A new stable surface in the Heisenberg group Fellow: Sebastiano - - PowerPoint PPT Presentation

MAnET Metric Analysis Meeting 18 – 19 January, 2017

A new stable surface in the Heisenberg group

Fellow: Sebastiano Nicolussi Golo Advisors: Francesco Serra Cassano Enrico Le Donne

Introduction

The Heisenberg group, H: three dimensional, simply connected, nilpotent, non-Abelian Lie group. In exponential coordinates, we identify H = R3.

◮ Group operation

(a, b, c) ∗ (x, y, z) =

• a + x, b + y, c + z + 1

2(ay − bx)

• ,

◮ Left-invariant vector fields

X(x, y, z) = ∂x − 1 2y∂z Y (x, y, z) = ∂y + 1 2x∂z Z(x, y, z) = ∂z,

Introduction

◮ Intrinsic graph of f : R2 → R:

Γf =

• (0, η, τ) ∗ (f(η, τ), 0, 0) : (η, τ) ∈ R2

=

• (f, η, τ − 1

2ηf) : (η, τ) ∈ R2

• ,

◮ Intrinsic derivative of f

∇ff = ∂ηf + f∂τf, which describes the tangent space of Γf,

◮ C 1 W is the set of all C1-intrinsic functions, i.e., f : R2 → R

such that both f and ∇ff are continuous.

◮ Intrinsic area of Γf over ω ⊂ R2

• ω
• 1 + (∇ff)2 dη dτ.

Introduction

Remarks

◮ There is a natural notion of subRiemannian perimeter for

subsets E ⊂ H. If E has locally finite perimeter, then its reduced boundary ∂∗E is “essentially” the intrinsic graph of a C1-intrinsic function.

Introduction

Remarks

◮ There is a natural notion of subRiemannian perimeter for

subsets E ⊂ H. If E has locally finite perimeter, then its reduced boundary ∂∗E is “essentially” the intrinsic graph of a C1-intrinsic function.

◮ There are C1-intrinsic functions f ∈ C 1 W whose intrinsic graph

Γf is a fractal in R3.

Introduction

Remarks

◮ There is a natural notion of subRiemannian perimeter for

subsets E ⊂ H. If E has locally finite perimeter, then its reduced boundary ∂∗E is “essentially” the intrinsic graph of a C1-intrinsic function.

◮ There are C1-intrinsic functions f ∈ C 1 W whose intrinsic graph

Γf is a fractal in R3.

◮ The set C 1 W is NOT a vector space. Indeed,

f ∈ C 1

W ⇒ f + 1 ∈ BVintrinsic,loc

Introduction

Remarks

◮ There is a natural notion of subRiemannian perimeter for

subsets E ⊂ H. If E has locally finite perimeter, then its reduced boundary ∂∗E is “essentially” the intrinsic graph of a C1-intrinsic function.

◮ There are C1-intrinsic functions f ∈ C 1 W whose intrinsic graph

Γf is a fractal in R3.

◮ The set C 1 W is NOT a vector space. Indeed,

f ∈ C 1

W ⇒ f + 1 ∈ BVintrinsic,loc ◮ By minimal surface in H we mean a topological surface that

minimizes the area among all its bounded variations.

The problem

Bernstein’s Problem: Under which conditions on f is the following sentence true? If Γf is a minimal surface in H, then Γf is a vertical plane. We are interested in this problem because we want to better understand the space C 1

W and the theory of perimeters in H. ◮ If f ∈ C 1(R2), then it’s true!1 ◮ If f ∈ C 0(R2) (even Lipschitz intrinsic), then it’s false!2

• 1M. Galli and M. Ritoré. “Area-stationary and stable surfaces of class C1 in

the sub-Riemannian Heisenberg group H1”. In: Adv. Math. 285 (2015), pp. 737–765.

• 2R. Monti, F. Serra Cassano, and D. Vittone. “A negative answer to the

Bernstein problem for intrinsic graphs in the Heisenberg group”. In: Boll. Unione Mat. Ital. (9) 1.3 (2008), pp. 709–727.

The problem

Bernstein’s Problem: Under which conditions on f is the following sentence true? If Γf is a minimal surface in H, then Γf is a vertical plane. We are interested in this problem because we want to better understand the space C 1

W and the theory of perimeters in H. ◮ If f ∈ C 1(R2), then it’s true!1 ◮ If f ∈ C 0(R2) (even Lipschitz intrinsic), then it’s false!2 ◮ What if f ∈ C 1 W?

• 1M. Galli and M. Ritoré. “Area-stationary and stable surfaces of class C1 in

the sub-Riemannian Heisenberg group H1”. In: Adv. Math. 285 (2015), pp. 737–765.

• 2R. Monti, F. Serra Cassano, and D. Vittone. “A negative answer to the

Bernstein problem for intrinsic graphs in the Heisenberg group”. In: Boll. Unione Mat. Ital. (9) 1.3 (2008), pp. 709–727.

