The sub-Finsler isoperimetric problem in the Heisenberg group.

Valentina Franceschi

Dipartimento di Matematica Tullio Levi-Civita

Universita di Padova

SRGI conference: Sub-Riemannian Geometry and Interactions

based on joint works with R. Monti, A. Righini, M. Sigalotti

1 / 22

Outline

1 Introduction: anisotropic isoperimetric problems

2 The sub-Finsler isoperimetric problem in the Heisenberg group

3 The characteristic set

2 / 22

Anisotropic perimeter measures

Figure: Equilibrium configurations of solid crystals (with sufficiently small grains).

(Image taken from UC San Diego News Center).

E ⊂ Rn regular set (∼ the crystal).

Free energy - anisotropic perimeter [Wulff (1901), Herring (1951), Taylor (1978)]:

Pφ(E) =

∫∂E

φ∗(νE ) dHn−1

νE inner unit normal to ∂E ;

φ∗: surface tension of the interface between the anisotropicmaterial E and the exterior.

Solutions are called Wulff shapes.

3 / 22

Anisotropic perimeter measures

Figure: Equilibrium configurations of solid crystals (with sufficiently small grains).

(Image taken from UC San Diego News Center).

E ⊂ Rn regular set (∼ the crystal).

Free energy - anisotropic perimeter [Wulff (1901), Herring (1951), Taylor (1978)]:

Pφ(E) =

∫∂E

φ∗(νE ) dHn−1 → minimize under volume constraint

νE inner unit normal to ∂E ;

φ∗: surface tension of the interface between the anisotropicmaterial E and the exterior.

Solutions are called Wulff shapes.3 / 22

Wulff shapes

Pφ(E) =

∫∂E

φ∗(νE ) dHn−1

φ∗ = | · | Euclidean norm: Pφ is the standardperimeter. Wulff shapes are round balls.

x

t

y

isoperimetric

Finsler case: φ∗ norm, dual to φ, i.e.,

φ∗(w) = maxφ(v)=1

〈w , v〉,

Example: φ(ξ) = `p(x , y) = (|x |p + |y |p)1p , p > 1 =⇒ φ∗ = `q , q =

p

p − 1.

{`1 = 1}<latexit sha1_base64="dYEO6dPjd0SsyYDWMe5VpNjA5cI=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5JUQS9C0YvHCvYDmlg22027dLMJuxulxP4ULx4U8eov8ea/cdvmoK0PBh7vzTAzL0g4U9pxvq3Cyura+kZxs7S1vbO7Z5f3WypOJaFNEvNYdgKsKGeCNjXTnHYSSXEUcNoORtdTv/1ApWKxuNPjhPoRHggWMoK1kXp22cs8yvm9x0Sox5euN+nZFafqzICWiZuTCuRo9Owvrx+TNKJCE46V6rpOov0MS80Ip5OSlyqaYDLCA9o1VOCIKj+bnT5Bx0bpozCWpoRGM/X3RIYjpcZRYDojrIdq0ZuK/3ndVIcXfsZEkmoqyHxRmHKkYzTNAfWZpETzsSGYSGZuRWSIJSbapFUyIbiLLy+TVq3qnlZrt2eV+lUeRxEO4QhOwIVzqMMNNKAJBB7hGV7hzXqyXqx362PeWrDymQP4A+vzBzJ+k/Q=</latexit>

{`1 = 1}<latexit sha1_base64="jUcYvrphzas5vM+eqA2vZqRzsPk=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisYD+giWWznbZLN5uwuxFK6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzwkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7ZBqFFxiw3AjsJ0opFEosBWObqd+6wmV5rF8MOMEg4gOJO9zRo2VfD/zUYhH79rzJ91S2a24M5Bl4uWkDDnq3dKX34tZGqE0TFCtO56bmCCjynAmcFL0U40JZSM6wI6lkkaog2x284ScWqVH+rGyJQ2Zqb8nMhppPY5C2xlRM9SL3lT8z+ukpn8VZFwmqUHJ5ov6qSAmJtMASI8rZEaMLaFMcXsrYUOqKDM2pqINwVt8eZk0qxXvvFK9vyjXbvI4CnAMJ3AGHlxCDe6gDg1gkMAzvMKbkzovzrvzMW9dcfKZI/gD5/MHWyKRPA==</latexit>

Wulff shapes are balls for the φ-norm. [Wulff (1901), Fonseca (1991),

Fonseca & Muller (1991)].

AIM: study the degenerate case of sub-Finsler perimeter measures.

4 / 22

Wulff shapes

Pφ(E) =

∫∂E

φ∗(νE ) dHn−1

φ∗ = | · | Euclidean norm: Pφ is the standardperimeter. Wulff shapes are round balls.

x

t

y

isoperimetric

Finsler case: φ∗ norm, dual to φ, i.e.,

φ∗(w) = maxφ(v)=1

〈w , v〉,

Example: φ(ξ) = `p(x , y) = (|x |p + |y |p)1p , p > 1 =⇒ φ∗ = `q , q =

p

p − 1.

{`1 = 1}<latexit sha1_base64="dYEO6dPjd0SsyYDWMe5VpNjA5cI=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5JUQS9C0YvHCvYDmlg22027dLMJuxulxP4ULx4U8eov8ea/cdvmoK0PBh7vzTAzL0g4U9pxvq3Cyura+kZxs7S1vbO7Z5f3WypOJaFNEvNYdgKsKGeCNjXTnHYSSXEUcNoORtdTv/1ApWKxuNPjhPoRHggWMoK1kXp22cs8yvm9x0Sox5euN+nZFafqzICWiZuTCuRo9Owvrx+TNKJCE46V6rpOov0MS80Ip5OSlyqaYDLCA9o1VOCIKj+bnT5Bx0bpozCWpoRGM/X3RIYjpcZRYDojrIdq0ZuK/3ndVIcXfsZEkmoqyHxRmHKkYzTNAfWZpETzsSGYSGZuRWSIJSbapFUyIbiLLy+TVq3qnlZrt2eV+lUeRxEO4QhOwIVzqMMNNKAJBB7hGV7hzXqyXqx362PeWrDymQP4A+vzBzJ+k/Q=</latexit>

{`1 = 1}<latexit sha1_base64="jUcYvrphzas5vM+eqA2vZqRzsPk=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisYD+giWWznbZLN5uwuxFK6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzwkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7ZBqFFxiw3AjsJ0opFEosBWObqd+6wmV5rF8MOMEg4gOJO9zRo2VfD/zUYhH79rzJ91S2a24M5Bl4uWkDDnq3dKX34tZGqE0TFCtO56bmCCjynAmcFL0U40JZSM6wI6lkkaog2x284ScWqVH+rGyJQ2Zqb8nMhppPY5C2xlRM9SL3lT8z+ukpn8VZFwmqUHJ5ov6qSAmJtMASI8rZEaMLaFMcXsrYUOqKDM2pqINwVt8eZk0qxXvvFK9vyjXbvI4CnAMJ3AGHlxCDe6gDg1gkMAzvMKbkzovzrvzMW9dcfKZI/gD5/MHWyKRPA==</latexit>

Wulff shapes are balls for the φ-norm. [Wulff (1901), Fonseca (1991),

Fonseca & Muller (1991)].

AIM: study the degenerate case of sub-Finsler perimeter measures.

4 / 22

Wulff shapes

Pφ(E) =

∫∂E

φ∗(νE ) dHn−1

φ∗ = | · | Euclidean norm: Pφ is the standardperimeter. Wulff shapes are round balls.

x

t

y

isoperimetric

Finsler case: φ∗ norm, dual to φ, i.e.,

φ∗(w) = maxφ(v)=1

〈w , v〉,

Example: φ(ξ) = `p(x , y) = (|x |p + |y |p)1p , p > 1 =⇒ φ∗ = `q , q =

p

p − 1.

{`1 = 1}<latexit sha1_base64="dYEO6dPjd0SsyYDWMe5VpNjA5cI=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5JUQS9C0YvHCvYDmlg22027dLMJuxulxP4ULx4U8eov8ea/cdvmoK0PBh7vzTAzL0g4U9pxvq3Cyura+kZxs7S1vbO7Z5f3WypOJaFNEvNYdgKsKGeCNjXTnHYSSXEUcNoORtdTv/1ApWKxuNPjhPoRHggWMoK1kXp22cs8yvm9x0Sox5euN+nZFafqzICWiZuTCuRo9Owvrx+TNKJCE46V6rpOov0MS80Ip5OSlyqaYDLCA9o1VOCIKj+bnT5Bx0bpozCWpoRGM/X3RIYjpcZRYDojrIdq0ZuK/3ndVIcXfsZEkmoqyHxRmHKkYzTNAfWZpETzsSGYSGZuRWSIJSbapFUyIbiLLy+TVq3qnlZrt2eV+lUeRxEO4QhOwIVzqMMNNKAJBB7hGV7hzXqyXqx362PeWrDymQP4A+vzBzJ+k/Q=</latexit>

{`1 = 1}<latexit sha1_base64="jUcYvrphzas5vM+eqA2vZqRzsPk=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisYD+giWWznbZLN5uwuxFK6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzwkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7ZBqFFxiw3AjsJ0opFEosBWObqd+6wmV5rF8MOMEg4gOJO9zRo2VfD/zUYhH79rzJ91S2a24M5Bl4uWkDDnq3dKX34tZGqE0TFCtO56bmCCjynAmcFL0U40JZSM6wI6lkkaog2x284ScWqVH+rGyJQ2Zqb8nMhppPY5C2xlRM9SL3lT8z+ukpn8VZFwmqUHJ5ov6qSAmJtMASI8rZEaMLaFMcXsrYUOqKDM2pqINwVt8eZk0qxXvvFK9vyjXbvI4CnAMJ3AGHlxCDe6gDg1gkMAzvMKbkzovzrvzMW9dcfKZI/gD5/MHWyKRPA==</latexit>

Wulff shapes are balls for the φ-norm. [Wulff (1901), Fonseca (1991),

Fonseca & Muller (1991)].

AIM: study the degenerate case of sub-Finsler perimeter measures.

4 / 22

Wulff shapes

Pφ(E) =

∫∂E

φ∗(νE ) dHn−1

φ∗ = | · | Euclidean norm: Pφ is the standardperimeter. Wulff shapes are round balls.

x

t

y

isoperimetric

Finsler case: φ∗ norm, dual to φ, i.e.,

φ∗(w) = maxφ(v)=1

〈w , v〉,

Example: φ(ξ) = `p(x , y) = (|x |p + |y |p)1p , p > 1 =⇒ φ∗ = `q , q =

p

p − 1.

