;
I
Harmonic map flow, Struwe.
n = 2 special: existence of conformal parametrization, criticality of the
Dirichlet energy.
No result in higher dimension.
Theorem (De Lellis - Ramic, 2016)
YES for n ≤ 6 if
(C) Γ lies on the boundary of a bounded, uniformly convex open set.
(N) Σ1 and Σ2 do not intersect in the interior.
NB: smoothness up to the boundary. Hence no multiplicity.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
4 / 30
Main theorems
Yes for n = 2 (without embeddedness but control on topology):
I
Douglas-Rado approach, Tomi-Tromba, Tromba, etc.;
I
Harmonic map flow, Struwe.
n = 2 special: existence of conformal parametrization, criticality of the
Dirichlet energy.
No result in higher dimension.
Theorem (De Lellis - Ramic, 2016)
YES for n ≤ 6 if
(C) Γ lies on the boundary of a bounded, uniformly convex open set.
(N) Σ1 and Σ2 do not intersect in the interior.
NB: smoothness up to the boundary. Hence no multiplicity.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
4 / 30
Main theorems
Yes for n = 2 (without embeddedness but control on topology):
I
Douglas-Rado approach, Tomi-Tromba, Tromba, etc.;
I
Harmonic map flow, Struwe.
n = 2 special: existence of conformal parametrization, criticality of the
Dirichlet energy.
No result in higher dimension.
Theorem (De Lellis - Ramic, 2016)
YES for n ≤ 6 if
(C) Γ lies on the boundary of a bounded, uniformly convex open set.
(N) Σ1 and Σ2 do not intersect in the interior.
NB: smoothness up to the boundary. Hence no multiplicity.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
4 / 30
Main theorems
Note: strict stability can be replaced by local minimality.
In fact, an important step of the proof is:
Lemma
A strictly stable smooth minimal Σ is a local minimum in the flat
topology.
Small addition to an argument of White (Hausdorff distance replaced
by flat distance).
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
5 / 30
Main theorems
Note: strict stability can be replaced by local minimality.
In fact, an important step of the proof is:
Lemma
A strictly stable smooth minimal Σ is a local minimum in the flat
topology.
Small addition to an argument of White (Hausdorff distance replaced
by flat distance).
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
5 / 30
Main theorems
Note: strict stability can be replaced by local minimality.
In fact, an important step of the proof is:
Lemma
A strictly stable smooth minimal Σ is a local minimum in the flat
topology.
Small addition to an argument of White (Hausdorff distance replaced
by flat distance).
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
5 / 30
Main theorems
Theorem (De Lellis - Ramic 2016)
YES for n ≥ 7 IF
(C) Same convexity assumption as before;
(N) Same nonintersection assumption;
(S1) Σ1 and Σ2 are smooth;
(S2) Σ3 is allowed to have an (n − 7)-dimensional singular set.
Note:
I
(S2) natural because of the celebrated example of Bombieri, De
Giorgi and Giusti.
I
(S1) not natural because of the same example!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
6 / 30
Main theorems
Theorem (De Lellis - Ramic 2016)
YES for n ≥ 7 IF
(C) Same convexity assumption as before;
(N) Same nonintersection assumption;
(S1) Σ1 and Σ2 are smooth;
(S2) Σ3 is allowed to have an (n − 7)-dimensional singular set.
Note:
I
(S2) natural because of the celebrated example of Bombieri, De
Giorgi and Giusti.
I
(S1) not natural because of the same example!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
6 / 30
Main theorems
Theorem (De Lellis - Ramic 2016)
YES for n ≥ 7 IF
(C) Same convexity assumption as before;
(N) Same nonintersection assumption;
(S1) Σ1 and Σ2 are smooth;
(S2) Σ3 is allowed to have an (n − 7)-dimensional singular set.
Note:
I
(S2) natural because of the celebrated example of Bombieri, De
Giorgi and Giusti.
I
(S1) not natural because of the same example!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
6 / 30
Main theorems
First obstruction: the proof of the White-type local minimality result
needs smoothness.
Assuming local minimality the unnatural assumption (S1) can be
removed (not done in the paper for a technical reason: much longer
proof, see below).
(N) can be removed (not done in the paper for the same technical
reason)
Final remarks:
I
all results proved in a general ambient Riemannian manifold.
I
similar methods work to produce free boundary minimal surfaces
(actually easier!), generalizing 2-d works of Grüter-Jost.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
7 / 30
Main theorems
First obstruction: the proof of the White-type local minimality result
needs smoothness.
Assuming local minimality the unnatural assumption (S1) can be
removed (not done in the paper for a technical reason: much longer
proof, see below).
(N) can be removed (not done in the paper for the same technical
reason)
Final remarks:
I
all results proved in a general ambient Riemannian manifold.
I
similar methods work to produce free boundary minimal surfaces
(actually easier!), generalizing 2-d works of Grüter-Jost.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
7 / 30
Main theorems
First obstruction: the proof of the White-type local minimality result
needs smoothness.
Assuming local minimality the unnatural assumption (S1) can be
removed (not done in the paper for a technical reason: much longer
proof, see below).
(N) can be removed (not done in the paper for the same technical
reason)
Final remarks:
I
all results proved in a general ambient Riemannian manifold.
