### The Isoperimetric Problem in the Heisenberg group H

^{n}First Taiwan Geometry Symposium, NCTS South

November 20, 2010

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 1 / 44

Talk outline

The Euclidean Isoperimetric Problem.

Relevant terminology, conepts, definitions in/of the Heisenberg group
H* ^{n}*.

conjecture.

Existence of the Isoperimetric Profile in Carnot-Groups: Leonardi-Rigot.

Smooth Cylindrical Case and some properties of the Isoperimetric profile: Leonardi-Masnou, Ritoré-Rosales.

Partial Symmetry Case: Danielli-Garofalo-Nhieu.

*The C*^{2}solution to the Isoperimetric problem inH^{1}: Ritore-Rosales.
The non-smooth cases: Monti, Monti-Rickly.

An improvement of Danielli-Garofalo-Nhieu due to Ritoré: calibration argument.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 2 / 44

Talk outline

The Euclidean Isoperimetric Problem.

Relevant terminology, conepts, definitions in/of the Heisenberg group
H* ^{n}*.

The Isoperimetric Problem inH* ^{n}*–Pansu’s Isoperimetric inequality and
conjecture.

Existence of the Isoperimetric Profile in Carnot-Groups: Leonardi-Rigot.

Smooth Cylindrical Case and some properties of the Isoperimetric profile: Leonardi-Masnou, Ritoré-Rosales.

Partial Symmetry Case: Danielli-Garofalo-Nhieu.

*The C*^{2}solution to the Isoperimetric problem inH^{1}: Ritore-Rosales.
The non-smooth cases: Monti, Monti-Rickly.

An improvement of Danielli-Garofalo-Nhieu due to Ritoré: calibration argument.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 2 / 44

Talk outline

The Euclidean Isoperimetric Problem.

Relevant terminology, conepts, definitions in/of the Heisenberg group
H* ^{n}*.

The Isoperimetric Problem inH* ^{n}*–Pansu’s Isoperimetric inequality and
conjecture.

Existence of the Isoperimetric Profile in Carnot-Groups:

Leonardi-Rigot.

profile: Leonardi-Masnou, Ritoré-Rosales. Partial Symmetry Case: Danielli-Garofalo-Nhieu.

*The C*^{2}solution to the Isoperimetric problem inH^{1}: Ritore-Rosales.
The non-smooth cases: Monti, Monti-Rickly.

An improvement of Danielli-Garofalo-Nhieu due to Ritoré: calibration argument.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 2 / 44

Talk outline

The Euclidean Isoperimetric Problem.

Relevant terminology, conepts, definitions in/of the Heisenberg group
H* ^{n}*.

The Isoperimetric Problem inH* ^{n}*–Pansu’s Isoperimetric inequality and
conjecture.

Existence of the Isoperimetric Profile in Carnot-Groups:

Leonardi-Rigot.

Smooth Cylindrical Case and some properties of the Isoperimetric profile: Leonardi-Masnou, Ritoré-Rosales.

Partial Symmetry Case: Danielli-Garofalo-Nhieu.

*The C*^{2}solution to the Isoperimetric problem inH^{1}: Ritore-Rosales.
The non-smooth cases: Monti, Monti-Rickly.

An improvement of Danielli-Garofalo-Nhieu due to Ritoré: calibration argument.

*Hn* November 20, 2010 2 / 44

Talk outline

The Euclidean Isoperimetric Problem.

Relevant terminology, conepts, definitions in/of the Heisenberg group
H* ^{n}*.

The Isoperimetric Problem inH* ^{n}*–Pansu’s Isoperimetric inequality and
conjecture.

Existence of the Isoperimetric Profile in Carnot-Groups:

Leonardi-Rigot.

Partial Symmetry Case: Danielli-Garofalo-Nhieu.

The non-smooth cases: Monti, Monti-Rickly.

An improvement of Danielli-Garofalo-Nhieu due to Ritoré: calibration argument.

*Hn* November 20, 2010 2 / 44

Talk outline

The Euclidean Isoperimetric Problem.

Relevant terminology, conepts, definitions in/of the Heisenberg group
H* ^{n}*.

The Isoperimetric Problem inH* ^{n}*–Pansu’s Isoperimetric inequality and
conjecture.

Existence of the Isoperimetric Profile in Carnot-Groups:

Leonardi-Rigot.

Partial Symmetry Case: Danielli-Garofalo-Nhieu.

*The C*^{2}solution to the Isoperimetric problem inH^{1}: Ritore-Rosales.

The non-smooth cases: Monti, Monti-Rickly.

An improvement of Danielli-Garofalo-Nhieu due to Ritoré: calibration argument.

*Hn* November 20, 2010 2 / 44

Talk outline

The Euclidean Isoperimetric Problem.

Relevant terminology, conepts, definitions in/of the Heisenberg group
H* ^{n}*.

The Isoperimetric Problem inH* ^{n}*–Pansu’s Isoperimetric inequality and
conjecture.