{`1 = 1}<latexit sha1_base64="dYEO6dPjd0SsyYDWMe5VpNjA5cI=">AAAB+nicbVBNS8NAEJ3Ur1q/Uj16WSyCp5JUQS9C0YvHCvYDmlg22027dLMJuxulxP4ULx4U8eov8ea/cdvmoK0PBh7vzTAzL0g4U9pxvq3Cyura+kZxs7S1vbO7Z5f3WypOJaFNEvNYdgKsKGeCNjXTnHYSSXEUcNoORtdTv/1ApWKxuNPjhPoRHggWMoK1kXp22cs8yvm9x0Sox5euN+nZFafqzICWiZuTCuRo9Owvrx+TNKJCE46V6rpOov0MS80Ip5OSlyqaYDLCA9o1VOCIKj+bnT5Bx0bpozCWpoRGM/X3RIYjpcZRYDojrIdq0ZuK/3ndVIcXfsZEkmoqyHxRmHKkYzTNAfWZpETzsSGYSGZuRWSIJSbapFUyIbiLLy+TVq3qnlZrt2eV+lUeRxEO4QhOwIVzqMMNNKAJBB7hGV7hzXqyXqx362PeWrDymQP4A+vzBzJ+k/Q=</latexit>

{`1 = 1}<latexit sha1_base64="jUcYvrphzas5vM+eqA2vZqRzsPk=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisYD+giWWznbZLN5uwuxFK6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzwkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7ZBqFFxiw3AjsJ0opFEosBWObqd+6wmV5rF8MOMEg4gOJO9zRo2VfD/zUYhH79rzJ91S2a24M5Bl4uWkDDnq3dKX34tZGqE0TFCtO56bmCCjynAmcFL0U40JZSM6wI6lkkaog2x284ScWqVH+rGyJQ2Zqb8nMhppPY5C2xlRM9SL3lT8z+ukpn8VZFwmqUHJ5ov6qSAmJtMASI8rZEaMLaFMcXsrYUOqKDM2pqINwVt8eZk0qxXvvFK9vyjXbvI4CnAMJ3AGHlxCDe6gDg1gkMAzvMKbkzovzrvzMW9dcfKZI/gD5/MHWyKRPA==</latexit>

Wulff shapes are balls for the φ-norm. [Wulff (1901), Fonseca (1991),

Fonseca & Muller (1991)].

AIM: study the degenerate case of sub-Finsler perimeter measures.

4 / 22

Outline

1 Introduction: anisotropic isoperimetric problems

2 The sub-Finsler isoperimetric problem in the Heisenberg group

3 The characteristic set

5 / 22

Sub-Finsler perimeter in the Heisenberg group

Heisenberg group H1: R3 with coordinates (ξ, z), ξ = (x , y).

Non-commutative group law: (ξ, z) ∗ (ξ′, z ′) = (ξ + ξ′, z + z ′ + ω(ξ, ξ′)), where

ω(ξ, ξ′) =1

2(xy ′ − x ′y), ξ = (x , y), ξ′ = (x ′, y ′)

Horizontal distribution D(H1) generated by

X = ∂x −y

2∂z , Y = ∂y +

x

2∂z

Identify D(H1) with R2: V = aX + bY ∈ D(H1)↔ (a, b) ∈ R2.

sub-Finsler perimeter (E regular enough)

For a given norm φ in R2, let

Pφ(E) =

∫∂E

φ∗(NE ) dHn−1, NE = 〈νE ,X 〉X + 〈νE ,Y 〉Y

6 / 22

The sub-Finsler isoperimetric problem in the Heisenberg group

For a volume m > 0 we study

Isoperimetric problem

inf{Pφ(E) : E ⊂ H1 measurable, L3(E) = m}.

Solutions are called φ-isoperimetric sets.

Here, Pφ is more generally defined on Lebsegue measurable sets:

sub-Finsler perimeter (general case)

Let E ⊂ H1 be a Lebesgue measurable set. The φ-perimeter of E is

Pφ(E) = sup

{∫E

div(V ) dp : V ∈ Dc(H1) ' R2, maxp∈H1

φ(V (p)) ≤ 1

}.

. We can fix m = 1: Pφ and L3 are respectively 3 and 4 homogeneous w.r.t.δλ(ξ, z) = (λξ, λ2z) =⇒ if E is φ-isoperimetric sets, also δλ(E) is so.

. Existence of isoperimetric sets is proved following [Leonardi & Rigot (2003)].

7 / 22

The sub-Finsler isoperimetric problem in the Heisenberg group

For a volume m > 0 we study

Isoperimetric problem

inf{Pφ(E) : E ⊂ H1 measurable, L3(E) = m}.

Solutions are called φ-isoperimetric sets.

Here, Pφ is more generally defined on Lebsegue measurable sets:

sub-Finsler perimeter (general case)

Let E ⊂ H1 be a Lebesgue measurable set. The φ-perimeter of E is

Pφ(E) = sup

{∫E

div(V ) dp : V ∈ Dc(H1) ' R2, maxp∈H1

φ(V (p)) ≤ 1

}.

. We can fix m = 1: Pφ and L3 are respectively 3 and 4 homogeneous w.r.t.δλ(ξ, z) = (λξ, λ2z) =⇒ if E is φ-isoperimetric sets, also δλ(E) is so.

. Existence of isoperimetric sets is proved following [Leonardi & Rigot (2003)].

7 / 22

The sub-Riemannian case: φ = φ∗ = | · |Pansu’s conjecture [Pansu (1982)]

Up to dilations and translations ∃! isoperimetric set E|·| whose boundary isfoliated by geodesics between two points on the z-axis, all associated with thesame covector.

geodesics are horizontal lifts (z = ω(ξ, ξ)) of circles

{ξ ∈ R2 : |ξ − ξ0| = c};

∂E|·| is C 2 (it is NOT C 3).

∂E|·| has constant horizontal curvature.

Isoperimetric sets are NOT CC balls [Monti (2000)].

Partial proofs:

[Ritore & Rosales (2008)], assuming isoperimetric sets to be C2;

[Monti (2008)]: in a class of symmetric sets;

[Monti & Rickly (2009)]: assuming isoperimetric sets to be convex;

[Ritore (2012)]: for competitors in a vertical cylinder;

[F., Montefalcone, Monti (2017)]: in a Riem. approximation, assuming isoperimetricsets to be a topological sphere.

8 / 22

The sub-Riemannian case: φ = φ∗ = | · |Pansu’s conjecture [Pansu (1982)]

Up to dilations and translations ∃! isoperimetric set E|·| whose boundary isfoliated by geodesics between two points on the z-axis, all associated with thesame covector.

geodesics are horizontal lifts (z = ω(ξ, ξ)) of circles

{ξ ∈ R2 : |ξ − ξ0| = c};

∂E|·| is C 2 (it is NOT C 3).

∂E|·| has constant horizontal curvature.

Isoperimetric sets are NOT CC balls [Monti (2000)].

Partial proofs:

[Ritore & Rosales (2008)], assuming isoperimetric sets to be C2;

[Monti (2008)]: in a class of symmetric sets;

[Monti & Rickly (2009)]: assuming isoperimetric sets to be convex;

[Ritore (2012)]: for competitors in a vertical cylinder;

[F., Montefalcone, Monti (2017)]: in a Riem. approximation, assuming isoperimetricsets to be a topological sphere.

8 / 22

The sub-Riemannian case: φ = φ∗ = | · |Pansu’s conjecture [Pansu (1982)]

Up to dilations and translations ∃! isoperimetric set E|·| whose boundary isfoliated by geodesics between two points on the z-axis, all associated with thesame covector.

geodesics are horizontal lifts (z = ω(ξ, ξ)) of circles

{ξ ∈ R2 : |ξ − ξ0| = c};

∂E|·| is C 2 (it is NOT C 3).

∂E|·| has constant horizontal curvature.

Isoperimetric sets are NOT CC balls [Monti (2000)].

Partial proofs:

[Ritore & Rosales (2008)], assuming isoperimetric sets to be C2;

[Monti (2008)]: in a class of symmetric sets;

[Monti & Rickly (2009)]: assuming isoperimetric sets to be convex;

[Ritore (2012)]: for competitors in a vertical cylinder;

[F., Montefalcone, Monti (2017)]: in a Riem. approximation, assuming isoperimetricsets to be a topological sphere.

8 / 22

General sub-Finsler case: φ given norm in R2

Works by [Sanchez (2017)]; [Pozuelo & Ritore (2020)].

Generalization of Pansu’s bubble: φ-bubble

We let the φ-circle centered at ξ0 ∈ R2, of radius r > 0 be

Cφ(ξ0, r) = {ξ ∈ R2 : φ(ξ − ξ0) = r}.

We call φ-bubble the bounded set Eφ whose boundary is foliated byhorizontal lifts of φ-circles of a given radius, passing through the origin.

-2

-1

0

1

2

x

-2

-1

0

1

2

y

0

1

2

z

-2

-1

0

1

2

x

-2

-1

0

1

2

y

0.0

0.5

1.0

1.5

2.0

z

Figure: First two pictures: The `p-bubbles with p = 3 (left) and p = 100 (right): therelative horizontal lifts of `p-circles is drawn in blue. Third picture: The metric ball forthe Carnot-Caratheodory distance relative to the `∞ norm. Image taken from [Barilari,Boscain, Le Donne, Sigalotti (2017)]

9 / 22

General sub-Finsler case: φ given norm in R2

Works by [Sanchez (2017)]; [Pozuelo & Ritore (2020)].

Generalization of Pansu’s bubble: φ-bubble

We let the φ-circle centered at ξ0 ∈ R2, of radius r > 0 be

Cφ(ξ0, r) = {ξ ∈ R2 : φ(ξ − ξ0) = r}.

We call φ-bubble the bounded set Eφ whose boundary is foliated byhorizontal lifts of φ-circles of a given radius, passing through the origin.

-2

-1

0

1

2

x

-2

-1

0

1

2

y

0

1

2

z

-2

-1

0

1

2

x

-2

-1

0

1

2

y

0.0

0.5

1.0

1.5

2.0

z

Figure: First two pictures: The `p-bubbles with p = 3 (left) and p = 100 (right): therelative horizontal lifts of `p-circles is drawn in blue. Third picture: The metric ball forthe Carnot-Caratheodory distance relative to the `∞ norm. Image taken from [Barilari,Boscain, Le Donne, Sigalotti (2017)]

9 / 22

Main result: the regular case

Notation: φ is of class C k , for k ∈ N if φ ∈ C k(R2 \ {0}).