I
similar methods work to produce free boundary minimal surfaces
(actually easier!), generalizing 2-d works of Grüter-Jost.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
7 / 30
Main theorems
First obstruction: the proof of the White-type local minimality result
needs smoothness.
Assuming local minimality the unnatural assumption (S1) can be
removed (not done in the paper for a technical reason: much longer
proof, see below).
(N) can be removed (not done in the paper for the same technical
reason)
Final remarks:
I
all results proved in a general ambient Riemannian manifold.
I
similar methods work to produce free boundary minimal surfaces
(actually easier!), generalizing 2-d works of Grüter-Jost.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
7 / 30
Main theorems
First obstruction: the proof of the White-type local minimality result
needs smoothness.
Assuming local minimality the unnatural assumption (S1) can be
removed (not done in the paper for a technical reason: much longer
proof, see below).
(N) can be removed (not done in the paper for the same technical
reason)
Final remarks:
I
all results proved in a general ambient Riemannian manifold.
I
similar methods work to produce free boundary minimal surfaces
(actually easier!), generalizing 2-d works of Grüter-Jost.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
7 / 30
Min-Max construction
Basic (old) idea: min-max method.
Consider all continuous 1-parameter families of surfaces St joining Σ1
and Σ2 .
Local minimality of Σ1 -Σ2 :
max Voln (St ) ≥ |{z}
∆ + max{Voln (Σ1 ), Voln (Σ2 )} .
fixed!
Minimizing max Voln (St ) over all paths {St } you find a saddle point (the
mountain pass).
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
8 / 30
Min-Max construction
Basic (old) idea: min-max method.
Consider all continuous 1-parameter families of surfaces St joining Σ1
and Σ2 .
Local minimality of Σ1 -Σ2 :
max Voln (St ) ≥ |{z}
∆ + max{Voln (Σ1 ), Voln (Σ2 )} .
fixed!
Minimizing max Voln (St ) over all paths {St } you find a saddle point (the
mountain pass).
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
8 / 30
Min-Max construction
Basic (old) idea: min-max method.
Consider all continuous 1-parameter families of surfaces St joining Σ1
and Σ2 .
Local minimality of Σ1 -Σ2 :
max Voln (St ) ≥ |{z}
∆ + max{Voln (Σ1 ), Voln (Σ2 )} .
fixed!
Minimizing max Voln (St ) over all paths {St } you find a saddle point (the
mountain pass).
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
8 / 30
Min-Max construction
Basic (old) idea: min-max method.
Consider all continuous 1-parameter families of surfaces St joining Σ1
and Σ2 .
Local minimality of Σ1 -Σ2 :
max Voln (St ) ≥ |{z}
∆ + max{Voln (Σ1 ), Voln (Σ2 )} .
fixed!
Minimizing max Voln (St ) over all paths {St } you find a saddle point (the
mountain pass).
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
8 / 30
Min-Max construction
More precisely:
I
Denote by M0 the minmax value.
I
Let {Stj } be a minimizing sequence of paths.
I
If Voln (Sj j ) → M0 , then {Sj j } is a minmax sequence.
t
t
Problem
Show the existence of a min-max sequence which converges to a
minimal surface Σ.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
9 / 30
Min-Max construction
More precisely:
I
Denote by M0 the minmax value.
I
Let {Stj } be a minimizing sequence of paths.
I
If Voln (Sj j ) → M0 , then {Sj j } is a minmax sequence.
t
t
Problem
Show the existence of a min-max sequence which converges to a
minimal surface Σ.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
9 / 30
Min-Max construction
More precisely:
I
Denote by M0 the minmax value.
I
Let {Stj } be a minimizing sequence of paths.
I
If Voln (Sj j ) → M0 , then {Sj j } is a minmax sequence.
t
t
Problem
Show the existence of a min-max sequence which converges to a
minimal surface Σ.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
9 / 30
Min-Max construction
More precisely:
I
Denote by M0 the minmax value.
I
Let {Stj } be a minimizing sequence of paths.
I
If Voln (Sj j ) → M0 , then {Sj j } is a minmax sequence.
t
t
Problem
Show the existence of a min-max sequence which converges to a
minimal surface Σ.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
9 / 30
Min-Max construction
More precisely:
I
Denote by M0 the minmax value.
I
Let {Stj } be a minimizing sequence of paths.
I
If Voln (Sj j ) → M0 , then {Sj j } is a minmax sequence.
t
t
Problem
Show the existence of a min-max sequence which converges to a
minimal surface Σ.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
9 / 30
A third minimal surface Σ3
Observe:
I
Regularity (and Boundary regularity!) of Σ ⇒ Σ taken with
multiplicity 1.
I
⇒ Voln (Σ) = M0 .
I
M0 > max{Voln (Σ1 ), Voln (Σ2 )} ⇒ Σ is a third minimal surface
distinct from Σ1 and Σ2 .
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
10 / 30
A third minimal surface Σ3
Observe:
I
Regularity (and Boundary regularity!) of Σ ⇒ Σ taken with
multiplicity 1.
I
⇒ Voln (Σ) = M0 .
I
M0 > max{Voln (Σ1 ), Voln (Σ2 )} ⇒ Σ is a third minimal surface
distinct from Σ1 and Σ2 .
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
10 / 30
A third minimal surface Σ3
Observe:
I
Regularity (and Boundary regularity!) of Σ ⇒ Σ taken with
multiplicity 1.