Existence of the Isoperimetric Profile in Carnot-Groups:

Leonardi-Rigot.

Partial Symmetry Case: Danielli-Garofalo-Nhieu.

*The C*^{2}solution to the Isoperimetric problem inH^{1}: Ritore-Rosales.

The non-smooth cases: Monti, Monti-Rickly.

argument.

*Hn* November 20, 2010 2 / 44

Talk outline

The Euclidean Isoperimetric Problem.

Relevant terminology, conepts, definitions in/of the Heisenberg group
H* ^{n}*.

The Isoperimetric Problem inH* ^{n}*–Pansu’s Isoperimetric inequality and
conjecture.

Existence of the Isoperimetric Profile in Carnot-Groups:

Leonardi-Rigot.

Partial Symmetry Case: Danielli-Garofalo-Nhieu.

*The C*^{2}solution to the Isoperimetric problem inH^{1}: Ritore-Rosales.

The non-smooth cases: Monti, Monti-Rickly.

An improvement of Danielli-Garofalo-Nhieu due to Ritoré: calibration argument.

*Hn* November 20, 2010 2 / 44

The Euclidean Isoperimetric Problem.

Relevant terminology, conepts, definitions in/of the Heisenberg group
H* ^{n}*.

The Isoperimetric Problem inH* ^{n}*–Pansu’s Isoperimetric inequality and
conjecture.

Existence of the Isoperimetric Profile in Carnot-Groups:

Leonardi-Rigot.

Partial Symmetry Case: Danielli-Garofalo-Nhieu.

*The C*^{2}solution to the Isoperimetric problem inH^{1}: Ritore-Rosales.

The non-smooth cases: Monti, Monti-Rickly.

An improvement of Danielli-Garofalo-Nhieu due to Ritoré: calibration argument.

*Hn* November 20, 2010 2 / 44

The Euclidean Isoperimetric Problem.

We begin with the following folklore which attributed the Isoperimetric Problem to Queen Dido, founder of the city of Carthage in North Africa.

Figure:Dido, Queen of Carthage. Engraving by Mathäus Merian the Elder 1630.

According to Virgil’s saga “Fleeing the vengeance of her brother, Dido (356-260 BC) lands on the coast of North Africa. For the bargain which Dido agrees to with a local potentate is this: she may have that portion of land which she is able to enclose with the hide of a bull. She then cut the hide into a seris of long thin strips and marked out a vast circumference. This area then eventually became the city of Carthage”.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 3 / 44

The Euclidean Isoperimetric Problem.

Queen Dido’s problem/solution is a variant of what is now known as isoperimetric type problems. In more precise term, Dido’s problem is formulated as follows.

Among all bounded, connected open regions in the plane with a fixed perimeter, determine the one(s) that has the maximum volume.

connected open regions in the plane with a fixed volume, determine the one(s) that has the minimum perimeter.

Dido’s solution is correct: (although part of her region is bounded by a sea shore which we assume it to be a straight line when compared to the relative length of the bull’s hide): the extremal regions are precisely one half of the open circular planar discs.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 4 / 44

The Euclidean Isoperimetric Problem.

Queen Dido’s problem/solution is a variant of what is now known as isoperimetric type problems. In more precise term, Dido’s problem is formulated as follows.

Among all bounded, connected open regions in the plane with a fixed perimeter, determine the one(s) that has the maximum volume.

The above problem is also equivalent to: Among all bounded, connected open regions in the plane with a fixed volume, determine the one(s) that has the minimum perimeter.

Dido’s solution is correct: (although part of her region is bounded by a sea shore which we assume it to be a straight line when compared to the relative length of the bull’s hide): the extremal regions are precisely one half of the open circular planar discs.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 4 / 44

Queen Dido’s problem/solution is a variant of what is now known as isoperimetric type problems. In more precise term, Dido’s problem is formulated as follows.

Among all bounded, connected open regions in the plane with a fixed perimeter, determine the one(s) that has the maximum volume.

The above problem is also equivalent to: Among all bounded, connected open regions in the plane with a fixed volume, determine the one(s) that has the minimum perimeter.

Dido’s solution is correct: (although part of her region is bounded by a sea shore which we assume it to be a straight line when compared to the relative length of the bull’s hide): the extremal regions are precisely one half of the open circular planar discs.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 4 / 44

The Euclidean Isoperimetric Problem.

The above problem is also equivalent to: Among all bounded, connected open regions in the plane with a fixed volume, determine the one(s) that has the minimum perimeter.

*Hn* November 20, 2010 4 / 44

The Euclidean Isoperimetric Problem.

Over the centuries, the isoperimetric problem (in various forms) has stimulated substantial mathematical research in numerous areas:

Geometric measure theory: The precise setting for the study of
classical questions in the calculus of variations and the proof of
existence of an isoperimetric profile. The tools are compactness
*theorems for BV functions. Consequently, a priori solutions are only*
guaranteed within the class of sets of finite perimeter.

of constant mean curvature. The classification of such surfaces provides a characterization of isoperimetric profiles.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 5 / 44

The Euclidean Isoperimetric Problem.