Theorem (F., Monti, Righini & Sigalotti)

Let φ be a norm of class C 2 such that φ∗ is of class C 2.

Let E ⊂ H1 be a φ-isoperimetric set of class C 2.

Then E = Eφ, up to left-translations and anisotropic dilations.

Generalizes [Ritore & Rosales (2008)], holding for φ = φ∗ = | · |.

Plan

. Main steps of the proof.

. Relation with ambient geodesics ?

. Regularity of φ-bubbles ?

. What if φ, φ∗ are not C 2 ?

10 / 22

Main result: the regular case

Notation: φ is of class C k , for k ∈ N if φ ∈ C k(R2 \ {0}).

Theorem (F., Monti, Righini & Sigalotti)

Let φ be a norm of class C 2 such that φ∗ is of class C 2.

Let E ⊂ H1 be a φ-isoperimetric set of class C 2.

Then E = Eφ, up to left-translations and anisotropic dilations.

Generalizes [Ritore & Rosales (2008)], holding for φ = φ∗ = | · |.

Plan

. Main steps of the proof.

. Relation with ambient geodesics ?

. Regularity of φ-bubbles ?

. What if φ, φ∗ are not C 2 ?

10 / 22

Main Steps of the Proof:Step 1 . Outside the characteristic set of E

C(E) = {p ∈ ∂E : Tp∂E = D(p)},

minimality yields Constant Curvature Equation.

For a z-graph {(ξ, z) : z = f (ξ)} ⊂ ∂E \ C(E):

div(∇φ∗

(∂x f +

y

2, ∂y f −

x

2

))= const.

{z = f(⇠)} ⇢ @E \ C(E)<latexit sha1_base64="XZ/4197vPYoxA8RJuDY2vFDBZ/U=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsxURTdCsRRcVrAPaErJpJk2NJMZkoxYh36JG3/FjQtFBFf6N2baLrT1QODknHu59x4v4kxpx/m2lpZXVtfWMxvZza3tnV17b7+hwlgSWichD2XLw4pyJmhdM81pK5IUBx6nTW9YSf3mHZWKheJWjyLaCXBfMJ8RrI3Utc9Q8nDp59E9K6AxUrGnqIYowlIzzGEVmW/ARKwgCrAeEKNV8tVC1845RWcCuEjcGcmBGWpd+xP1QhIHVGjCsVJt14l0J0nHEE7HWRQrGmEyxH3aNlTggKpOMjlvDI+N0oN+KM0TGk7U3x0JDpQaBZ6pTJdU814q/ue1Y+1fdBImolhTQaaD/JhDHcI0K9hjkhLNR4ZgIpnZFZIBlphok2jWhODOn7xIGqWie1Is3ZzmylezODLgEByBPHDBOSiDa1ADdUDAI3gGr+DNerJerHfrY1q6ZM16DsAfWF8/aI6h/w==</latexit>

⇠<latexit sha1_base64="Y9AsMRvmdxTQqcE8JuUntKKm/5U=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120i7dbMLuRiyhP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6mfqtR1Sax/LBjBP0IzqQPOSMGivdd594r1R2K+4MZJl4OSlDjnqv9NXtxyyNUBomqNYdz02Mn1FlOBM4KXZTjQllIzrAjqWSRqj9bHbqhJxapU/CWNmShszU3xMZjbQeR4HtjKgZ6kVvKv7ndVITXvkZl0lqULL5ojAVxMRk+jfpc4XMiLEllClubyVsSBVlxqZTtCF4iy8vk2a14p1XqncX5dp1HkcBjuEEzsCDS6jBLdShAQwG8Ayv8OYI58V5dz7mrStOPnMEf+B8/gBdM43Z</latexit>

z<latexit sha1_base64="VLEo6VgUnu2TnOxoOkqsMPXvyTo=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI9ELx4hkUcCGzI79MLI7OxmZtYECV/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7cxvPaLSPJb3ZpygH9GB5CFn1Fip/tQrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZssjjycwCmcgwdXUIU7qEEDGCA8wyu8OQ/Oi/PufCxac042cwx/4Hz+AOqPjQI=</latexit>

Integrating (CCE) along Legendre curves (i.e., γ(t) ∈ Tγ(t)∂E ∩D(γ(t)))

=⇒ ∂E \ C(E) is foliated by horizontal lifts of φ-circles of given radius.

11 / 22

Main result: the regular caseStep 1

Remark: Step 1 uses only the regularity of E and of φ∗.

It can be extended to the case where φ∗ is piecewise C 2 in the following sense:

∃k ∈ N, A1, . . . ,Ak ∈ R2 s.t. φ∗ is C 2 on R2 \ ∪kj=1span(Aj).

{`p = 1}<latexit sha1_base64="ZTC+qR7AGyMZe2VycIwlBcgZ44c=">AAAB83icbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisYD+giWWznbZLN5uwuxFK6N/w4kERr/4Zb/4bt20O2vpg4PHeDDPzwkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7ZBqFFxiw3AjsJ0opFEosBWObqd+6wmV5rF8MOMEg4gOJO9zRo2VfD/zUYjH5NrzJ91S2a24M5Bl4uWkDDnq3dKX34tZGqE0TFCtO56bmCCjynAmcFL0U40JZSM6wI6lkkaog2x284ScWqVH+rGyJQ2Zqb8nMhppPY5C2xlRM9SL3lT8z+ukpn8VZFwmqUHJ5ov6qSAmJtMASI8rZEaMLaFMcXsrYUOqKDM2pqINwVt8eZk0qxXvvFK9vyjXbvI4CnAMJ3AGHlxCDe6gDg1gkMAzvMKbkzovzrvzMW9dcfKZI/gD5/MHu5qRew==</latexit>

{`q = 1}<latexit sha1_base64="WzY8tJwQJyu5uSs0ZQNvp5UuP2o=">AAAB83icbVDLSgNBEOyNrxhfUY9eBoPgKexGQS9C0IvHCOYB2TXMTnqTIbMPZ2aFsOQ3vHhQxKs/482/cZLsQRMLGoqqbrq7/ERwpW372yqsrK6tbxQ3S1vbO7t75f2DlopTybDJYhHLjk8VCh5hU3MtsJNIpKEvsO2PbqZ++wml4nF0r8cJeiEdRDzgjGojuW7mohAPj1eOO+mVK3bVnoEsEycnFcjR6JW/3H7M0hAjzQRVquvYifYyKjVnAiclN1WYUDaiA+waGtEQlZfNbp6QE6P0SRBLU5EmM/X3REZDpcahbzpDqodq0ZuK/3ndVAeXXsajJNUYsfmiIBVEx2QaAOlziUyLsSGUSW5uJWxIJWXaxFQyITiLLy+TVq3qnFVrd+eV+nUeRxGO4BhOwYELqMMtNKAJDBJ4hld4s1LrxXq3PuatBSufOYQ/sD5/AL0ikXw=</latexit>

Figure: If φ = `p with p > 2, the dual norm `q , q = p/(p − 1) < 2 is only C2 out ofthe coordinate axes.

φ∗ piecewise C 2 and E of class C 2 =⇒ ∂E \ C(E) is foliated byhorizontal lifts of φ-circles all of the same radius.

12 / 22

Main result: the regular caseStep 2

Step 2 . The characteristic set. Remark: We use all the regularity of φ, φ∗, E .

Theorem (F., Monti, Righini & Sigalotti)

Let φ, φ∗ be of class C 2 and let E ⊂ H1 be a φ-isoperimetric set of class C 2.

Then C(E) consists of isolated points.

Lemma

Let p0 ∈ C(E).

There exists r > 0 such that for p ∈ ∂E ∩ B(p0, r), p 6= p0, the maximalhorizontal lift of the φ-circle in ∂E through p meets p0.

p0 2 C(E)<latexit sha1_base64="JBxyvG4DH8Eq9DmNKnwcIhybChA=">AAAB/HicbVBNS8NAFHypX7V+RXv0sliEeilJFfRYLILHCrYWmlI22027dLMJuxshlPpXvHhQxKs/xJv/xk2bg7YOLAwz7/Fmx485U9pxvq3C2vrG5lZxu7Szu7d/YB8edVSUSELbJOKR7PpYUc4EbWumOe3GkuLQ5/TBnzQz/+GRSsUica/TmPZDPBIsYARrIw3scjxwPCa8EOsxwRw1qzdnA7vi1Jw50Cpxc1KBHK2B/eUNI5KEVGjCsVI914l1f4qlZoTTWclLFI0xmeAR7RkqcEhVfzoPP0OnRhmiIJLmCY3m6u+NKQ6VSkPfTGYh1bKXif95vUQHV/0pE3GiqSCLQ0HCkY5Q1gQaMkmJ5qkhmEhmsiIyxhITbfoqmRLc5S+vkk695p7X6ncXlcZ1XkcRjuEEquDCJTTgFlrQBgIpPMMrvFlP1ov1bn0sRgtWvlOGP7A+fwAt0pPO</latexit>

p<latexit sha1_base64="YCRLqY3FYFMPr169IIibebwVZHg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZtIvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucP22eM+A==</latexit>

Sub-Riemannian case: [Cheng, Hwang, Malchiodi & Yang (2005), Ritore & Rosales (2008)]

using Jacobi fields and Riemannian connections.

13 / 22

Main result: the regular caseStep 2

Step 2 . The characteristic set. Remark: We use all the regularity of φ, φ∗, E .

Theorem (F., Monti, Righini & Sigalotti)

Let φ, φ∗ be of class C 2 and let E ⊂ H1 be a φ-isoperimetric set of class C 2.

Then C(E) consists of isolated points.

Lemma

Let p0 ∈ C(E).