I
⇒ Voln (Σ) = M0 .
I
M0 > max{Voln (Σ1 ), Voln (Σ2 )} ⇒ Σ is a third minimal surface
distinct from Σ1 and Σ2 .
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
10 / 30
A third minimal surface Σ3
Observe:
I
Regularity (and Boundary regularity!) of Σ ⇒ Σ taken with
multiplicity 1.
I
⇒ Voln (Σ) = M0 .
I
M0 > max{Voln (Σ1 ), Voln (Σ2 )} ⇒ Σ is a third minimal surface
distinct from Σ1 and Σ2 .
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
10 / 30
Closed minimal submanifolds
The idea goes back at least to Birkhoff in the case of geodesics to
produce simple closed geodesics in surfaces diffeomorphic to S2 .
A classical problem: generalize Birkhoff’s construction to produce
minimal closed (hyper)-surfaces in Riemannian manifolds.
Abstract geometric measure theory: producing a stationary varifold is
rather simple (Almgren’s pull-tight lemma). The version with fixed
boundaries is a minor modification.
REAL ISSUE : regularity!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
11 / 30
Closed minimal submanifolds
The idea goes back at least to Birkhoff in the case of geodesics to
produce simple closed geodesics in surfaces diffeomorphic to S2 .
A classical problem: generalize Birkhoff’s construction to produce
minimal closed (hyper)-surfaces in Riemannian manifolds.
Abstract geometric measure theory: producing a stationary varifold is
rather simple (Almgren’s pull-tight lemma). The version with fixed
boundaries is a minor modification.
REAL ISSUE : regularity!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
11 / 30
Closed minimal submanifolds
The idea goes back at least to Birkhoff in the case of geodesics to
produce simple closed geodesics in surfaces diffeomorphic to S2 .
A classical problem: generalize Birkhoff’s construction to produce
minimal closed (hyper)-surfaces in Riemannian manifolds.
Abstract geometric measure theory: producing a stationary varifold is
rather simple (Almgren’s pull-tight lemma). The version with fixed
boundaries is a minor modification.
REAL ISSUE : regularity!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
11 / 30
Closed minimal submanifolds
The idea goes back at least to Birkhoff in the case of geodesics to
produce simple closed geodesics in surfaces diffeomorphic to S2 .
A classical problem: generalize Birkhoff’s construction to produce
minimal closed (hyper)-surfaces in Riemannian manifolds.
Abstract geometric measure theory: producing a stationary varifold is
rather simple (Almgren’s pull-tight lemma). The version with fixed
boundaries is a minor modification.
REAL ISSUE : regularity!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
11 / 30
Pitts’ breakthrough
Pitts’ Monograph (1981): Full regularity for n ≤ 5.
Three fundamental ingredients:
I
Rather loose concept of continuity in the parameter, to allow many
deformations.
I
Local minimality/local stability of the min-max surface.
I
Schoen-Simon-Yau curvature estimates for stable hypersurfaces.
Last ingredient reason for n ≤ 5. Schoen-Simon compactness
theorem ⇒ extension of Pitts’ theory to any n.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
12 / 30
Pitts’ breakthrough
Pitts’ Monograph (1981): Full regularity for n ≤ 5.
Three fundamental ingredients:
I
Rather loose concept of continuity in the parameter, to allow many
deformations.
I
Local minimality/local stability of the min-max surface.
I
Schoen-Simon-Yau curvature estimates for stable hypersurfaces.
Last ingredient reason for n ≤ 5. Schoen-Simon compactness
theorem ⇒ extension of Pitts’ theory to any n.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
12 / 30
Pitts’ breakthrough
Pitts’ Monograph (1981): Full regularity for n ≤ 5.
Three fundamental ingredients:
I
Rather loose concept of continuity in the parameter, to allow many
deformations.
I
Local minimality/local stability of the min-max surface.
I
Schoen-Simon-Yau curvature estimates for stable hypersurfaces.
Last ingredient reason for n ≤ 5. Schoen-Simon compactness
theorem ⇒ extension of Pitts’ theory to any n.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
12 / 30
Pitts’ breakthrough
Pitts’ Monograph (1981): Full regularity for n ≤ 5.
Three fundamental ingredients:
I
Rather loose concept of continuity in the parameter, to allow many
deformations.
I
Local minimality/local stability of the min-max surface.
I
Schoen-Simon-Yau curvature estimates for stable hypersurfaces.
Last ingredient reason for n ≤ 5. Schoen-Simon compactness
theorem ⇒ extension of Pitts’ theory to any n.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
12 / 30
Pitts’ breakthrough
Pitts’ Monograph (1981): Full regularity for n ≤ 5.
Three fundamental ingredients:
I
Rather loose concept of continuity in the parameter, to allow many
deformations.
I
Local minimality/local stability of the min-max surface.
I
Schoen-Simon-Yau curvature estimates for stable hypersurfaces.
Last ingredient reason for n ≤ 5. Schoen-Simon compactness
theorem ⇒ extension of Pitts’ theory to any n.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
12 / 30
Pitts’ and alternative families
Ingredients 2 and 3 rather efficiently treated (see Schoen-Simon, De
Lellis-Colding).
Pitts’ families: “discretized” one-parameter families of currents, very
hard to work with. Big source of technical problems and hard GMT.