Over the centuries, the isoperimetric problem (in various forms) has stimulated substantial mathematical research in numerous areas:

Geometric measure theory: The precise setting for the study of
classical questions in the calculus of variations and the proof of
existence of an isoperimetric profile. The tools are compactness
*theorems for BV functions. Consequently, a priori solutions are only*
guaranteed within the class of sets of finite perimeter.

Differential Geometry: (Smooth) isoperimetric solutions are surfaces of constant mean curvature. The classification of such surfaces provides a characterization of isoperimetric profiles.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 5 / 44

Over the centuries, the isoperimetric problem (in various forms) has stimulated substantial mathematical research in numerous areas:

Geometric measure theory: The precise setting for the study of
classical questions in the calculus of variations and the proof of
existence of an isoperimetric profile. The tools are compactness
*theorems for BV functions. Consequently, a priori solutions are only*
guaranteed within the class of sets of finite perimeter.

Differential Geometry: (Smooth) isoperimetric solutions are surfaces of constant mean curvature. The classification of such surfaces provides a characterization of isoperimetric profiles.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 5 / 44

The Euclidean Isoperimetric Problem.

PDE: The introduction of dynamic altorihms of volume-constrained
curvature flows which provides a way to smoothly deform a given
*region so that the isoperimetric ratio P(E)*^{n−1}^{n}*/|E| cecreases*
monotonically. If the flow exists for all time, the deformed regions
converge, in a suitable sense, to a solution of the isoperimetric
problem.

Functional Analysis: An equivalent way of formulating the isoperimetric problem consists in viewing it as a best constant problem for a Sobolev inequality, relating mean values of a given smooth function with those of its derivatives.

Geometric function theory: Symmetrization procedures that replace a given mathematical object or region with one admitting a larger symmetry group while retaining certain properties.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 6 / 44

PDE: The introduction of dynamic altorihms of volume-constrained
curvature flows which provides a way to smoothly deform a given
*region so that the isoperimetric ratio P(E)*^{n−1}^{n}*/|E| cecreases*
monotonically. If the flow exists for all time, the deformed regions
converge, in a suitable sense, to a solution of the isoperimetric
problem.

Functional Analysis: An equivalent way of formulating the isoperimetric problem consists in viewing it as a best constant problem for a Sobolev inequality, relating mean values of a given smooth function with those of its derivatives.

Geometric function theory: Symmetrization procedures that replace a given mathematical object or region with one admitting a larger symmetry group while retaining certain properties.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 6 / 44

The Euclidean Isoperimetric Problem.

PDE: The introduction of dynamic altorihms of volume-constrained
curvature flows which provides a way to smoothly deform a given
*region so that the isoperimetric ratio P(E)*^{n−1}^{n}*/|E| cecreases*
monotonically. If the flow exists for all time, the deformed regions
converge, in a suitable sense, to a solution of the isoperimetric
problem.

Functional Analysis: An equivalent way of formulating the isoperimetric problem consists in viewing it as a best constant problem for a Sobolev inequality, relating mean values of a given smooth function with those of its derivatives.

Geometric function theory: Symmetrization procedures that replace a given mathematical object or region with one admitting a larger symmetry group while retaining certain properties.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 6 / 44

We recall the classical isoperimetric inequality in the Euclidean space.

Theorem 1

*For every Borel set*Ω ⊂ R^{n}*, n ≥ 2, with finite perimeter P(Ω),*

*min{|Ω|,|R** ^{n}*\

*Ω|} ≤ C*

*iso*(R

^{n}*)P(*Ω)

^{n−1}*, (1)*

^{n}*where*

*C** _{iso}*(R

*) = 1*

^{n}*n*

*ω*

_{n−1}

^{n−1}^{1}

,

Here,*ω**k**is the surface measure of the unit sphere S** ^{k}*inR

*. Equality holds in (1) if and only if almost everywhere*

^{k+1}*Ω = B(x,R) (i.e. a ball) for some x ∈ R*

*and*

^{n}*R > 0. In the case where ∂Ω is smooth say C*

^{1}

*then P(*Ω) coincide with surface measure of

*∂Ω. In the non-smooth case P(Ω) = Var(χ*Ω,R

*) where*

^{n}*χ*Ωis the indicator function ofΩ and

*Var(u) = sup*
(Z

R^{n}*u(x)*

*n*

X

*i=1*

*∂**x**i**G**i**dx*

¯

¯

*¯G**i**∈ C**o*^{∞}(R^{n}*), G*^{2}_{1}*+ · · · + C*^{2}*n*≤ 1
)

*is the variation of u in*R* ^{n}*. Note thatΩ need not to be bounded in a priori.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 7 / 44

The Euclidean Isoperimetric Problem.