There exists r > 0 such that for p ∈ ∂E ∩ B(p0, r), p 6= p0, the maximalhorizontal lift of the φ-circle in ∂E through p meets p0.

p0 2 C(E)<latexit sha1_base64="JBxyvG4DH8Eq9DmNKnwcIhybChA=">AAAB/HicbVBNS8NAFHypX7V+RXv0sliEeilJFfRYLILHCrYWmlI22027dLMJuxshlPpXvHhQxKs/xJv/xk2bg7YOLAwz7/Fmx485U9pxvq3C2vrG5lZxu7Szu7d/YB8edVSUSELbJOKR7PpYUc4EbWumOe3GkuLQ5/TBnzQz/+GRSsUica/TmPZDPBIsYARrIw3scjxwPCa8EOsxwRw1qzdnA7vi1Jw50Cpxc1KBHK2B/eUNI5KEVGjCsVI914l1f4qlZoTTWclLFI0xmeAR7RkqcEhVfzoPP0OnRhmiIJLmCY3m6u+NKQ6VSkPfTGYh1bKXif95vUQHV/0pE3GiqSCLQ0HCkY5Q1gQaMkmJ5qkhmEhmsiIyxhITbfoqmRLc5S+vkk695p7X6ncXlcZ1XkcRjuEEquDCJTTgFlrQBgIpPMMrvFlP1ov1bn0sRgtWvlOGP7A+fwAt0pPO</latexit>

p<latexit sha1_base64="YCRLqY3FYFMPr169IIibebwVZHg=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZtIvV9yqOwdZJV5OKpCj0S9/9QYxSyOUhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFrqaQRaj+bHzolZ1YZkDBWtqQhc/X3REYjrSdRYDsjakZ62ZuJ/3nd1ITXfsZlkhqUbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucP22eM+A==</latexit>

Sub-Riemannian case: [Cheng, Hwang, Malchiodi & Yang (2005), Ritore & Rosales (2008)]

using Jacobi fields and Riemannian connections.13 / 22

Main result: the regular caseConclusion

. Conclusion: if φ, φ∗,E are of class C 2, E = Eφ.

Q1: Relation with ambient geodesics ?

Q2: Regularity of φ-bubbles ?

Q3: What if φ, φ∗ are not C 2 ?

Q1: Relation with ambient geodesics

Let φ† be the norm defined as φ†(ξ) = φ∗(ξ⊥), ξ ∈ R2,

where ξ⊥ = (x , y)⊥ = (−y , x), ξ = (x , y) ∈ R2.

Proposition (F., Monti, Righini & Sigalotti, cf. Berestovskiı (1994))

Let φ, φ∗ be of class C 2, and E ⊂ H1 be φ-isoperimetric of class C 2.

Then ∂E \ C(E) is foliated by geodesics of H1 relative to the norm φ†.

14 / 22

Main result: the regular caseConclusion

. Conclusion: if φ, φ∗,E are of class C 2, E = Eφ.

Q1: Relation with ambient geodesics ?

Q2: Regularity of φ-bubbles ?

Q3: What if φ, φ∗ are not C 2 ?

Q1: Relation with ambient geodesics

Let φ† be the norm defined as φ†(ξ) = φ∗(ξ⊥), ξ ∈ R2,

where ξ⊥ = (x , y)⊥ = (−y , x), ξ = (x , y) ∈ R2.

Proposition (F., Monti, Righini & Sigalotti, cf. Berestovskiı (1994))

Let φ, φ∗ be of class C 2, and E ⊂ H1 be φ-isoperimetric of class C 2.

Then ∂E \ C(E) is foliated by geodesics of H1 relative to the norm φ†.

14 / 22

Main result: the regular caseConclusion

. Conclusion: if φ, φ∗,E are of class C 2, E = Eφ.

�

Q1: Relation with ambient geodesics ?

Q2: Regularity of φ-bubbles ?

Q3: What if φ, φ∗ are not C 2 ?

Q1: Relation with ambient geodesics

Let φ† be the norm defined as φ†(ξ) = φ∗(ξ⊥), ξ ∈ R2,

where ξ⊥ = (x , y)⊥ = (−y , x), ξ = (x , y) ∈ R2.

Proposition (F., Monti, Righini & Sigalotti, cf. Berestovskiı (1994))

Let φ, φ∗ be of class C 2, and E ⊂ H1 be φ-isoperimetric of class C 2.

Then ∂E \ C(E) is foliated by geodesics of H1 relative to the norm φ†.

14 / 22

Main result: the regular caseConclusion

. Conclusion: if φ, φ∗,E are of class C 2, E = Eφ.

�

-2

-1

0

1

2

x

-2

-1

0

1

2

y

0

1

2

z

Q1: Relation with ambient geodesics ?

Q2: Regularity of φ-bubbles ?

Q3: What if φ, φ∗ are not C 2 ?

Q1: Relation with ambient geodesics

Let φ† be the norm defined as φ†(ξ) = φ∗(ξ⊥), ξ ∈ R2,

where ξ⊥ = (x , y)⊥ = (−y , x), ξ = (x , y) ∈ R2.

Proposition (F., Monti, Righini & Sigalotti, cf. Berestovskiı (1994))

Let φ, φ∗ be of class C 2, and E ⊂ H1 be φ-isoperimetric of class C 2.

Then ∂E \ C(E) is foliated by geodesics of H1 relative to the norm φ†.

14 / 22

Main result: the regular caseConclusion

. Conclusion: if φ, φ∗,E are of class C 2, E = Eφ.

�

-2

-1

0

1

2

x

-2

-1

0

1

2

y

0

1

2

z

Q1: Relation with ambient geodesics ?

Q2: Regularity of φ-bubbles ?

Q3: What if φ, φ∗ are not C 2 ?

Q1: Relation with ambient geodesics

Let φ† be the norm defined as φ†(ξ) = φ∗(ξ⊥), ξ ∈ R2,

where ξ⊥ = (x , y)⊥ = (−y , x), ξ = (x , y) ∈ R2.

Proposition (F., Monti, Righini & Sigalotti, cf. Berestovskiı (1994))

Let φ, φ∗ be of class C 2, and E ⊂ H1 be φ-isoperimetric of class C 2.

Then ∂E \ C(E) is foliated by geodesics of H1 relative to the norm φ†.

14 / 22

Q2: Regularity of φ-bubbles

Notation: We say that a norm φ in R2 is of class C `+, ` ∈ N if φ ∈ C `(R2 \ {0})and φ-circles have strictly positive curvature.

Theorem (F., Monti, Righini & Sigalotti)

If φ is of class C 4+, then ∂Eφ is an embedded surface of class C 2.

Theorem (Pozuelo & Ritore)

If φ is of class C 2+, then ∂Eφ is an embedded surface of class C 2.

. In [Pozuelo & Ritore], φ does not need to be symmetric w.r.t. the origin. Aconstruction of Pansu’s bubbles is proposed in this case. They show theirminimality among competitors in a suitable cylinder via calibration techniques.

T<latexit sha1_base64="wuUeoCr40AnIpr753sXMM0828Vs=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BLx4TyAuSJcxOepMxs7PLzKwQQr7AiwdFvPpJ3vwbJ8keNLGgoajqprsrSATXxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8P/fbT6g0j2XDTBL0IzqUPOSMGivVG/1iyS27C5B14mWkBBlq/eJXbxCzNEJpmKBadz03Mf6UKsOZwFmhl2pMKBvTIXYtlTRC7U8Xh87IhVUGJIyVLWnIQv09MaWR1pMosJ0RNSO96s3F/7xuasJbf8plkhqUbLkoTAUxMZl/TQZcITNiYgllittbCRtRRZmx2RRsCN7qy+ukVSl7V+VK/bpUvcviyMMZnMMleHADVXiAGjSBAcIzvMKb8+i8OO/Ox7I152Qzp/AHzucPsPeM3A==</latexit>

→ φ-bubble in the case {φ = 1} = T . Image takenfrom [Pozuelo & Ritore].

15 / 22

Q2: Regularity of φ-bubbles

Notation: We say that a norm φ in R2 is of class C `+, ` ∈ N if φ ∈ C `(R2 \ {0})and φ-circles have strictly positive curvature.

Theorem (F., Monti, Righini & Sigalotti)

If φ is of class C 4+, then ∂Eφ is an embedded surface of class C 2.

Theorem (Pozuelo & Ritore)

If φ is of class C 2+, then ∂Eφ is an embedded surface of class C 2.

. In [Pozuelo & Ritore], φ does not need to be symmetric w.r.t. the origin. Aconstruction of Pansu’s bubbles is proposed in this case. They show theirminimality among competitors in a suitable cylinder via calibration techniques.

T<latexit sha1_base64="wuUeoCr40AnIpr753sXMM0828Vs=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BLx4TyAuSJcxOepMxs7PLzKwQQr7AiwdFvPpJ3vwbJ8keNLGgoajqprsrSATXxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8P/fbT6g0j2XDTBL0IzqUPOSMGivVG/1iyS27C5B14mWkBBlq/eJXbxCzNEJpmKBadz03Mf6UKsOZwFmhl2pMKBvTIXYtlTRC7U8Xh87IhVUGJIyVLWnIQv09MaWR1pMosJ0RNSO96s3F/7xuasJbf8plkhqUbLkoTAUxMZl/TQZcITNiYgllittbCRtRRZmx2RRsCN7qy+ukVSl7V+VK/bpUvcviyMMZnMMleHADVXiAGjSBAcIzvMKb8+i8OO/Ox7I152Qzp/AHzucPsPeM3A==</latexit>

→ φ-bubble in the case {φ = 1} = T . Image takenfrom [Pozuelo & Ritore].

15 / 22

Q3: No C 2 regularity: General norms

For a general norm φ in R2 (no regularity assumptions), we deduce a corollaryof our main theorem, assuming the following conjecture to hold true

Conjecture: If φ is of class C∞+ , φ-isoperimetric sets are of class C 2.

Corollary (F., Monti, Righini & Sigalotti)

Assume that the previous conjecture holds true. Then for any norm φ in R2 theφ-bubble Eφ ⊂ H1 is φ-isoperimetric.

. Approximation procedure.