De Lellis-Tasnady 2009: alternative proposal
Definition
A generalized family {St } varies smoothly in the real parameter t
except for finitely many t where the smoothness fails at finitely many
points of St .
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
13 / 30
Pitts’ and alternative families
Ingredients 2 and 3 rather efficiently treated (see Schoen-Simon, De
Lellis-Colding).
Pitts’ families: “discretized” one-parameter families of currents, very
hard to work with. Big source of technical problems and hard GMT.
De Lellis-Tasnady 2009: alternative proposal
Definition
A generalized family {St } varies smoothly in the real parameter t
except for finitely many t where the smoothness fails at finitely many
points of St .
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
13 / 30
Pitts’ and alternative families
Ingredients 2 and 3 rather efficiently treated (see Schoen-Simon, De
Lellis-Colding).
Pitts’ families: “discretized” one-parameter families of currents, very
hard to work with. Big source of technical problems and hard GMT.
De Lellis-Tasnady 2009: alternative proposal
Definition
A generalized family {St } varies smoothly in the real parameter t
except for finitely many t where the smoothness fails at finitely many
points of St .
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
13 / 30
Pitts’ and alternative families
These alternative families are very natural way of “sweeping”
manifolds. Basic example: level sets of a Morse function!
The dots mark singular t’s and the respective space singularities.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
14 / 30
Pitts’ and alternative families
These alternative families are very natural way of “sweeping”
manifolds. Basic example: level sets of a Morse function!
The dots mark singular t’s and the respective space singularities.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
14 / 30
Alternative families
Theorem (De Lellis - Tasnady 2009)
The alternative families give a quicker and less technical approach to
Pitts’ existence of closed minimal hypersurfaces in any (closed)
Riemannian manifold.
Unfortunately: not clear if as powerful as Pitts’ theory (main weakness:
Marques-Neves proof of Willmore needs Pitts’ theory).
De Lellis - Ramic: adaptation at the boundary of De Lellis-Tasnady.
Technical conditions (N) (nonintersection of Σ1 and Σ2 ) and (S1)
(smoothness of Σ1 , Σ2 )
⇒ simple proof of existence of paths connecting Σ1 and Σ2 (use level
sets of an appropriate function!).
Assumptions (N) and (S1) not necessary, though.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
15 / 30
Alternative families
Theorem (De Lellis - Tasnady 2009)
The alternative families give a quicker and less technical approach to
Pitts’ existence of closed minimal hypersurfaces in any (closed)
Riemannian manifold.
Unfortunately: not clear if as powerful as Pitts’ theory (main weakness:
Marques-Neves proof of Willmore needs Pitts’ theory).
De Lellis - Ramic: adaptation at the boundary of De Lellis-Tasnady.
Technical conditions (N) (nonintersection of Σ1 and Σ2 ) and (S1)
(smoothness of Σ1 , Σ2 )
⇒ simple proof of existence of paths connecting Σ1 and Σ2 (use level
sets of an appropriate function!).
Assumptions (N) and (S1) not necessary, though.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
15 / 30
Alternative families
Theorem (De Lellis - Tasnady 2009)
The alternative families give a quicker and less technical approach to
Pitts’ existence of closed minimal hypersurfaces in any (closed)
Riemannian manifold.
Unfortunately: not clear if as powerful as Pitts’ theory (main weakness:
Marques-Neves proof of Willmore needs Pitts’ theory).
De Lellis - Ramic: adaptation at the boundary of De Lellis-Tasnady.
Technical conditions (N) (nonintersection of Σ1 and Σ2 ) and (S1)
(smoothness of Σ1 , Σ2 )
⇒ simple proof of existence of paths connecting Σ1 and Σ2 (use level
sets of an appropriate function!).
Assumptions (N) and (S1) not necessary, though.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
15 / 30
Alternative families
Theorem (De Lellis - Tasnady 2009)
The alternative families give a quicker and less technical approach to
Pitts’ existence of closed minimal hypersurfaces in any (closed)
Riemannian manifold.
Unfortunately: not clear if as powerful as Pitts’ theory (main weakness:
Marques-Neves proof of Willmore needs Pitts’ theory).
De Lellis - Ramic: adaptation at the boundary of De Lellis-Tasnady.
Technical conditions (N) (nonintersection of Σ1 and Σ2 ) and (S1)
(smoothness of Σ1 , Σ2 )
⇒ simple proof of existence of paths connecting Σ1 and Σ2 (use level
sets of an appropriate function!).
Assumptions (N) and (S1) not necessary, though.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
15 / 30
Alternative families
Theorem (De Lellis - Tasnady 2009)
The alternative families give a quicker and less technical approach to
Pitts’ existence of closed minimal hypersurfaces in any (closed)
Riemannian manifold.
Unfortunately: not clear if as powerful as Pitts’ theory (main weakness:
Marques-Neves proof of Willmore needs Pitts’ theory).
De Lellis - Ramic: adaptation at the boundary of De Lellis-Tasnady.
Technical conditions (N) (nonintersection of Σ1 and Σ2 ) and (S1)
(smoothness of Σ1 , Σ2 )
⇒ simple proof of existence of paths connecting Σ1 and Σ2 (use level
sets of an appropriate function!).
Assumptions (N) and (S1) not necessary, though.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
15 / 30
Quick sketch of existence of good paths
Σ1 above, Σ2 below.
Move Σ1 slightly upward.