We recall the classical isoperimetric inequality in the Euclidean space.

Theorem 1

*For every Borel set*Ω ⊂ R^{n}*, n ≥ 2, with finite perimeter P(Ω),*

*min{|Ω|,|R** ^{n}*\

*Ω|} ≤ C*

*iso*(R

^{n}*)P(*Ω)

^{n−1}*, (1)*

^{n}*where*

*C** _{iso}*(R

*) = 1*

^{n}*n*

*ω*

_{n−1}

^{n−1}^{1}

,

Here,*ω**k**is the surface measure of the unit sphere S** ^{k}*inR

*. Equality holds in (1) if and only if almost everywhere*

^{k+1}*Ω = B(x,R) (i.e. a ball) for some x ∈ R*

*and*

^{n}*R > 0. In the case where ∂Ω is smooth say C*

^{1}

*then P(*Ω) coincide with surface measure of

*∂Ω. In the non-smooth case P(Ω) = Var(χ*Ω,R

*) where*

^{n}*χ*Ωis the indicator function ofΩ and

*Var(u) = sup*
(Z

R^{n}*u(x)*

*n*

X

*i=1*

*∂**x**i**G**i**dx*

¯

¯

*¯G**i**∈ C**o*^{∞}(R^{n}*), G*^{2}_{1}*+ · · · + C*^{2}*n*≤ 1
)

*is the variation of u in*R* ^{n}*. Note thatΩ need not to be bounded in a priori.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 7 / 44

The Euclidean Isoperimetric Problem.

*Roughtly speaking, the isoperimetric problem consists in finding the*
*smallest constant C**iso*(R* ^{n}*) and classifying setsΩ such that inequality (1)
becomes an equality. This problem is equivalent to the two following
formulations:

*Among all bounded, connected open sets of fixed perimeter L, find*
*one with largest volume V .*

*Among all bounded, connected open sets with fixed volume V , find*
*one with smallest perimeter L.*

However, it was not until 1841 that Jacob Steiner gave the first proof (which contain gaps) but later on completed by many

mathematicians. Steiner proved that if such a region exists in the plane, it must be a circle.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 8 / 44

The Euclidean Isoperimetric Problem.

*Roughtly speaking, the isoperimetric problem consists in finding the*
*smallest constant C**iso*(R* ^{n}*) and classifying setsΩ such that inequality (1)
becomes an equality. This problem is equivalent to the two following
formulations:

*Among all bounded, connected open sets of fixed perimeter L, find*
*one with largest volume V .*

*Among all bounded, connected open sets with fixed volume V , find*
*one with smallest perimeter L.*

Of course, the solution (inR^{2}anyway) was known long long time ago.

However, it was not until 1841 that Jacob Steiner gave the first proof (which contain gaps) but later on completed by many

mathematicians. Steiner proved that if such a region exists in the plane, it must be a circle.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 8 / 44

*Roughtly speaking, the isoperimetric problem consists in finding the*
*smallest constant C**iso*(R* ^{n}*) and classifying setsΩ such that inequality (1)
becomes an equality. This problem is equivalent to the two following
formulations:

*Among all bounded, connected open sets of fixed perimeter L, find*
*one with largest volume V .*

*Among all bounded, connected open sets with fixed volume V , find*
*one with smallest perimeter L.*

Of course, the solution (inR^{2}anyway) was known long long time ago.

However, it was not until 1841 that Jacob Steiner gave the first proof (which contain gaps) but later on completed by many

mathematicians. Steiner proved that if such a region exists in the plane, it must be a circle.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 8 / 44

The Euclidean Isoperimetric Problem.

The idea of his proof can be outlined in the following three steps. Assume therefore that there is a regionG in the plane such that among all other regions with the same perimeter ofG , then G must be a disc.

Step I: The regionG must be convex. For if not, using reflection, we can construct another region with the same perimeter but enclose a larger area, this contradict our assumption onG .

Figure:Steiner’s proof, step I.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 9 / 44

The idea of his proof can be outlined in the following three steps. Assume therefore that there is a regionG in the plane such that among all other regions with the same perimeter ofG , then G must be a disc.

Step I: The regionG must be convex. For if not, using reflection, we can construct another region with the same perimeter but enclose a larger area, this contradict our assumption onG .

Figure:Steiner’s proof, step I.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 9 / 44

The Euclidean Isoperimetric Problem.

The idea of his proof can be outlined in the following three steps. Assume therefore that there is a regionG in the plane such that among all other regions with the same perimeter ofG , then G must be a disc.

Step I: The regionG must be convex. For if not, using reflection, we can construct another region with the same perimeter but enclose a larger area, this contradict our assumption onG .

Figure:Steiner’s proof, step I.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 9 / 44

The Euclidean Isoperimetric Problem.