1 ∀ε > 0, ∃ φε of class C∞+ with φ∗ε of class C∞:

(1− η(ε))φε ≤ φ ≤ (1 + η(ε))φε, η(ε)ε→0+

→ 0.{� = 1}

<latexit sha1_base64="wNBbMNTLK12Ymlsqaz0fGT+FReE=">AAAB8XicdVBNSwMxEJ31s9avqkcvwSJ4Kpsqtj0IRS8eK9gP7C4lm6ZtaDa7JFmhLP0XXjwo4tV/481/Y7atoKIPBh7vzTAzL4gF18Z1P5yl5ZXVtfXcRn5za3tnt7C339JRoihr0khEqhMQzQSXrGm4EawTK0bCQLB2ML7K/PY9U5pH8tZMYuaHZCj5gFNirHTnpV484hfYm/YKRbfkui7GGGUEV85dS2q1ahlXEc4siyIs0OgV3r1+RJOQSUMF0bqL3dj4KVGGU8GmeS/RLCZ0TIasa6kkIdN+Ort4io6t0keDSNmSBs3U7xMpCbWehIHtDIkZ6d9eJv7ldRMzqPopl3FimKTzRYNEIBOh7H3U54pRIyaWEKq4vRXREVGEGhtS3obw9Sn6n7TKJXxaKt+cFeuXizhycAhHcAIYKlCHa2hAEyhIeIAneHa08+i8OK/z1iVnMXMAP+C8fQKEF5DT</latexit>

{�" = 1}<latexit sha1_base64="h7sRxhWdR48IIQZIuoznSe5wZh8=">AAAB/3icdVDLSsNAFJ3UV62vqODGzWARXJUkDW1dCEU3LivYBzShTKbTduhkEmYmhRK78FfcuFDErb/hzr9x0lZQ0QMXDufcy733BDGjUlnWh5FbWV1b38hvFra2d3b3zP2DlowSgUkTRywSnQBJwignTUUVI51YEBQGjLSD8VXmtydESBrxWzWNiR+iIacDipHSUs888lIvHtGeN0GCxJKyiF/Y3qxnFq3Sea3iuBVolSyrajt2RpyqW3ahrZUMRbBEo2e+e/0IJyHhCjMkZde2YuWnSCiKGZkVvESSGOExGpKuphyFRPrp/P4ZPNVKHw4ioYsrOFe/T6QolHIaBrozRGokf3uZ+JfXTdSg5qeUx4kiHC8WDRIGVQSzMGCfCoIVm2qCsKD6VohHSCCsdGQFHcLXp/B/0nJKdrnk3LjF+uUyjjw4BifgDNigCurgGjRAE2BwBx7AE3g27o1H48V4XbTmjOXMIfgB4+0TqZWWhg==</latexit>

2 Then: (1− η(ε))Pφ(F ) ≤ Pφε(F ) ≤ (1 + η(ε))Pφ(F ) for any meas. F ,

and Eφε → Eφ Hausdorff (hence L1).

3 Conclude by semicontinuity of Pφ and main Theorem.

16 / 22

Q3: No C 2 regularity: The crystalline case

Notation: A norm φ : R2 → [0,∞) is called crystalline if {φ = 1} is a convexpolygon centrally symmetric with respect to the origin.

{� = 1}<latexit sha1_base64="wNBbMNTLK12Ymlsqaz0fGT+FReE=">AAAB8XicdVBNSwMxEJ31s9avqkcvwSJ4Kpsqtj0IRS8eK9gP7C4lm6ZtaDa7JFmhLP0XXjwo4tV/481/Y7atoKIPBh7vzTAzL4gF18Z1P5yl5ZXVtfXcRn5za3tnt7C339JRoihr0khEqhMQzQSXrGm4EawTK0bCQLB2ML7K/PY9U5pH8tZMYuaHZCj5gFNirHTnpV484hfYm/YKRbfkui7GGGUEV85dS2q1ahlXEc4siyIs0OgV3r1+RJOQSUMF0bqL3dj4KVGGU8GmeS/RLCZ0TIasa6kkIdN+Ort4io6t0keDSNmSBs3U7xMpCbWehIHtDIkZ6d9eJv7ldRMzqPopl3FimKTzRYNEIBOh7H3U54pRIyaWEKq4vRXREVGEGhtS3obw9Sn6n7TKJXxaKt+cFeuXizhycAhHcAIYKlCHa2hAEyhIeIAneHa08+i8OK/z1iVnMXMAP+C8fQKEF5DT</latexit>

v1<latexit sha1_base64="CjmP85F9x9vbcL2ENUsPPB1Vqnk=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120i7dbMLuplBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0N/NbY1Sax/LJTBL0IzqQPOSMGis9jnter1R2K+4cZJV4OSlDjnqv9NXtxyyNUBomqNYdz02Mn1FlOBM4LXZTjQllIzrAjqWSRqj9bH7qlJxbpU/CWNmShszV3xMZjbSeRIHtjKgZ6mVvJv7ndVIT3vgZl0lqULLFojAVxMRk9jfpc4XMiIkllClubyVsSBVlxqZTtCF4yy+vkma14l1Wqg9X5dptHkcBTuEMLsCDa6jBPdShAQwG8Ayv8OYI58V5dz4WrWtOPnMCf+B8/gAJ8o2i</latexit>

v2<latexit sha1_base64="lE8WDoWfUTC8KgvVYi3+vwjrTxc=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120y7dbMLupFBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJFIYdN1vZ219Y3Nru7BT3N3bPzgsHR03TZxqxhsslrFuB9RwKRRvoEDJ24nmNAokbwWju5nfGnNtRKyecJJwP6IDJULBKFrpcdyr9kplt+LOQVaJl5My5Kj3Sl/dfszSiCtkkhrT8dwE/YxqFEzyabGbGp5QNqID3rFU0YgbP5ufOiXnVumTMNa2FJK5+nsio5ExkyiwnRHFoVn2ZuJ/XifF8MbPhEpS5IotFoWpJBiT2d+kLzRnKCeWUKaFvZWwIdWUoU2naEPwll9eJc1qxbusVB+uyrXbPI4CnMIZXIAH11CDe6hDAxgM4Ble4c2Rzovz7nwsWtecfOYE/sD5/AELdo2j</latexit>

v0 = v2N<latexit sha1_base64="/iW5nARCLKZ5hjmV/4BQy2sLM5g=">AAAB8XicbVBNSwMxEJ31s9avqkcvwSJ4KrtV0ItQ9OJJKtgPbJclm2bb0GyyJNlCWfovvHhQxKv/xpv/xrTdg7Y+GHi8N8PMvDDhTBvX/XZWVtfWNzYLW8Xtnd29/dLBYVPLVBHaIJJL1Q6xppwJ2jDMcNpOFMVxyGkrHN5O/daIKs2keDTjhPox7gsWMYKNlZ5GgXs9CrLq/SQold2KOwNaJl5OypCjHpS+uj1J0pgKQzjWuuO5ifEzrAwjnE6K3VTTBJMh7tOOpQLHVPvZ7OIJOrVKD0VS2RIGzdTfExmOtR7Hoe2MsRnoRW8q/ud1UhNd+RkTSWqoIPNFUcqRkWj6PuoxRYnhY0swUczeisgAK0yMDaloQ/AWX14mzWrFO69UHy7KtZs8jgIcwwmcgQeXUIM7qEMDCAh4hld4c7Tz4rw7H/PWFSefOYI/cD5/APLSkHE=</latexit>

e1<latexit sha1_base64="ncHTOm5TWUpsqD6Xz54liGZOCiM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gH2vX664VXcOskq8nFQgR6Nf/uoNYpZGKA0TVOuu5ybGz6gynAmclnqpxoSyMR1i11JJI9R+Nj91Ss6sMiBhrGxJQ+bq74mMRlpPosB2RtSM9LI3E//zuqkJr/2MyyQ1KNliUZgKYmIy+5sMuEJmxMQSyhS3txI2oooyY9Mp2RC85ZdXSatW9S6qtfvLSv0mj6MIJ3AK5+DBFdThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifP+/9jZE=</latexit>

e2<latexit sha1_base64="pUK9EvQlzDaqpbNMSw4weVJEtcE=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gP1av1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwDxgY2S</latexit>

e2N<latexit sha1_base64="wYp9N2gtr+ViSNjaZ6WrPdndsCw=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04kkq2A9oQ9lsJ+3aTTbsboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8IBFcG9f9dlZW19Y3Ngtbxe2d3b390sFhU8tUMWwwKaRqB1Sj4DE2DDcC24lCGgUCW8HoZuq3nlBpLuMHM07Qj+gg5iFn1Fipib2sejfplcpuxZ2BLBMvJ2XIUe+Vvrp9ydIIY8ME1brjuYnxM6oMZwInxW6qMaFsRAfYsTSmEWo/m107IadW6ZNQKluxITP190RGI63HUWA7I2qGetGbiv95ndSEV37G4yQ1GLP5ojAVxEgyfZ30uUJmxNgSyhS3txI2pIoyYwMq2hC8xZeXSbNa8c4r1fuLcu06j6MAx3ACZ+DBJdTgFurQAAaP8Ayv8OZI58V5dz7mrStOPnMEf+B8/gBS8472</latexit> v1, . . . , v2N = ordered vertices;

ei = vi − vi−1, i = 1, . . . , 2N edges,

ei = (ei,1, ei,2).

We consider the left-invariant vector fields

Xi := ei,1X + ei,2Y , i = 1, . . . , 2N,

Theorem

Let φ be crystalline and E ⊂ H1 be φ-isoperimetric.

Let A ⊂ H1 be an open set such that ∂E ∩ A = {z = f (ξ)} with f ∈ C 2.

Then there exists i = 1, . . . ,N s. t. ∂E ∩ A is foliated by integral curves of Xi .

17 / 22

Q3: No C 2 regularity: The crystalline case

Notation: A norm φ : R2 → [0,∞) is called crystalline if {φ = 1} is a convexpolygon centrally symmetric with respect to the origin.

{� = 1}<latexit sha1_base64="wNBbMNTLK12Ymlsqaz0fGT+FReE=">AAAB8XicdVBNSwMxEJ31s9avqkcvwSJ4Kpsqtj0IRS8eK9gP7C4lm6ZtaDa7JFmhLP0XXjwo4tV/481/Y7atoKIPBh7vzTAzL4gF18Z1P5yl5ZXVtfXcRn5za3tnt7C339JRoihr0khEqhMQzQSXrGm4EawTK0bCQLB2ML7K/PY9U5pH8tZMYuaHZCj5gFNirHTnpV484hfYm/YKRbfkui7GGGUEV85dS2q1ahlXEc4siyIs0OgV3r1+RJOQSUMF0bqL3dj4KVGGU8GmeS/RLCZ0TIasa6kkIdN+Ort4io6t0keDSNmSBs3U7xMpCbWehIHtDIkZ6d9eJv7ldRMzqPopl3FimKTzRYNEIBOh7H3U54pRIyaWEKq4vRXREVGEGhtS3obw9Sn6n7TKJXxaKt+cFeuXizhycAhHcAIYKlCHa2hAEyhIeIAneHa08+i8OK/z1iVnMXMAP+C8fQKEF5DT</latexit>

v1<latexit sha1_base64="CjmP85F9x9vbcL2ENUsPPB1Vqnk=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120i7dbMLuplBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0N/NbY1Sax/LJTBL0IzqQPOSMGis9jnter1R2K+4cZJV4OSlDjnqv9NXtxyyNUBomqNYdz02Mn1FlOBM4LXZTjQllIzrAjqWSRqj9bH7qlJxbpU/CWNmShszV3xMZjbSeRIHtjKgZ6mVvJv7ndVIT3vgZl0lqULLFojAVxMRk9jfpc4XMiIkllClubyVsSBVlxqZTtCF4yy+vkma14l1Wqg9X5dptHkcBTuEMLsCDa6jBPdShAQwG8Ayv8OYI58V5dz4WrWtOPnMCf+B8/gAJ8o2i</latexit>