Σ1 moved up
Σ2
Take a Morse function f which equals 1 at Σ1 and 0 at Σ0 .
“Collapse” the boundaries of the level sets
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
16 / 30
Quick sketch of existence of good paths
Σ1 above, Σ2 below.
Move Σ1 slightly upward.
Σ1 moved up
Σ2
Take a Morse function f which equals 1 at Σ1 and 0 at Σ0 .
“Collapse” the boundaries of the level sets
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
16 / 30
Quick sketch of existence of good paths
Σ1 above, Σ2 below.
Move Σ1 slightly upward.
Σ1 moved up
Σ2
Take a Morse function f which equals 1 at Σ1 and 0 at Σ0 .
“Collapse” the boundaries of the level sets
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
16 / 30
The real issue
The min-max methods produces now a third minimal surface which is
regular in the interior.
Problem
Prove regularity at the boundary.
In the rest of the talk I will explain how to solve the problem.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
17 / 30
The real issue
The min-max methods produces now a third minimal surface which is
regular in the interior.
Problem
Prove regularity at the boundary.
In the rest of the talk I will explain how to solve the problem.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
17 / 30
The real issue
The min-max methods produces now a third minimal surface which is
regular in the interior.
Problem
Prove regularity at the boundary.
In the rest of the talk I will explain how to solve the problem.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
17 / 30
Almgren’s combinatorial lemma
Fix Σ3 , produced by the min-max algorithm (not yet known to be
regular).
Pitts: based on Almgren (beautiful combination of analysis and
combinatorics).
Lemma
At most points (i.e. except for finitely many) it is impossible to deform
continuously Σ3 decreasing its area.
⇒ At most points Σ3 is stable
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
18 / 30
Almgren’s combinatorial lemma
Fix Σ3 , produced by the min-max algorithm (not yet known to be
regular).
Pitts: based on Almgren (beautiful combination of analysis and
combinatorics).
Lemma
At most points (i.e. except for finitely many) it is impossible to deform
continuously Σ3 decreasing its area.
⇒ At most points Σ3 is stable
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
18 / 30
Almgren’s combinatorial lemma
Fix Σ3 , produced by the min-max algorithm (not yet known to be
regular).
Pitts: based on Almgren (beautiful combination of analysis and
combinatorics).
Lemma
At most points (i.e. except for finitely many) it is impossible to deform
continuously Σ3 decreasing its area.
⇒ At most points Σ3 is stable
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
18 / 30
Almgren’s combinatorial lemma
Fix Σ3 , produced by the min-max algorithm (not yet known to be
regular).
Pitts: based on Almgren (beautiful combination of analysis and
combinatorics).
Lemma
At most points (i.e. except for finitely many) it is impossible to deform
continuously Σ3 decreasing its area.
⇒ At most points Σ3 is stable
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
18 / 30
Stable varifolds vs. stable minimal surfaces
Unfortunately stable varifolds are not regular (Ex.: pair of planes).
Almgren-Pitts’:
Lemma
A suitable ε − δ version of stability holds (locally) for the minmax
t
sequence Sj j converging to Σ3 .
Intuitively:
Irregular stable varifolds (like pair of planes) cannot be limit of
embedded surfaces, if the limit is taken in a sufficiently strong sense.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
19 / 30
Stable varifolds vs. stable minimal surfaces
Unfortunately stable varifolds are not regular (Ex.: pair of planes).
Almgren-Pitts’:
Lemma
A suitable ε − δ version of stability holds (locally) for the minmax
t
sequence Sj j converging to Σ3 .
Intuitively:
Irregular stable varifolds (like pair of planes) cannot be limit of
embedded surfaces, if the limit is taken in a sufficiently strong sense.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
19 / 30
Stable varifolds vs. stable minimal surfaces
Unfortunately stable varifolds are not regular (Ex.: pair of planes).
Almgren-Pitts’:
Lemma
A suitable ε − δ version of stability holds (locally) for the minmax
t
sequence Sj j converging to Σ3 .
Intuitively:
Irregular stable varifolds (like pair of planes) cannot be limit of
embedded surfaces, if the limit is taken in a sufficiently strong sense.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
19 / 30
Pitts’ strategy
Pitts’ key idea:
I
Fix small convex neighborhood U of a point p;
I
Deform Sj j to a new min-max sequence S̄j j which achieves the
minimum area in U.
I
I
t
t
S̄tjj is a (classical) minimal stable surface in U (celebrated
regularity theory for boundaries of Caccioppoli sets, started with
De Giorgi!).
S̄tjj converges to a stable varifold Σ̄3 : Σ̄3 and Σ3 coincide outside
U. unique continuation ⇒ Σ3 = Σ̄3 .
NB: unique continuation fails for general varifolds. Subtle
argument needed.
I
Schoen-Simon compactness theorem ⇒ Σ̄3 is regular in U.
Thus Σ3 is regular in U.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
20 / 30
Pitts’ strategy
Pitts’ key idea:
I
Fix small convex neighborhood U of a point p;
I
Deform Sj j to a new min-max sequence S̄j j which achieves the
minimum area in U.
I
I
t
t
S̄tjj is a (classical) minimal stable surface in U (celebrated
regularity theory for boundaries of Caccioppoli sets, started with
De Giorgi!).