Step II: Any straight line
that divides the perimeter
ofG in half must also divide
the area ofG in half. Since
G is convex, each half of the
bounding curve lies entirely
on one side of the line
*through A and B (see the*
figures). Suppose the line
*AB does not divide the area*
ofG in half, reflect the
*larger area across AB to*
obtain another region
having the same perimeter
ofG but with a larger area.

Again, we obtain a contradiction.

Figure:The argument only works for a convex region.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 10 / 44

The Euclidean Isoperimetric Problem.

Step II: Any straight line
that divides the perimeter
ofG in half must also divide
the area ofG in half. Since
G is convex, each half of the
bounding curve lies entirely
on one side of the line
*through A and B (see the*
figures). Suppose the line
*AB does not divide the area*
ofG in half, reflect the
*larger area across AB to*
obtain another region
having the same perimeter
ofG but with a larger area.

Again, we obtain a contradiction.

Figure:Steiner’s proof, step II.

Figure:The argument only works for a convex region.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 10 / 44

Step II: Any straight line
that divides the perimeter
ofG in half must also divide
the area ofG in half. Since
G is convex, each half of the
bounding curve lies entirely
on one side of the line
*through A and B (see the*
figures). Suppose the line
*AB does not divide the area*
ofG in half, reflect the
*larger area across AB to*
obtain another region
having the same perimeter
ofG but with a larger area.

Again, we obtain a contradiction.

Figure:Steiner’s proof, step II.

Figure:The argument only works for a convex region.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 10 / 44

The Euclidean Isoperimetric Problem.

Step III: Now we concentrate on “half of
*the figure”. Pick any point C on this half*
*curve and join it to A and B to obtain two*

*“lunes” AMC and BNC and a triangle*
*ACB. The angle at C can be either < π/2,*

*= π/2 or > π/2. Imagine that the lunes*
are made of non-deformable material
*and they are hinged at C. Now the area of*
the region is the area of the two lunes
plus the area of the triangle. The area of
*the triangle is computed by the base AC*
and height*BD. By adjusting the angle at*
*C by moving the lunes, we see that the*
*largest height is obtained when B = D,*
*that is the angle at C is**π/2.*

*Hence, from elementary geometry, the angle ACB is a right angle if and only*
*if C lies on a semi-circle with diameter AB.*

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 11 / 44

Step III: Now we concentrate on “half of
*the figure”. Pick any point C on this half*
*curve and join it to A and B to obtain two*

*“lunes” AMC and BNC and a triangle*
*ACB. The angle at C can be either < π/2,*

*= π/2 or > π/2. Imagine that the lunes*
are made of non-deformable material
*and they are hinged at C. Now the area of*
the region is the area of the two lunes
plus the area of the triangle. The area of
*the triangle is computed by the base AC*
*and height BD. By adjusting the angle at*
*C by moving the lunes, we see that the*
*largest height is obtained when B = D,*
*that is the angle at C is**π/2.*

*Hence, from elementary geometry, the angle ACB is a right angle if and only*
*if C lies on a semi-circle with diameter AB.*

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 11 / 44

The Euclidean Isoperimetric Problem.

Step III: Now we concentrate on “half of
*the figure”. Pick any point C on this half*
*curve and join it to A and B to obtain two*

*“lunes” AMC and BNC and a triangle*
*ACB. The angle at C can be either < π/2,*

*= π/2 or > π/2. Imagine that the lunes*
are made of non-deformable material
*and they are hinged at C. Now the area of*
the region is the area of the two lunes
plus the area of the triangle. The area of
*the triangle is computed by the base AC*
*and height BD. By adjusting the angle at*
*C by moving the lunes, we see that the*
*largest height is obtained when B = D,*
*that is the angle at C is**π/2.*

*Hence, from elementary geometry, the angle ACB is a right angle if and only*
*if C lies on a semi-circle with diameter AB.*

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 11 / 44

Since Steiner’s proof there are many many proofs for the isoperimetric
inequality. We present a few for theR^{2}case.

*Proof by complex function theory. Let z = x + iy and*

*dA = dx ∧ dy =*^{1}_{2}*idz ∧ dz. Using the fact that winding number of ∂Ω is*
one, Green and Fubini’s theorem we find

4πA = Z

Ω2πidz ∧ dz = Z

Ω

Z

*∂Ω*

*d**ξ*

*ξ − z**dz ∧ dz =*
Z

*∂Ω*

Z

*∂Ω*

*ξ − z*

*ξ − z**dz d**ξ ≤ L*^{2}.
The case of equality is easy to analyze in the above. The interplay
between geometric extremal problems (e.g. isoperimetric problem)
and sharp analytic inequalities is witnessed in the following analytic
proof of the planar isoperimetric inequality. First, let’s recall Wirtinger’s
*inequality. If f is in the Sobolev space W*^{1,2}([0, 2π]) satisfying

R_{2π}

0 *f (t)dt = 0 then*
Z 2*π*

0 *|f (t)|*^{2}*dt ≤*
Z 2*π*

0 *|f*^{0}*(t)|*^{2}*dt ,* (2)

*with equality holds only when f (t) = Acos(t) + B sin(t). The proof of*
Wirtinger’s inequality is an easy exercise in Fourier series.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 12 / 44

The Euclidean Isoperimetric Problem.