v2<latexit sha1_base64="lE8WDoWfUTC8KgvVYi3+vwjrTxc=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120y7dbMLupFBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJFIYdN1vZ219Y3Nru7BT3N3bPzgsHR03TZxqxhsslrFuB9RwKRRvoEDJ24nmNAokbwWju5nfGnNtRKyecJJwP6IDJULBKFrpcdyr9kplt+LOQVaJl5My5Kj3Sl/dfszSiCtkkhrT8dwE/YxqFEzyabGbGp5QNqID3rFU0YgbP5ufOiXnVumTMNa2FJK5+nsio5ExkyiwnRHFoVn2ZuJ/XifF8MbPhEpS5IotFoWpJBiT2d+kLzRnKCeWUKaFvZWwIdWUoU2naEPwll9eJc1qxbusVB+uyrXbPI4CnMIZXIAH11CDe6hDAxgM4Ble4c2Rzovz7nwsWtecfOYE/sD5/AELdo2j</latexit>

v0 = v2N<latexit sha1_base64="/iW5nARCLKZ5hjmV/4BQy2sLM5g=">AAAB8XicbVBNSwMxEJ31s9avqkcvwSJ4KrtV0ItQ9OJJKtgPbJclm2bb0GyyJNlCWfovvHhQxKv/xpv/xrTdg7Y+GHi8N8PMvDDhTBvX/XZWVtfWNzYLW8Xtnd29/dLBYVPLVBHaIJJL1Q6xppwJ2jDMcNpOFMVxyGkrHN5O/daIKs2keDTjhPox7gsWMYKNlZ5GgXs9CrLq/SQold2KOwNaJl5OypCjHpS+uj1J0pgKQzjWuuO5ifEzrAwjnE6K3VTTBJMh7tOOpQLHVPvZ7OIJOrVKD0VS2RIGzdTfExmOtR7Hoe2MsRnoRW8q/ud1UhNd+RkTSWqoIPNFUcqRkWj6PuoxRYnhY0swUczeisgAK0yMDaloQ/AWX14mzWrFO69UHy7KtZs8jgIcwwmcgQeXUIM7qEMDCAh4hld4c7Tz4rw7H/PWFSefOYI/cD5/APLSkHE=</latexit>

e1<latexit sha1_base64="ncHTOm5TWUpsqD6Xz54liGZOCiM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gH2vX664VXcOskq8nFQgR6Nf/uoNYpZGKA0TVOuu5ybGz6gynAmclnqpxoSyMR1i11JJI9R+Nj91Ss6sMiBhrGxJQ+bq74mMRlpPosB2RtSM9LI3E//zuqkJr/2MyyQ1KNliUZgKYmIy+5sMuEJmxMQSyhS3txI2oooyY9Mp2RC85ZdXSatW9S6qtfvLSv0mj6MIJ3AK5+DBFdThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifP+/9jZE=</latexit>

e2<latexit sha1_base64="pUK9EvQlzDaqpbNMSw4weVJEtcE=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gP1av1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwDxgY2S</latexit>

e2N<latexit sha1_base64="wYp9N2gtr+ViSNjaZ6WrPdndsCw=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04kkq2A9oQ9lsJ+3aTTbsboQS+h+8eFDEq//Hm//GbZuDtj4YeLw3w8y8IBFcG9f9dlZW19Y3Ngtbxe2d3b390sFhU8tUMWwwKaRqB1Sj4DE2DDcC24lCGgUCW8HoZuq3nlBpLuMHM07Qj+gg5iFn1Fipib2sejfplcpuxZ2BLBMvJ2XIUe+Vvrp9ydIIY8ME1brjuYnxM6oMZwInxW6qMaFsRAfYsTSmEWo/m107IadW6ZNQKluxITP190RGI63HUWA7I2qGetGbiv95ndSEV37G4yQ1GLP5ojAVxEgyfZ30uUJmxNgSyhS3txI2pIoyYwMq2hC8xZeXSbNa8c4r1fuLcu06j6MAx3ACZ+DBJdTgFurQAAaP8Ayv8OZI58V5dz7mrStOPnMEf+B8/gBS8472</latexit> v1, . . . , v2N = ordered vertices;

ei = vi − vi−1, i = 1, . . . , 2N edges,

ei = (ei,1, ei,2).

We consider the left-invariant vector fields

Xi := ei,1X + ei,2Y , i = 1, . . . , 2N,

Theorem

Let φ be crystalline and E ⊂ H1 be φ-isoperimetric.

Let A ⊂ H1 be an open set such that ∂E ∩ A = {z = f (ξ)} with f ∈ C 2.

Then there exists i = 1, . . . ,N s. t. ∂E ∩ A is foliated by integral curves of Xi .

17 / 22

Q3: No C 2 regularity: The crystalline case

Example: φ(x , y) = `∞(x , y) = max{|x |, |y |}. The φ-bubble is obtained byhorizontal lifts of squares.

{� = 1}<latexit sha1_base64="wNBbMNTLK12Ymlsqaz0fGT+FReE=">AAAB8XicdVBNSwMxEJ31s9avqkcvwSJ4Kpsqtj0IRS8eK9gP7C4lm6ZtaDa7JFmhLP0XXjwo4tV/481/Y7atoKIPBh7vzTAzL4gF18Z1P5yl5ZXVtfXcRn5za3tnt7C339JRoihr0khEqhMQzQSXrGm4EawTK0bCQLB2ML7K/PY9U5pH8tZMYuaHZCj5gFNirHTnpV484hfYm/YKRbfkui7GGGUEV85dS2q1ahlXEc4siyIs0OgV3r1+RJOQSUMF0bqL3dj4KVGGU8GmeS/RLCZ0TIasa6kkIdN+Ort4io6t0keDSNmSBs3U7xMpCbWehIHtDIkZ6d9eJv7ldRMzqPopl3FimKTzRYNEIBOh7H3U54pRIyaWEKq4vRXREVGEGhtS3obw9Sn6n7TKJXxaKt+cFeuXizhycAhHcAIYKlCHa2hAEyhIeIAneHa08+i8OK/z1iVnMXMAP+C8fQKEF5DT</latexit>

v1<latexit sha1_base64="CjmP85F9x9vbcL2ENUsPPB1Vqnk=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120i7dbMLuplBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0N/NbY1Sax/LJTBL0IzqQPOSMGis9jnter1R2K+4cZJV4OSlDjnqv9NXtxyyNUBomqNYdz02Mn1FlOBM4LXZTjQllIzrAjqWSRqj9bH7qlJxbpU/CWNmShszV3xMZjbSeRIHtjKgZ6mVvJv7ndVIT3vgZl0lqULLFojAVxMRk9jfpc4XMiIkllClubyVsSBVlxqZTtCF4yy+vkma14l1Wqg9X5dptHkcBTuEMLsCDa6jBPdShAQwG8Ayv8OYI58V5dz4WrWtOPnMCf+B8/gAJ8o2i</latexit>

v2<latexit sha1_base64="lE8WDoWfUTC8KgvVYi3+vwjrTxc=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120y7dbMLupFBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJFIYdN1vZ219Y3Nru7BT3N3bPzgsHR03TZxqxhsslrFuB9RwKRRvoEDJ24nmNAokbwWju5nfGnNtRKyecJJwP6IDJULBKFrpcdyr9kplt+LOQVaJl5My5Kj3Sl/dfszSiCtkkhrT8dwE/YxqFEzyabGbGp5QNqID3rFU0YgbP5ufOiXnVumTMNa2FJK5+nsio5ExkyiwnRHFoVn2ZuJ/XifF8MbPhEpS5IotFoWpJBiT2d+kLzRnKCeWUKaFvZWwIdWUoU2naEPwll9eJc1qxbusVB+uyrXbPI4CnMIZXIAH11CDe6hDAxgM4Ble4c2Rzovz7nwsWtecfOYE/sD5/AELdo2j</latexit>

e1<latexit sha1_base64="ncHTOm5TWUpsqD6Xz54liGZOCiM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gH2vX664VXcOskq8nFQgR6Nf/uoNYpZGKA0TVOuu5ybGz6gynAmclnqpxoSyMR1i11JJI9R+Nj91Ss6sMiBhrGxJQ+bq74mMRlpPosB2RtSM9LI3E//zuqkJr/2MyyQ1KNliUZgKYmIy+5sMuEJmxMQSyhS3txI2oooyY9Mp2RC85ZdXSatW9S6qtfvLSv0mj6MIJ3AK5+DBFdThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifP+/9jZE=</latexit>

e2<latexit sha1_base64="pUK9EvQlzDaqpbNMSw4weVJEtcE=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gP1av1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwDxgY2S</latexit>

Through the previous theorem, we recoversome “local” information, but we are notable to “glow” all the pieces.

Foliation by Y ∝ e2 - Foliation by X ∝ e1

18 / 22

Q3: No C 2 regularity: The crystalline case

Example: φ(x , y) = `∞(x , y) = max{|x |, |y |}. The φ-bubble is obtained byhorizontal lifts of squares.