S̄tjj converges to a stable varifold Σ̄3 : Σ̄3 and Σ3 coincide outside
U. unique continuation ⇒ Σ3 = Σ̄3 .
NB: unique continuation fails for general varifolds. Subtle
argument needed.
I
Schoen-Simon compactness theorem ⇒ Σ̄3 is regular in U.
Thus Σ3 is regular in U.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
20 / 30
Pitts’ strategy
Pitts’ key idea:
I
Fix small convex neighborhood U of a point p;
I
Deform Sj j to a new min-max sequence S̄j j which achieves the
minimum area in U.
I
I
t
t
S̄tjj is a (classical) minimal stable surface in U (celebrated
regularity theory for boundaries of Caccioppoli sets, started with
De Giorgi!).
S̄tjj converges to a stable varifold Σ̄3 : Σ̄3 and Σ3 coincide outside
U. unique continuation ⇒ Σ3 = Σ̄3 .
NB: unique continuation fails for general varifolds. Subtle
argument needed.
I
Schoen-Simon compactness theorem ⇒ Σ̄3 is regular in U.
Thus Σ3 is regular in U.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
20 / 30
Pitts’ strategy
Pitts’ key idea:
I
Fix small convex neighborhood U of a point p;
I
Deform Sj j to a new min-max sequence S̄j j which achieves the
minimum area in U.
I
I
t
t
S̄tjj is a (classical) minimal stable surface in U (celebrated
regularity theory for boundaries of Caccioppoli sets, started with
De Giorgi!).
S̄tjj converges to a stable varifold Σ̄3 : Σ̄3 and Σ3 coincide outside
U. unique continuation ⇒ Σ3 = Σ̄3 .
NB: unique continuation fails for general varifolds. Subtle
argument needed.
I
Schoen-Simon compactness theorem ⇒ Σ̄3 is regular in U.
Thus Σ3 is regular in U.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
20 / 30
Pitts’ strategy
Pitts’ key idea:
I
Fix small convex neighborhood U of a point p;
I
Deform Sj j to a new min-max sequence S̄j j which achieves the
minimum area in U.
I
I
t
t
S̄tjj is a (classical) minimal stable surface in U (celebrated
regularity theory for boundaries of Caccioppoli sets, started with
De Giorgi!).
S̄tjj converges to a stable varifold Σ̄3 : Σ̄3 and Σ3 coincide outside
U. unique continuation ⇒ Σ3 = Σ̄3 .
NB: unique continuation fails for general varifolds. Subtle
argument needed.
I
Schoen-Simon compactness theorem ⇒ Σ̄3 is regular in U.
Thus Σ3 is regular in U.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
20 / 30
Pitts’ strategy
Pitts’ key idea:
I
Fix small convex neighborhood U of a point p;
I
Deform Sj j to a new min-max sequence S̄j j which achieves the
minimum area in U.
I
I
t
t
S̄tjj is a (classical) minimal stable surface in U (celebrated
regularity theory for boundaries of Caccioppoli sets, started with
De Giorgi!).
S̄tjj converges to a stable varifold Σ̄3 : Σ̄3 and Σ3 coincide outside
U. unique continuation ⇒ Σ3 = Σ̄3 .
NB: unique continuation fails for general varifolds. Subtle
argument needed.
I
Schoen-Simon compactness theorem ⇒ Σ̄3 is regular in U.
Thus Σ3 is regular in U.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
20 / 30
Pitts’ strategy
Pitts’ key idea:
I
Fix small convex neighborhood U of a point p;
I
Deform Sj j to a new min-max sequence S̄j j which achieves the
minimum area in U.
I
I
t
t
S̄tjj is a (classical) minimal stable surface in U (celebrated
regularity theory for boundaries of Caccioppoli sets, started with
De Giorgi!).
S̄tjj converges to a stable varifold Σ̄3 : Σ̄3 and Σ3 coincide outside
U. unique continuation ⇒ Σ3 = Σ̄3 .
NB: unique continuation fails for general varifolds. Subtle
argument needed.
I
Schoen-Simon compactness theorem ⇒ Σ̄3 is regular in U.
Thus Σ3 is regular in U.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
20 / 30
Pitts’ strategy
Pitts’ key idea:
I
Fix small convex neighborhood U of a point p;
I
Deform Sj j to a new min-max sequence S̄j j which achieves the
minimum area in U.
I
I
t
t
S̄tjj is a (classical) minimal stable surface in U (celebrated
regularity theory for boundaries of Caccioppoli sets, started with
De Giorgi!).
S̄tjj converges to a stable varifold Σ̄3 : Σ̄3 and Σ3 coincide outside
U. unique continuation ⇒ Σ3 = Σ̄3 .
NB: unique continuation fails for general varifolds. Subtle
argument needed.
I
Schoen-Simon compactness theorem ⇒ Σ̄3 is regular in U.
Thus Σ3 is regular in U.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
20 / 30
Pitts’ strategy at the boundary
Pitts’ strategy is meant for the interior regularity.
“Checklist” at the boundary:
I
Almgren-Pitts combinatorial lemma: OK.
I
Existence of “local replacements”: OK (few technical
adjustments).
I
Regularity at the boundary: even better! No singularity at the
boundary even for n ≥ 7.
I
Schoen-Simon compactness theorem at the boundary: missing!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
21 / 30
Pitts’ strategy at the boundary
Pitts’ strategy is meant for the interior regularity.