*Let ds denote the element of arc length and assume that∂Ω is a Lipschitz*
curve which is the boundary of a domainΩ ⊂ R^{2}*. Denote by x = (x*1*, x*2) the
position vector. By translating the regionΩ which preserves area and
perimeter, we may assume thatR

*∂Ω**xds = 0. The divergence theorem and*
*Wirtinger’s inequality applied to x*1*, x*2then gives

*2A =*
Z

Ω*div(x) dA =*
Z

*∂Ω**< x,~n > ds ≤*
Z

*∂Ω**|x|ds ≤*p
*L*

µZ

*∂Ω**|x|*^{2}*ds*

¶^{1}_{2}

≤p
*L*

·µ *L*
2π

¶2Z

*∂Ω*

¯

¯

¯

¯
*dx*
*ds*

¯

¯

¯

¯

2

*ds*

¸

1 2

≤*L*^{2}
2π.
Equality holds if and only ifΩ is a disc.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 13 / 44

The Euclidean Isoperimetric Problem.

The proof for higher dimension Euclidean spaces are more technical and we will not recall them here. However a few remarks should be made.

The case ofR^{3}was proved by Schwarz in 1884. His argument can be
described in two steps:

symmetrization) that reduce the problem to finding the solution among a class of rotationally symmetric objects.

A geometric construction to rule out rotationally symmetric candidates different from the sphere.

The full generality of Theorem 1 was established by de Giorgi in 1958. There are many proofs of the Euclidean isoperimetric problem. There are also analogues of the same type of problem an in different settings, i.e., Riemannian manifolds. Our goal is to investigate the

Sub-Riemannian counterpart of this problem and we start with the Heisenberg group.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 14 / 44

The Euclidean Isoperimetric Problem.

The proof for higher dimension Euclidean spaces are more technical and we will not recall them here. However a few remarks should be made.

The case ofR^{3}was proved by Schwarz in 1884. His argument can be
described in two steps:

A symmetrization process (known nowadays as Schwarz

symmetrization) that reduce the problem to finding the solution among a class of rotationally symmetric objects.

A geometric construction to rule out rotationally symmetric candidates different from the sphere.

The full generality of Theorem 1 was established by de Giorgi in 1958. There are many proofs of the Euclidean isoperimetric problem. There are also analogues of the same type of problem an in different settings, i.e., Riemannian manifolds. Our goal is to investigate the

Sub-Riemannian counterpart of this problem and we start with the Heisenberg group.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 14 / 44

The Euclidean Isoperimetric Problem.

The proof for higher dimension Euclidean spaces are more technical and we will not recall them here. However a few remarks should be made.

The case ofR^{3}was proved by Schwarz in 1884. His argument can be
described in two steps:

A symmetrization process (known nowadays as Schwarz

symmetrization) that reduce the problem to finding the solution among a class of rotationally symmetric objects.

A geometric construction to rule out rotationally symmetric candidates different from the sphere.

There are many proofs of the Euclidean isoperimetric problem. There are also analogues of the same type of problem an in different settings, i.e., Riemannian manifolds. Our goal is to investigate the

Sub-Riemannian counterpart of this problem and we start with the Heisenberg group.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 14 / 44

The Euclidean Isoperimetric Problem.

The case ofR^{3}was proved by Schwarz in 1884. His argument can be
described in two steps:

A symmetrization process (known nowadays as Schwarz

A geometric construction to rule out rotationally symmetric candidates different from the sphere.

The full generality of Theorem 1 was established by de Giorgi in 1958.

There are many proofs of the Euclidean isoperimetric problem. There are also analogues of the same type of problem an in different settings, i.e., Riemannian manifolds. Our goal is to investigate the

Sub-Riemannian counterpart of this problem and we start with the Heisenberg group.

*Hn* November 20, 2010 14 / 44

The case ofR^{3}was proved by Schwarz in 1884. His argument can be
described in two steps:

A symmetrization process (known nowadays as Schwarz

A geometric construction to rule out rotationally symmetric candidates different from the sphere.

The full generality of Theorem 1 was established by de Giorgi in 1958.

There are many proofs of the Euclidean isoperimetric problem. There are also analogues of the same type of problem an in different settings, i.e., Riemannian manifolds. Our goal is to investigate the

Sub-Riemannian counterpart of this problem and we start with the Heisenberg group.

*Hn* November 20, 2010 14 / 44

The Euclidean Isoperimetric Problem.

The case ofR^{3}was proved by Schwarz in 1884. His argument can be
described in two steps:

A symmetrization process (known nowadays as Schwarz

A geometric construction to rule out rotationally symmetric candidates different from the sphere.