{� = 1}<latexit sha1_base64="wNBbMNTLK12Ymlsqaz0fGT+FReE=">AAAB8XicdVBNSwMxEJ31s9avqkcvwSJ4Kpsqtj0IRS8eK9gP7C4lm6ZtaDa7JFmhLP0XXjwo4tV/481/Y7atoKIPBh7vzTAzL4gF18Z1P5yl5ZXVtfXcRn5za3tnt7C339JRoihr0khEqhMQzQSXrGm4EawTK0bCQLB2ML7K/PY9U5pH8tZMYuaHZCj5gFNirHTnpV484hfYm/YKRbfkui7GGGUEV85dS2q1ahlXEc4siyIs0OgV3r1+RJOQSUMF0bqL3dj4KVGGU8GmeS/RLCZ0TIasa6kkIdN+Ort4io6t0keDSNmSBs3U7xMpCbWehIHtDIkZ6d9eJv7ldRMzqPopl3FimKTzRYNEIBOh7H3U54pRIyaWEKq4vRXREVGEGhtS3obw9Sn6n7TKJXxaKt+cFeuXizhycAhHcAIYKlCHa2hAEyhIeIAneHa08+i8OK/z1iVnMXMAP+C8fQKEF5DT</latexit>

v1<latexit sha1_base64="CjmP85F9x9vbcL2ENUsPPB1Vqnk=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120i7dbMLuplBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0N/NbY1Sax/LJTBL0IzqQPOSMGis9jnter1R2K+4cZJV4OSlDjnqv9NXtxyyNUBomqNYdz02Mn1FlOBM4LXZTjQllIzrAjqWSRqj9bH7qlJxbpU/CWNmShszV3xMZjbSeRIHtjKgZ6mVvJv7ndVIT3vgZl0lqULLFojAVxMRk9jfpc4XMiIkllClubyVsSBVlxqZTtCF4yy+vkma14l1Wqg9X5dptHkcBTuEMLsCDa6jBPdShAQwG8Ayv8OYI58V5dz4WrWtOPnMCf+B8/gAJ8o2i</latexit>

v2<latexit sha1_base64="lE8WDoWfUTC8KgvVYi3+vwjrTxc=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120y7dbMLupFBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJFIYdN1vZ219Y3Nru7BT3N3bPzgsHR03TZxqxhsslrFuB9RwKRRvoEDJ24nmNAokbwWju5nfGnNtRKyecJJwP6IDJULBKFrpcdyr9kplt+LOQVaJl5My5Kj3Sl/dfszSiCtkkhrT8dwE/YxqFEzyabGbGp5QNqID3rFU0YgbP5ufOiXnVumTMNa2FJK5+nsio5ExkyiwnRHFoVn2ZuJ/XifF8MbPhEpS5IotFoWpJBiT2d+kLzRnKCeWUKaFvZWwIdWUoU2naEPwll9eJc1qxbusVB+uyrXbPI4CnMIZXIAH11CDe6hDAxgM4Ble4c2Rzovz7nwsWtecfOYE/sD5/AELdo2j</latexit>

e1<latexit sha1_base64="ncHTOm5TWUpsqD6Xz54liGZOCiM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gH2vX664VXcOskq8nFQgR6Nf/uoNYpZGKA0TVOuu5ybGz6gynAmclnqpxoSyMR1i11JJI9R+Nj91Ss6sMiBhrGxJQ+bq74mMRlpPosB2RtSM9LI3E//zuqkJr/2MyyQ1KNliUZgKYmIy+5sMuEJmxMQSyhS3txI2oooyY9Mp2RC85ZdXSatW9S6qtfvLSv0mj6MIJ3AK5+DBFdThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifP+/9jZE=</latexit>

e2<latexit sha1_base64="pUK9EvQlzDaqpbNMSw4weVJEtcE=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gP1av1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwDxgY2S</latexit>

Through the previous theorem, we recoversome “local” information, but we are notable to “glow” all the pieces.

Foliation by Y ∝ e2 - Foliation by X ∝ e1

18 / 22

Q3: No C 2 regularity: The crystalline case

Example: φ(x , y) = `∞(x , y) = max{|x |, |y |}. The φ-bubble is obtained byhorizontal lifts of squares.

{� = 1}<latexit sha1_base64="wNBbMNTLK12Ymlsqaz0fGT+FReE=">AAAB8XicdVBNSwMxEJ31s9avqkcvwSJ4Kpsqtj0IRS8eK9gP7C4lm6ZtaDa7JFmhLP0XXjwo4tV/481/Y7atoKIPBh7vzTAzL4gF18Z1P5yl5ZXVtfXcRn5za3tnt7C339JRoihr0khEqhMQzQSXrGm4EawTK0bCQLB2ML7K/PY9U5pH8tZMYuaHZCj5gFNirHTnpV484hfYm/YKRbfkui7GGGUEV85dS2q1ahlXEc4siyIs0OgV3r1+RJOQSUMF0bqL3dj4KVGGU8GmeS/RLCZ0TIasa6kkIdN+Ort4io6t0keDSNmSBs3U7xMpCbWehIHtDIkZ6d9eJv7ldRMzqPopl3FimKTzRYNEIBOh7H3U54pRIyaWEKq4vRXREVGEGhtS3obw9Sn6n7TKJXxaKt+cFeuXizhycAhHcAIYKlCHa2hAEyhIeIAneHa08+i8OK/z1iVnMXMAP+C8fQKEF5DT</latexit>

v1<latexit sha1_base64="CjmP85F9x9vbcL2ENUsPPB1Vqnk=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120i7dbMLuplBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0N/NbY1Sax/LJTBL0IzqQPOSMGis9jnter1R2K+4cZJV4OSlDjnqv9NXtxyyNUBomqNYdz02Mn1FlOBM4LXZTjQllIzrAjqWSRqj9bH7qlJxbpU/CWNmShszV3xMZjbSeRIHtjKgZ6mVvJv7ndVIT3vgZl0lqULLFojAVxMRk9jfpc4XMiIkllClubyVsSBVlxqZTtCF4yy+vkma14l1Wqg9X5dptHkcBTuEMLsCDa6jBPdShAQwG8Ayv8OYI58V5dz4WrWtOPnMCf+B8/gAJ8o2i</latexit>

v2<latexit sha1_base64="lE8WDoWfUTC8KgvVYi3+vwjrTxc=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120y7dbMLupFBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJFIYdN1vZ219Y3Nru7BT3N3bPzgsHR03TZxqxhsslrFuB9RwKRRvoEDJ24nmNAokbwWju5nfGnNtRKyecJJwP6IDJULBKFrpcdyr9kplt+LOQVaJl5My5Kj3Sl/dfszSiCtkkhrT8dwE/YxqFEzyabGbGp5QNqID3rFU0YgbP5ufOiXnVumTMNa2FJK5+nsio5ExkyiwnRHFoVn2ZuJ/XifF8MbPhEpS5IotFoWpJBiT2d+kLzRnKCeWUKaFvZWwIdWUoU2naEPwll9eJc1qxbusVB+uyrXbPI4CnMIZXIAH11CDe6hDAxgM4Ble4c2Rzovz7nwsWtecfOYE/sD5/AELdo2j</latexit>

e1<latexit sha1_base64="ncHTOm5TWUpsqD6Xz54liGZOCiM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gH2vX664VXcOskq8nFQgR6Nf/uoNYpZGKA0TVOuu5ybGz6gynAmclnqpxoSyMR1i11JJI9R+Nj91Ss6sMiBhrGxJQ+bq74mMRlpPosB2RtSM9LI3E//zuqkJr/2MyyQ1KNliUZgKYmIy+5sMuEJmxMQSyhS3txI2oooyY9Mp2RC85ZdXSatW9S6qtfvLSv0mj6MIJ3AK5+DBFdThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifP+/9jZE=</latexit>

e2<latexit sha1_base64="pUK9EvQlzDaqpbNMSw4weVJEtcE=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gP1av1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwDxgY2S</latexit>

Through the previous theorem, we recoversome “local” information, but we are notable to “glow” all the pieces.

Foliation by Y ∝ e2 - Foliation by X ∝ e1

18 / 22

Q3: No C 2 regularity: The crystalline case

Example: φ(x , y) = `∞(x , y) = max{|x |, |y |}. The φ-bubble is obtained byhorizontal lifts of squares.

{� = 1}<latexit sha1_base64="wNBbMNTLK12Ymlsqaz0fGT+FReE=">AAAB8XicdVBNSwMxEJ31s9avqkcvwSJ4Kpsqtj0IRS8eK9gP7C4lm6ZtaDa7JFmhLP0XXjwo4tV/481/Y7atoKIPBh7vzTAzL4gF18Z1P5yl5ZXVtfXcRn5za3tnt7C339JRoihr0khEqhMQzQSXrGm4EawTK0bCQLB2ML7K/PY9U5pH8tZMYuaHZCj5gFNirHTnpV484hfYm/YKRbfkui7GGGUEV85dS2q1ahlXEc4siyIs0OgV3r1+RJOQSUMF0bqL3dj4KVGGU8GmeS/RLCZ0TIasa6kkIdN+Ort4io6t0keDSNmSBs3U7xMpCbWehIHtDIkZ6d9eJv7ldRMzqPopl3FimKTzRYNEIBOh7H3U54pRIyaWEKq4vRXREVGEGhtS3obw9Sn6n7TKJXxaKt+cFeuXizhycAhHcAIYKlCHa2hAEyhIeIAneHa08+i8OK/z1iVnMXMAP+C8fQKEF5DT</latexit>

v1<latexit sha1_base64="CjmP85F9x9vbcL2ENUsPPB1Vqnk=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120i7dbMLuplBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6jhVDBssFrFqB1Sj4BIbhhuB7UQhjQKBrWB0N/NbY1Sax/LJTBL0IzqQPOSMGis9jnter1R2K+4cZJV4OSlDjnqv9NXtxyyNUBomqNYdz02Mn1FlOBM4LXZTjQllIzrAjqWSRqj9bH7qlJxbpU/CWNmShszV3xMZjbSeRIHtjKgZ6mVvJv7ndVIT3vgZl0lqULLFojAVxMRk9jfpc4XMiIkllClubyVsSBVlxqZTtCF4yy+vkma14l1Wqg9X5dptHkcBTuEMLsCDa6jBPdShAQwG8Ayv8OYI58V5dz4WrWtOPnMCf+B8/gAJ8o2i</latexit>

v2<latexit sha1_base64="lE8WDoWfUTC8KgvVYi3+vwjrTxc=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120y7dbMLupFBCf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXJFIYdN1vZ219Y3Nru7BT3N3bPzgsHR03TZxqxhsslrFuB9RwKRRvoEDJ24nmNAokbwWju5nfGnNtRKyecJJwP6IDJULBKFrpcdyr9kplt+LOQVaJl5My5Kj3Sl/dfszSiCtkkhrT8dwE/YxqFEzyabGbGp5QNqID3rFU0YgbP5ufOiXnVumTMNa2FJK5+nsio5ExkyiwnRHFoVn2ZuJ/XifF8MbPhEpS5IotFoWpJBiT2d+kLzRnKCeWUKaFvZWwIdWUoU2naEPwll9eJc1qxbusVB+uyrXbPI4CnMIZXIAH11CDe6hDAxgM4Ble4c2Rzovz7nwsWtecfOYE/sD5/AELdo2j</latexit>

e1<latexit sha1_base64="ncHTOm5TWUpsqD6Xz54liGZOCiM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gH2vX664VXcOskq8nFQgR6Nf/uoNYpZGKA0TVOuu5ybGz6gynAmclnqpxoSyMR1i11JJI9R+Nj91Ss6sMiBhrGxJQ+bq74mMRlpPosB2RtSM9LI3E//zuqkJr/2MyyQ1KNliUZgKYmIy+5sMuEJmxMQSyhS3txI2oooyY9Mp2RC85ZdXSatW9S6qtfvLSv0mj6MIJ3AK5+DBFdThDhrQBAZDeIZXeHOE8+K8Ox+L1oKTzxzDHzifP+/9jZE=</latexit>

e2<latexit sha1_base64="pUK9EvQlzDaqpbNMSw4weVJEtcE=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0gP1av1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vklat6l1Ua/eXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwDxgY2S</latexit>

Through the previous theorem, we recoversome “local” information, but we are notable to “glow” all the pieces.