The problem

Variational approach: if Γf is a minimal surface, then, for all ψ ∈ C ∞

c (ω), the following two conditions are satisfied:

• 1. If(ψ) =

d dǫ

• ω
• 1 + (∇f+ǫψ(f + ǫψ))2 dη dτ
• ǫ=0 = 0
• 2. IIf(ψ) =

d2 dǫ2

• ω
• 1 + (∇f+ǫψ(f + ǫψ))2 dη dτ
• ǫ=0 ≥ 0

The problem

Variational approach: if Γf is a minimal surface, then, for all ψ ∈ C ∞

c (ω), the following two conditions are satisfied:

• 1. If(ψ) =

d dǫ

• ω
• 1 + (∇f+ǫψ(f + ǫψ))2 dη dτ
• ǫ=0 = 0
• 2. IIf(ψ) =

d2 dǫ2

• ω
• 1 + (∇f+ǫψ(f + ǫψ))2 dη dτ
• ǫ=0 ≥ 0

This approach cannot work in general: If we only assume f ∈ C 1

W,

then it is not true that the graph of the function f + ǫψ has necessarily locally finite area! Indeed, f ∈ C 1

W ⇒ f + 1 ∈ BVintrinsic,loc

The problem

Variational approach: if Γf is a minimal surface, then, for all ψ ∈ C ∞

c (ω), the following two conditions are satisfied:

• 1. If(ψ) =

d dǫ

• ω
• 1 + (∇f+ǫψ(f + ǫψ))2 dη dτ
• ǫ=0 = 0
• 2. IIf(ψ) =

d2 dǫ2

• ω
• 1 + (∇f+ǫψ(f + ǫψ))2 dη dτ
• ǫ=0 ≥ 0

This approach cannot work in general: If we only assume f ∈ C 1

W,

then it is not true that the graph of the function f + ǫψ has necessarily locally finite area! Indeed, f ∈ C 1

W ⇒ f + 1 ∈ BVintrinsic,loc

But this is another story...

The problem

Variational approach: if Γf is a minimal surface, then, for all ψ ∈ C ∞

c (ω), the following two conditions are satisfied:

• 1. If(ψ) =

d dǫ

• ω
• 1 + (∇f+ǫψ(f + ǫψ))2 dη dτ
• ǫ=0 = 0
• 2. IIf(ψ) =

d2 dǫ2

• ω
• 1 + (∇f+ǫψ(f + ǫψ))2 dη dτ
• ǫ=0 ≥ 0

This approach cannot work in general: If we only assume f ∈ C 1

W,

then it is not true that the graph of the function f + ǫψ has necessarily locally finite area! Indeed, f ∈ C 1

W ⇒ f + 1 ∈ BVintrinsic,loc

But this is another story... We decided tackle the following problem: If f ∈ W 1,1

loc (R2) ∩ C 1 W and Γf is a minimal surface in H,

then Γf is a vertical plane.

First Variation

For f ∈ W 1,1

loc (R2) ∩ C 1 W and ψ ∈ C ∞ c (R2),

If(ψ) =

• R2

∇ff

• 1 + (∇ff)2 (∂ηψ + ∂τ(fψ)) dη dτ.

First Variation

For f ∈ W 1,1

loc (R2) ∩ C 1 W and ψ ∈ C ∞ c (R2),

If(ψ) =

• R2

∇ff

• 1 + (∇ff)2 (∂ηψ + ∂τ(fψ)) dη dτ.

If f ∈ C 2(R2), then If(ψ) = −

• R2 ∇f
• ∇ff
• 1 + (∇ff)2
• ψ dη dτ.

First Variation

For f ∈ W 1,1

loc (R2) ∩ C 1 W and ψ ∈ C ∞ c (R2),

If(ψ) =

• R2

∇ff

• 1 + (∇ff)2 (∂ηψ + ∂τ(fψ)) dη dτ.

If f ∈ C 2(R2), then If(ψ) = −

• R2 ∇f
• ∇ff
• 1 + (∇ff)2
• ψ dη dτ.

Thus, If = 0 if and only if ∇f∇ff = 0. This is a second order differential equation.

First Variation

For f ∈ W 1,1

loc (R2) ∩ C 1 W and ψ ∈ C ∞ c (R2),

If(ψ) =

• R2

∇ff

• 1 + (∇ff)2 (∂ηψ + ∂τ(fψ)) dη dτ.

If f ∈ C 2(R2), then If(ψ) = −

• R2 ∇f
• ∇ff
• 1 + (∇ff)2
• ψ dη dτ.

Thus, If = 0 if and only if ∇f∇ff = 0. This is a second order differential equation. However, we can interpret it as (Lagrangian interpretation) ∇ff is constant along the integral curves of the vector field ∇f = ∂η + f∂τ .

First Variation

Conjecture

If f ∈ W 1,1

loc ∩ C 1 W and If = 0, then ∇ff is constant along the

integral lines of ∇f, i.e., ∇f∇ff = 0. We are working on it. However, we know that, if f ∈ W 1,1

loc ∩ C 1 W and ∇ff is constant

along the integral lines of ∇f, i.e., ∇f∇ff = 0, then If = 0.