“Checklist” at the boundary:
I
Almgren-Pitts combinatorial lemma: OK.
I
Existence of “local replacements”: OK (few technical
adjustments).
I
Regularity at the boundary: even better! No singularity at the
boundary even for n ≥ 7.
I
Schoen-Simon compactness theorem at the boundary: missing!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
21 / 30
Pitts’ strategy at the boundary
Pitts’ strategy is meant for the interior regularity.
“Checklist” at the boundary:
I
Almgren-Pitts combinatorial lemma: OK.
I
Existence of “local replacements”: OK (few technical
adjustments).
I
Regularity at the boundary: even better! No singularity at the
boundary even for n ≥ 7.
I
Schoen-Simon compactness theorem at the boundary: missing!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
21 / 30
Pitts’ strategy at the boundary
Pitts’ strategy is meant for the interior regularity.
“Checklist” at the boundary:
I
Almgren-Pitts combinatorial lemma: OK.
I
Existence of “local replacements”: OK (few technical
adjustments).
I
Regularity at the boundary: even better! No singularity at the
boundary even for n ≥ 7.
I
Schoen-Simon compactness theorem at the boundary: missing!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
21 / 30
Pitts’ strategy at the boundary
Pitts’ strategy is meant for the interior regularity.
“Checklist” at the boundary:
I
Almgren-Pitts combinatorial lemma: OK.
I
Existence of “local replacements”: OK (few technical
adjustments).
I
Regularity at the boundary: even better! No singularity at the
boundary even for n ≥ 7.
I
Schoen-Simon compactness theorem at the boundary: missing!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
21 / 30
Maximum principle
The min-max surface is contained in the convex hull ch (Γ) of its
boundary Γ.
Γ ⊂ ∂Ω and Ω bounded uniformly convex.
⇒ ch (Γ) meets ∂Ω transversally.
For any point p ∈ ∂Ω ∩ ch(Γ), ch(Γ) is contained in a suitable “wedge”
centered at p.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
22 / 30
Maximum principle
The min-max surface is contained in the convex hull ch (Γ) of its
boundary Γ.
Γ ⊂ ∂Ω and Ω bounded uniformly convex.
⇒ ch (Γ) meets ∂Ω transversally.
For any point p ∈ ∂Ω ∩ ch(Γ), ch(Γ) is contained in a suitable “wedge”
centered at p.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
22 / 30
Maximum principle
The min-max surface is contained in the convex hull ch (Γ) of its
boundary Γ.
Γ ⊂ ∂Ω and Ω bounded uniformly convex.
⇒ ch (Γ) meets ∂Ω transversally.
For any point p ∈ ∂Ω ∩ ch(Γ), ch(Γ) is contained in a suitable “wedge”
centered at p.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
22 / 30
Maximum principle
The min-max surface is contained in the convex hull ch (Γ) of its
boundary Γ.
Γ ⊂ ∂Ω and Ω bounded uniformly convex.
⇒ ch (Γ) meets ∂Ω transversally.
For any point p ∈ ∂Ω ∩ ch(Γ), ch(Γ) is contained in a suitable “wedge”
centered at p.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
22 / 30
The wedge
tangent
plane
to ∂Ω
Camillo De Lellis (UZH)
Wedge
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
23 / 30
Blow-up strategy
“Classical approach”: if the needed compactness statement fails
(compactness ⇐⇒ Regularity) there is a global non planar stable
minimal surface Σ in the wedge such that
∂Σ = `
the tip of the wedge.
For n = 2 (Tasnady, PhD thesis 2009):
I
reflection of Σ gives a complete minimal surface in R3 ;
I
the Gauss map misses too many values.
I
Osserman’s theorem ⇒ Σ is planar.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
24 / 30
Blow-up strategy
“Classical approach”: if the needed compactness statement fails
(compactness ⇐⇒ Regularity) there is a global non planar stable
minimal surface Σ in the wedge such that
∂Σ = `
the tip of the wedge.
For n = 2 (Tasnady, PhD thesis 2009):
I
reflection of Σ gives a complete minimal surface in R3 ;
I
the Gauss map misses too many values.
I
Osserman’s theorem ⇒ Σ is planar.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
24 / 30
Blow-up strategy
“Classical approach”: if the needed compactness statement fails
(compactness ⇐⇒ Regularity) there is a global non planar stable
minimal surface Σ in the wedge such that
∂Σ = `
the tip of the wedge.
For n = 2 (Tasnady, PhD thesis 2009):
I
reflection of Σ gives a complete minimal surface in R3 ;
I
the Gauss map misses too many values.
I
Osserman’s theorem ⇒ Σ is planar.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
24 / 30
Blow-up strategy
“Classical approach”: if the needed compactness statement fails
(compactness ⇐⇒ Regularity) there is a global non planar stable
minimal surface Σ in the wedge such that
∂Σ = `
the tip of the wedge.
For n = 2 (Tasnady, PhD thesis 2009):
I
reflection of Σ gives a complete minimal surface in R3 ;
I
the Gauss map misses too many values.
I
Osserman’s theorem ⇒ Σ is planar.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
24 / 30
Blow-up strategy
“Classical approach”: if the needed compactness statement fails
(compactness ⇐⇒ Regularity) there is a global non planar stable
minimal surface Σ in the wedge such that
∂Σ = `
the tip of the wedge.