Foliation by Y ∝ e2 - Foliation by X ∝ e1

18 / 22

Outline

1 Introduction: anisotropic isoperimetric problems

2 The sub-Finsler isoperimetric problem in the Heisenberg group

3 The characteristic set

19 / 22

Study of the characteristic set in the C 2 settingGeneralizes [Cheng, Hwang, Malchiodi & Yang (2005), Ritore & Rosales (2008)].

. Qualitative structure of the characteristic set

Lemma (F., Monti, Righini & Sigalotti)

Let Σ ⊂ H1 be a C 2 surface with constant φ-curvature.

Then C(Σ) consists of isolated points and C 1 curves.

. Structure of characteristic curves

Notation: We say that a C 2 surface Σ is φ-critical if it is closed, it has constantφ-curvature at non-characteristic points and it is a critical point of theφ-perimeter functional (under volume constraint) in a neighborhood of anycharacteristic point.

Remark: C 2 pieces of φ-isoperimetric sets are φ-critical.

Theorem (F., Monti, Righini & Sigalotti)

Let φ and φ∗ be of class C 2 and let Σ ⊂ H1 be a complete and oriented surfaceof class C 2.

If Σ is φ-critical with non-vanishing φ-curvature then C(Σ) consists of isolatedpoints and C 2 curves that are either horizontal lines or horizontal lifts of simpleclosed curves.

20 / 22

Study of the characteristic set in the C 2 settingGeneralizes [Cheng, Hwang, Malchiodi & Yang (2005), Ritore & Rosales (2008)].

. Qualitative structure of the characteristic set

Lemma (F., Monti, Righini & Sigalotti)

Let Σ ⊂ H1 be a C 2 surface with constant φ-curvature.

Then C(Σ) consists of isolated points and C 1 curves.

. Structure of characteristic curves

Notation: We say that a C 2 surface Σ is φ-critical if it is closed, it has constantφ-curvature at non-characteristic points and it is a critical point of theφ-perimeter functional (under volume constraint) in a neighborhood of anycharacteristic point.

Remark: C 2 pieces of φ-isoperimetric sets are φ-critical.

Theorem (F., Monti, Righini & Sigalotti)

Let φ and φ∗ be of class C 2 and let Σ ⊂ H1 be a complete and oriented surfaceof class C 2.

If Σ is φ-critical with non-vanishing φ-curvature then C(Σ) consists of isolatedpoints and C 2 curves that are either horizontal lines or horizontal lifts of simpleclosed curves.

20 / 22

Study of the characteristic set in the C 2 setting

1 Characteristic curves are indeed C 2.

2 Use the foliation property to parametrize the surface around characteristiccurves via φ-circles:

p 2 C(⌃)<latexit sha1_base64="KqSngv1Y6K5NVNtQobLnzJB8/AQ=">AAAB/3icbVDLSgMxFM3UV62vUcGNm2AR6qbMVEGXxW5cVrQP6Awlk2ba0CQzJBmhjF34K25cKOLW33Dn35hpZ6GtBwKHc+7lnpwgZlRpx/m2Ciura+sbxc3S1vbO7p69f9BWUSIxaeGIRbIbIEUYFaSlqWakG0uCeMBIJxg3Mr/zQKSikbjXk5j4HA0FDSlG2kh9+yj2qPA40iOMGGxUvDs65Oisb5edqjMDXCZuTsogR7Nvf3mDCCecCI0ZUqrnOrH2UyQ1xYxMS16iSIzwGA1Jz1CBOFF+Oss/hadGGcAwkuYJDWfq740UcaUmPDCTWVK16GXif14v0eGVn1IRJ5oIPD8UJgzqCGZlwAGVBGs2MQRhSU1WiEdIIqxNZSVTgrv45WXSrlXd82rt9qJcv87rKIJjcAIqwAWXoA5uQBO0AAaP4Bm8gjfryXqx3q2P+WjByncOwR9Ynz/xKpVl</latexit>

� circles<latexit sha1_base64="URdyVneO8Eh59kYdUgujxERL6Jo=">AAAB/XicdVDLSgMxFM34rPU1PnZugkVwVSZVbLsrunFZwT6gM5RMmrahmQfJHbEOxV9x40IRt/6HO//GTFtBRQ9cOJxzb3Lv8WMpNDjOh7WwuLS8sppby69vbG5t2zu7TR0livEGi2Sk2j7VXIqQN0CA5O1YcRr4krf80UXmt2640iIKr2Eccy+gg1D0BaNgpK6978ZD4WIX+C2kTCgmuZ507YJTdByHEIIzQspnjiHVaqVEKphklkEBzVHv2u9uL2JJwENgkmrdIU4MXkoVCPPgJO8mmseUjeiAdwwNacC1l063n+Ajo/RwP1KmQsBT9ftESgOtx4FvOgMKQ/3by8S/vE4C/YqXijBOgIds9lE/kRginEWBe0JxBnJsCGVKmF0xG1JFGZjA8iaEr0vx/6RZKpKTYunqtFA7n8eRQwfoEB0jgsqohi5RHTUQQ3foAT2hZ+veerRerNdZ64I1n9lDP2C9fQJCzJXE</latexit>

� ⇢ C(E)<latexit sha1_base64="/G5J5Wvn963jXXzFEmuj5Nqgst8=">AAACBHicbVDLSgNBEJz1GeMr6jGXwSDES9iNgh6DQfQYwTwgG0LvZDYZMjO7zMwKYcnBi7/ixYMiXv0Ib/6Nk8dBEwsaiqpuuruCmDNtXPfbWVldW9/YzGxlt3d29/ZzB4cNHSWK0DqJeKRaAWjKmaR1wwynrVhREAGnzWBYnfjNB6o0i+S9GcW0I6AvWcgIGCt1c3n/BoQA7Osk0NT4AsyAAMfV4vVpN1dwS+4UeJl4c1JAc9S6uS+/F5FEUGkIB63bnhubTgrKMMLpOOsnmsZAhtCnbUslCKo76fSJMT6xSg+HkbIlDZ6qvydSEFqPRGA7J0fqRW8i/ue1ExNedlIm48RQSWaLwoRjE+FJIrjHFCWGjywBopi9FZMBKCDG5pa1IXiLLy+TRrnknZXKd+eFytU8jgzKo2NURB66QBV0i2qojgh6RM/oFb05T86L8+58zFpXnPnMEfoD5/MHVqSXPQ==</latexit>time t

<latexit sha1_base64="aWuM32EK9hYdQbyzo+jZ+lWawQs=">AAAB9XicbVDLTgJBEJzFF+IL9ehlIjHxRHbRRI9ELx4xkUcCK5kdGpgws7uZ6VXJhv/w4kFjvPov3vwbB9iDgpV0UqnqTndXEEth0HW/ndzK6tr6Rn6zsLW9s7tX3D9omCjRHOo8kpFuBcyAFCHUUaCEVqyBqUBCMxhdT/3mA2gjovAOxzH4ig1C0RecoZXuOwhPmKJQQCcUu8WSW3ZnoMvEy0iJZKh1i1+dXsQTBSFyyYxpe26Mfso0Ci5hUugkBmLGR2wAbUtDpsD46ezqCT2xSo/2I20rRDpTf0+kTBkzVoHtVAyHZtGbiv957QT7l34qwjhBCPl8UT+RFCM6jYD2hAaOcmwJ41rYWykfMs042qAKNgRv8eVl0qiUvbNy5fa8VL3K4siTI3JMTolHLkiV3JAaqRNONHkmr+TNeXRenHfnY96ac7KZQ/IHzucPiD2Shg==</latexit>

time s<latexit sha1_base64="nFVAHOrsthWKc1vGDUaIFFYBhJA=">AAAB9XicdVDLSgNBEJz1GeMr6tHLYBA8hZ0oJt5ELx4jmAckMcxOOjo4O7vM9KphyX948aCIV//Fm3/jbBJBRQsaiqpuuruCWEmLvv/hzczOzS8s5pbyyyura+uFjc2GjRIjoC4iFZlWwC0oqaGOEhW0YgM8DBQ0g5vTzG/egrEy0hc4jKEb8istB1JwdNJlB+EeU5Qh0BG1vULRL/m+zxijGWGVQ9+Ro6NqmVUpyyyHIpmi1iu8d/qRSELQKBS3ts38GLspNyiFglG+k1iIubjhV9B2VPMQbDcdXz2iu07p00FkXGmkY/X7RMpDa4dh4DpDjtf2t5eJf3ntBAfVbip1nCBoMVk0SBTFiGYR0L40IFANHeHCSHcrFdfccIEuqLwL4etT+j9plEtsv1Q+Pygen0zjyJFtskP2CCMVckzOSI3UiSCGPJAn8uzdeY/ei/c6aZ3xpjNb5Ae8t0/Tr5K7</latexit>

φ-circles meet the characteristic curveorthogonally.

There is a maximal time s for whichthe parametrization does not meet acharacteristic point. This is constantalong the whole characteristic curve.

This reads into the fact that thecharacteristic curve is an unboundedperiodic lift.

Remark: For φ = φ∗ = | · |, [Ritore & Rosales (2008)] use Jacobi fields techniques.

Open: We expect that characteristic curves are lifts of φ†-circles.

21 / 22

-2

-1

0

1

2

x

-2

-1

0

1

2

y

0

1

2

z

Open:

. The crystalline case: how to extend the “local” foliation to a “global” one.(crystalline characteristic set?);

. Extend the study of the characteristic set to the case of asymmetric norms(triangle type a la [Pozuelo & Ritore (2020)]);

. Extend the study of the characteristic set to the case where E , φ, φ∗ havelower regularity properties;

. Study symmetry properties of the φ-bubbles.

Thank you for your attention!

22 / 22

-2

-1

0

1

2

x

-2

-1

0

1

2

y

0

1

2

z

Open:

. The crystalline case: how to extend the “local” foliation to a “global” one.(crystalline characteristic set?);

. Extend the study of the characteristic set to the case of asymmetric norms(triangle type a la [Pozuelo & Ritore (2020)]);

. Extend the study of the characteristic set to the case where E , φ, φ∗ havelower regularity properties;

. Study symmetry properties of the φ-bubbles.

Thank you for your attention!

22 / 22