∇f∇ff = 0

∇f = ∂η + f∂τ

Lemma

Let f ∈ C 1

• W. ∇ff is constant along the integral curves of ∇f if

and only if there are continuous functions A, B : R → R such that

• 1. all the integral curves of ∇f are given by t → (t, g(t, ζ)),

where g(t, ζ) = A(ζ) 2 t2 + B(ζ)t + ζ;

∇f∇ff = 0

∇f = ∂η + f∂τ

Lemma

Let f ∈ C 1

• W. ∇ff is constant along the integral curves of ∇f if

and only if there are continuous functions A, B : R → R such that

• 1. all the integral curves of ∇f are given by t → (t, g(t, ζ)),

where g(t, ζ) = A(ζ) 2 t2 + B(ζ)t + ζ;

• 2. The functions A, B must satisfy some properties. In particular,

A must be non-decreasing;

∇f∇ff = 0

∇f = ∂η + f∂τ

Lemma

Let f ∈ C 1

• W. ∇ff is constant along the integral curves of ∇f if

and only if there are continuous functions A, B : R → R such that

• 1. all the integral curves of ∇f are given by t → (t, g(t, ζ)),

where g(t, ζ) = A(ζ) 2 t2 + B(ζ)t + ζ;

• 2. The functions A, B must satisfy some properties. In particular,

A must be non-decreasing;

• 3. f(t, g(t, ζ)) = ∂tg(t, z) = A(ζ)t + B(ζ);
• 4. ∇ff(t, g(t, ζ)) = ∂2

t g(t, z) = A(ζ).

∇f∇ff = 0

Consider the case (B = 0, A : R → R non-decreasing and continuous) f

• t, A(ζ)t2

2 + ζ

• = A(ζ)t.

We may construct the intrinsic graph of f in this way: Γf = {(0, 0, ζ) + t(A(ζ), 1, 0) : t, ζ ∈ R}

∇f∇ff = 0 : Examples

A(ζ) = 0 f(η, τ) = 0

∇f∇ff = 0 : Examples

A(ζ) = 1 f(η, τ) = η

∇f∇ff = 0 : Examples

A(ζ) = ζ f(η, τ) = η η2/2 + 1

Second Variation

For f ∈ W 1,1

loc (R2) ∩ C 1 W and ψ ∈ C ∞ c (R2),

IIf(ψ) =

• R2

(∂ηψ + f∂τψ + ψ∂τf)2 (1 + (∇ff)2)3/2 + ∂τ(ψ2)∇ff (1 + (∇ff)2)1/2 dη dτ Let’s assume that f satisfies ∇f∇ff = 0, with A, B ∈ C (R).

Second Variation

For f ∈ W 1,1

loc (R2) ∩ C 1 W and ψ ∈ C ∞ c (R2),

IIf(ψ) =

• R2

(∂ηψ + f∂τψ + ψ∂τf)2 (1 + (∇ff)2)3/2 + ∂τ(ψ2)∇ff (1 + (∇ff)2)1/2 dη dτ Let’s assume that f satisfies ∇f∇ff = 0, with A, B ∈ C (R).

Proposition

If A (and B) is absolutely continuous, then

• 1. f ∈ W 1,1

loc (R2) ∩ C 1 W;

• 2. If = 0;
• 3. if IIf ≥ 0, then A′ = B = 0 and thus Γf is a vertical plane.

A new stable surface

Theorem

Take B = 0 and A(ζ) =      ζ ≤ 0 Cantor function 0 ≤ ζ ≤ 1 1 ζ ≥ 1

A new stable surface

Theorem

Take B = 0 and A(ζ) =      ζ ≤ 0 Cantor function 0 ≤ ζ ≤ 1 1 ζ ≥ 1 Then

• 1. f ∈ W 1,2

loc (R2) ∩ C 1 W;

A new stable surface

Theorem

Take B = 0 and A(ζ) =      ζ ≤ 0 Cantor function 0 ≤ ζ ≤ 1 1 ζ ≥ 1 Then

• 1. f ∈ W 1,2

loc (R2) ∩ C 1 W;

• 2. If = 0;

A new stable surface

Theorem

Take B = 0 and A(ζ) =      ζ ≤ 0 Cantor function 0 ≤ ζ ≤ 1 1 ζ ≥ 1 Then

• 1. f ∈ W 1,2

loc (R2) ∩ C 1 W;

• 2. If = 0;
• 3. IIf ≥ 0.
• Γf is a stable surface

A new stable surface

Theorem

Take B = 0 and A(ζ) =      ζ ≤ 0 Cantor function 0 ≤ ζ ≤ 1 1 ζ ≥ 1 Then

• 1. f ∈ W 1,2

loc (R2) ∩ C 1 W;

• 2. If = 0;
• 3. IIf ≥ 0.
• Γf is a stable surface

However, we still don’t know the answer to the question Is Γf a minimal surface?

A new stable surface

Thank you for your attention!