The full generality of Theorem 1 was established by de Giorgi in 1958.

Sub-Riemannian counterpart of this problem and we start with the Heisenberg group.

*Hn* November 20, 2010 14 / 44

The Heisenberg groupH* ^{n}*and relevant concept/quantities.

The Heisenberg groupH* ^{n}*is a Lie group onR

*with the following group law:*

^{2n+1}*(x, y, t) ◦ (x*^{0}*, y*^{0}*, t*^{0}*) = (x + x*^{0}*, y + y*^{0}*, t + t*^{0}+1

2*(x*^{0}*· y − y*^{0}*· x)) .*
*where x = (x*1*, ..., x**n**), y = (y*1*, ..., y**n**), t ∈ R and the dot product is the*
standard dot product on Euclidean spaces.

Associated to this group law we work with the following standard
*left-invariant vector fields: (i = 1..n)*

*X**i*= *∂*

*∂x**i*−*y** _{i}*
2

*∂*

*∂t*, *Y**i*= *∂*

*∂y**i*+*x** _{i}*
2

*∂*

*∂t*, *T = [X**i**, Y**i*] = *∂*

*∂t*,
*together with an inner product <, > with respect to which these 2n + 1*
vector fields form an orthonormal system.H* ^{n}*equiped with <, > is then
a Riemannian manifold.

measure and therefore a Haar measure onH^{n}*. For sets E ⊂ H** ^{n}*, the

*volume of E is the Lebesgue measure of E and will be denoted by |E|.*

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 15 / 44

The Heisenberg groupH* ^{n}*and relevant concept/quantities.

The Heisenberg groupH* ^{n}*is a Lie group onR

*with the following group law:*

^{2n+1}*(x, y, t) ◦ (x*^{0}*, y*^{0}*, t*^{0}*) = (x + x*^{0}*, y + y*^{0}*, t + t*^{0}+1

2*(x*^{0}*· y − y*^{0}*· x)) .*
*where x = (x*1*, ..., x**n**), y = (y*1*, ..., y**n**), t ∈ R and the dot product is the*
standard dot product on Euclidean spaces.

Associated to this group law we work with the following standard
*left-invariant vector fields: (i = 1..n)*

*X**i*= *∂*

*∂x**i*−*y** _{i}*
2

*∂*

*∂t*, *Y**i*= *∂*

*∂y**i*+*x** _{i}*
2

*∂*

*∂t*, *T = [X**i**, Y**i*] = *∂*

*∂t*,
*together with an inner product <, > with respect to which these 2n + 1*
vector fields form an orthonormal system.H* ^{n}*equiped with <, > is then
a Riemannian manifold.

The Lebesgue measure onR* ^{2n+1}*is both left and right translation
measure and therefore a Haar measure onH

^{n}*. For sets E ⊂ H*

*, the*

^{n}*volume of E is the Lebesgue measure of E and will be denoted by |E|.*

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 15 / 44

The Heisenberg groupH* ^{n}*is a Lie group onR

*with the following group law:*

^{2n+1}*(x, y, t) ◦ (x*^{0}*, y*^{0}*, t*^{0}*) = (x + x*^{0}*, y + y*^{0}*, t + t*^{0}+1

2*(x*^{0}*· y − y*^{0}*· x)) .*
*where x = (x*1*, ..., x**n**), y = (y*1*, ..., y**n**), t ∈ R and the dot product is the*
standard dot product on Euclidean spaces.

Associated to this group law we work with the following standard
*left-invariant vector fields: (i = 1..n)*

*X**i*= *∂*

*∂x**i*−*y** _{i}*
2

*∂*

*∂t*, *Y**i*= *∂*

*∂y**i*+*x** _{i}*
2

*∂*

*∂t*, *T = [X**i**, Y**i*] = *∂*

*∂t*,
*together with an inner product <, > with respect to which these 2n + 1*
vector fields form an orthonormal system.H* ^{n}*equiped with <, > is then
a Riemannian manifold.

The Lebesgue measure onR* ^{2n+1}*is both left and right translation
measure and therefore a Haar measure onH

^{n}*. For sets E ⊂ H*

*, the*

^{n}*volume of E is the Lebesgue measure of E and will be denoted by |E|.*

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 15 / 44

The Heisenberg groupH* ^{n}*and relevant concept/quantities.

*Given an oriented C*^{2}embedded hypersurfaceS ⊂ H* ^{n}*(and after an

*orientation is chosen) we let N be the Riemannian unit normal to*S and

*we write N =*P

*n*

*i=1**(p*_{i}*X*_{i}*+ q**i**Y*_{i}*) + ωT*

*The projection of N onto the horizontal plane span{X*_{i}*, Y*_{i}*| i = 1, .., n} at each*
*point g ∈ S is called the horizontal normal and is denoted by*

*N** _{H}*=P

*n*

*i=1**p*_{i}*X*_{i}*+ q**i**Y** _{i}*.