For n = 2 (Tasnady, PhD thesis 2009):
I
reflection of Σ gives a complete minimal surface in R3 ;
I
the Gauss map misses too many values.
I
Osserman’s theorem ⇒ Σ is planar.
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
24 / 30
Blow-up strategy
tip of
the wedge
Camillo De Lellis (UZH)
Σ
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
25 / 30
Blow-up – Blow-down strategy
Another classical strategy (Fleming, cf. his talk!).
Monotonicity formula: tangent cone Σ∞ at infinity (NB: in the sense of
varifolds).
IF Σ∞ is planar (with multiplicity 1) THEN Σ is planar
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
26 / 30
Blow-up – Blow-down strategy
Another classical strategy (Fleming, cf. his talk!).
Monotonicity formula: tangent cone Σ∞ at infinity (NB: in the sense of
varifolds).
IF Σ∞ is planar (with multiplicity 1) THEN Σ is planar
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
26 / 30
Blow-up – Blow-down – Blow-up !
Take at a tangent cone Σ∞,p of Σ∞ at any point p at the tip.
If Σ∞,p is planar (with multiplicity 1) at most points THEN Σ∞ is planar
with multiplicity 1 (the reason is nontrivial...).
Almgren’s stratification theorem: at most points p Σ∞,p is the union of
finitely many halfplanes:
Σ∞,p =
m
X
πi
i=1
(with possible repetitions!).
Goal: m = 1!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
27 / 30
Blow-up – Blow-down – Blow-up !
Take at a tangent cone Σ∞,p of Σ∞ at any point p at the tip.
If Σ∞,p is planar (with multiplicity 1) at most points THEN Σ∞ is planar
with multiplicity 1 (the reason is nontrivial...).
Almgren’s stratification theorem: at most points p Σ∞,p is the union of
finitely many halfplanes:
Σ∞,p =
m
X
πi
i=1
(with possible repetitions!).
Goal: m = 1!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
27 / 30
Blow-up – Blow-down – Blow-up !
Take at a tangent cone Σ∞,p of Σ∞ at any point p at the tip.
If Σ∞,p is planar (with multiplicity 1) at most points THEN Σ∞ is planar
with multiplicity 1 (the reason is nontrivial...).
Almgren’s stratification theorem: at most points p Σ∞,p is the union of
finitely many halfplanes:
Σ∞,p =
m
X
πi
i=1
(with possible repetitions!).
Goal: m = 1!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
27 / 30
Blow-up – Blow-down – Blow-up !
Take at a tangent cone Σ∞,p of Σ∞ at any point p at the tip.
If Σ∞,p is planar (with multiplicity 1) at most points THEN Σ∞ is planar
with multiplicity 1 (the reason is nontrivial...).
Almgren’s stratification theorem: at most points p Σ∞,p is the union of
finitely many halfplanes:
Σ∞,p =
m
X
πi
i=1
(with possible repetitions!).
Goal: m = 1!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
27 / 30
Star of (half)-planes
Wedge
π3
v1
v2
π2
v3
π1
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
28 / 30
White’s trick
Allowing deformations which are nonzero on the tip `
δΣ∞,p =
m
X
vi Hn−1 ` .
i=1
Recall: Σ∞,p is the limit of some classical surfaces Sj with classical
boundaries.
I
Consequence 1 (elementary intersection theory): m is odd.
I
Consequence 2: lower semicontinuity of the first variation
m X vi ≤ 1
i=1
Elementary computation: m = 1!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
29 / 30
White’s trick
Allowing deformations which are nonzero on the tip `
δΣ∞,p =
m
X
vi Hn−1 ` .
i=1
Recall: Σ∞,p is the limit of some classical surfaces Sj with classical
boundaries.
I
Consequence 1 (elementary intersection theory): m is odd.
I
Consequence 2: lower semicontinuity of the first variation
m X vi ≤ 1
i=1
Elementary computation: m = 1!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
29 / 30
White’s trick
Allowing deformations which are nonzero on the tip `
δΣ∞,p =
m
X
vi Hn−1 ` .
i=1
Recall: Σ∞,p is the limit of some classical surfaces Sj with classical
boundaries.
I
Consequence 1 (elementary intersection theory): m is odd.
I
Consequence 2: lower semicontinuity of the first variation
m X vi ≤ 1
i=1
Elementary computation: m = 1!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
29 / 30
White’s trick
Allowing deformations which are nonzero on the tip `
δΣ∞,p =
m
X
vi Hn−1 ` .
i=1
Recall: Σ∞,p is the limit of some classical surfaces Sj with classical
boundaries.
I
Consequence 1 (elementary intersection theory): m is odd.
I
Consequence 2: lower semicontinuity of the first variation
m X vi ≤ 1
i=1
Elementary computation: m = 1!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
29 / 30
White’s trick
Allowing deformations which are nonzero on the tip `
δΣ∞,p =
m
X
vi Hn−1 ` .
i=1
Recall: Σ∞,p is the limit of some classical surfaces Sj with classical
boundaries.
I
Consequence 1 (elementary intersection theory): m is odd.
I
Consequence 2: lower semicontinuity of the first variation
m X vi ≤ 1
i=1
Elementary computation: m = 1!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
29 / 30
Thank you
for your attention!
Camillo De Lellis (UZH)
Min-Max for Plateau
Pisa,
Sept. 19-23 2016
30 / 30