The set

ΣS*def*

*= {g ∈ S | N**H**(g) = 0}*

*is called the characteristic set (singular set by some authors) of the*
hypersurface*S . For our purpose, it suffices to know that for any C*^{2}
hypersurface*S we have σ(Σ*S*) = 0 where dσ is the Riemannian volume*
onS .

*For points g ∉ Σ*S*we define the horizontal Gauss map (the horizontal unit*
*normal) by setting*

*ν**H**(g) =* *N*_{H}*(g)*

*|N**H**(g)|*=
X*n*
*i=1*

*p*_{i}*X**i**+ q*_{i}*Y**i*.

The vector field*ν**H*plays an important role in the analysis of hypersurfaces
in this setting.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 16 / 44

The Heisenberg groupH* ^{n}*and relevant concept/quantities.

*Given an oriented C*^{2}embedded hypersurfaceS ⊂ H* ^{n}*(and after an

*orientation is chosen) we let N be the Riemannian unit normal to*S and

*we write N =*P

*n*

*i=1**(p*_{i}*X*_{i}*+ q**i**Y*_{i}*) + ωT*

*The projection of N onto the horizontal plane span{X*_{i}*, Y*_{i}*| i = 1, .., n} at each*
*point g ∈ S is called the horizontal normal and is denoted by*

*N** _{H}*=P

*n*

*i=1**p*_{i}*X*_{i}*+ q**i**Y** _{i}*.
The set

ΣS*def*

*= {g ∈ S | N**H**(g) = 0}*

*is called the characteristic set (singular set by some authors) of the*
hypersurface*S . For our purpose, it suffices to know that for any C*^{2}
hypersurface*S we have σ(Σ*S*) = 0 where dσ is the Riemannian volume*
onS .

*normal) by setting*

*ν**H**(g) =* *N*_{H}*(g)*

*|N**H**(g)|*=
X*n*
*i=1*

*p*_{i}*X**i**+ q*_{i}*Y**i*.

The vector field*ν**H*plays an important role in the analysis of hypersurfaces
in this setting.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 16 / 44

The Heisenberg groupH* ^{n}*and relevant concept/quantities.

*Given an oriented C*^{2}embedded hypersurfaceS ⊂ H* ^{n}*(and after an

*orientation is chosen) we let N be the Riemannian unit normal to*S and

*we write N =*P

*n*

*i=1**(p*_{i}*X*_{i}*+ q**i**Y*_{i}*) + ωT*

*The projection of N onto the horizontal plane span{X*_{i}*, Y*_{i}*| i = 1, .., n} at each*
*point g ∈ S is called the horizontal normal and is denoted by*

*N** _{H}*=P

*n*

*i=1**p*_{i}*X*_{i}*+ q**i**Y** _{i}*.
The set

ΣS*def*

*= {g ∈ S | N**H**(g) = 0}*

*is called the characteristic set (singular set by some authors) of the*
hypersurface*S . For our purpose, it suffices to know that for any C*^{2}
hypersurface*S we have σ(Σ*S*) = 0 where dσ is the Riemannian volume*
onS .

*For points g ∉ Σ*S*we define the horizontal Gauss map (the horizontal unit*
*normal) by setting*

*ν**H**(g) =* *N*_{H}*(g)*

*|N**H**(g)|*=
X*n*
*i=1*

*p*_{i}*X*_{i}*+ q*_{i}*Y** _{i}*.

The vector field*ν**H*plays an important role in the analysis of hypersurfaces
in this setting.

First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg group*Hn* November 20, 2010 16 / 44

*Given an oriented C*^{2}embedded hypersurfaceS ⊂ H* ^{n}*(and after an

*orientation is chosen) we let N be the Riemannian unit normal to*S and

*we write N =*P

*n*

*i=1**(p*_{i}*X*_{i}*+ q**i**Y*_{i}*) + ωT*

*The projection of N onto the horizontal plane span{X*_{i}*, Y*_{i}*| i = 1, .., n} at each*
*point g ∈ S is called the horizontal normal and is denoted by*

*N** _{H}*=P

*n*

*i=1**p*_{i}*X*_{i}*+ q**i**Y** _{i}*.
The set

ΣS*def*

*= {g ∈ S | N**H**(g) = 0}*

*is called the characteristic set (singular set by some authors) of the*
hypersurface*S . For our purpose, it suffices to know that for any C*^{2}
hypersurface*S we have σ(Σ*S*) = 0 where dσ is the Riemannian volume*
onS .

*For points g ∉ Σ*S*we define the horizontal Gauss map (the horizontal unit*
*normal) by setting*

*ν**H**(g) =* *N*_{H}*(g)*

*|N**H**(g)|*=
X*n*
*i=1*

*p*_{i}*X*_{i}*+ q*_{i}*Y** _{i}*.

The vector field*ν**H*plays an important role in the analysis of hypersurfaces
in this setting.

*Hn* November 20, 2010 16 / 44