The Isoperimetric Problem in the Heisenberg group H
nFirst Taiwan Geometry Symposium, NCTS South
November 20, 2010
First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg groupHn November 20, 2010 1 / 44
Talk outline
The Euclidean Isoperimetric Problem.
Relevant terminology, conepts, definitions in/of the Heisenberg group Hn.
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 C2solution to the Isoperimetric problem inH1: 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 groupHn November 20, 2010 2 / 44
Talk outline
The Euclidean Isoperimetric Problem.
Relevant terminology, conepts, definitions in/of the Heisenberg group Hn.
The Isoperimetric Problem inHn–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 C2solution to the Isoperimetric problem inH1: 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 groupHn November 20, 2010 2 / 44
Talk outline
The Euclidean Isoperimetric Problem.
Relevant terminology, conepts, definitions in/of the Heisenberg group Hn.
The Isoperimetric Problem inHn–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 C2solution to the Isoperimetric problem inH1: 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 groupHn November 20, 2010 2 / 44
Talk outline
The Euclidean Isoperimetric Problem.
Relevant terminology, conepts, definitions in/of the Heisenberg group Hn.
The Isoperimetric Problem inHn–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 C2solution to the Isoperimetric problem inH1: 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 groupHn November 20, 2010 2 / 44
Talk outline
The Euclidean Isoperimetric Problem.
Relevant terminology, conepts, definitions in/of the Heisenberg group Hn.
The Isoperimetric Problem inHn–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 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 groupHn November 20, 2010 2 / 44
Talk outline
The Euclidean Isoperimetric Problem.
Relevant terminology, conepts, definitions in/of the Heisenberg group Hn.
The Isoperimetric Problem inHn–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 C2solution to the Isoperimetric problem inH1: 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 groupHn November 20, 2010 2 / 44
Talk outline
The Euclidean Isoperimetric Problem.
Relevant terminology, conepts, definitions in/of the Heisenberg group Hn.
The Isoperimetric Problem inHn–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 C2solution to the Isoperimetric problem inH1: Ritore-Rosales.
The non-smooth cases: Monti, Monti-Rickly.
argument.
First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg groupHn November 20, 2010 2 / 44
Talk outline
The Euclidean Isoperimetric Problem.
Relevant terminology, conepts, definitions in/of the Heisenberg group Hn.
The Isoperimetric Problem inHn–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 C2solution to the Isoperimetric problem inH1: 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 groupHn November 20, 2010 2 / 44
The Euclidean Isoperimetric Problem.
Relevant terminology, conepts, definitions in/of the Heisenberg group Hn.
The Isoperimetric Problem inHn–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 C2solution to the Isoperimetric problem inH1: 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 groupHn 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 groupHn 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 groupHn 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 groupHn 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 groupHn 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 groupHn 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 groupHn 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 groupHn 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 groupHn 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−1n /|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 groupHn 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−1n /|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 groupHn 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−1n /|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 groupHn November 20, 2010 6 / 44
We recall the classical isoperimetric inequality in the Euclidean space.
Theorem 1
For every Borel setΩ ⊂ Rn, n ≥ 2, with finite perimeter P(Ω),
min{|Ω|,|Rn\Ω|} ≤ Ciso(Rn)P(Ω)n−1n , (1)
where
Ciso(Rn) = 1 nωn−1n−11
,
Here,ωkis the surface measure of the unit sphere SkinRk+1. Equality holds in (1) if and only if almost everywhereΩ = B(x,R) (i.e. a ball) for some x ∈ Rnand R > 0. In the case where ∂Ω is smooth say C1then P(Ω) coincide with surface measure of∂Ω. In the non-smooth case P(Ω) = Var(χΩ,Rn) whereχΩis the indicator function ofΩ and
Var(u) = sup (Z
Rnu(x)
n
X
i=1
∂xiGidx
¯
¯
¯Gi∈ Co∞(Rn), G21+ · · · + C2n≤ 1 )
is the variation of u inRn. Note thatΩ need not to be bounded in a priori.
First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg groupHn November 20, 2010 7 / 44
The Euclidean Isoperimetric Problem.
We recall the classical isoperimetric inequality in the Euclidean space.
Theorem 1
For every Borel setΩ ⊂ Rn, n ≥ 2, with finite perimeter P(Ω),
min{|Ω|,|Rn\Ω|} ≤ Ciso(Rn)P(Ω)n−1n , (1)
where
Ciso(Rn) = 1 nωn−1n−11
,
Here,ωkis the surface measure of the unit sphere SkinRk+1. Equality holds in (1) if and only if almost everywhereΩ = B(x,R) (i.e. a ball) for some x ∈ Rnand R > 0. In the case where ∂Ω is smooth say C1then P(Ω) coincide with surface measure of∂Ω. In the non-smooth case P(Ω) = Var(χΩ,Rn) whereχΩis the indicator function ofΩ and
Var(u) = sup (Z
Rnu(x)
n
X
i=1
∂xiGidx
¯
¯
¯Gi∈ Co∞(Rn), G21+ · · · + C2n≤ 1 )
is the variation of u inRn. Note thatΩ need not to be bounded in a priori.
First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg groupHn November 20, 2010 7 / 44
The Euclidean Isoperimetric Problem.
Roughtly speaking, the isoperimetric problem consists in finding the smallest constant Ciso(Rn) 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 groupHn November 20, 2010 8 / 44
The Euclidean Isoperimetric Problem.
Roughtly speaking, the isoperimetric problem consists in finding the smallest constant Ciso(Rn) 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 (inR2anyway) 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 groupHn November 20, 2010 8 / 44
Roughtly speaking, the isoperimetric problem consists in finding the smallest constant Ciso(Rn) 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 (inR2anyway) 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 groupHn 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 groupHn 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 groupHn 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 groupHn 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 groupHn 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 groupHn 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 groupHn 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 heightBD. 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 isπ/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 groupHn 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 isπ/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 groupHn 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 isπ/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 groupHn November 20, 2010 11 / 44
Since Steiner’s proof there are many many proofs for the isoperimetric inequality. We present a few for theR2case.
Proof by complex function theory. Let z = x + iy and
dA = dx ∧ dy =12idz ∧ 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
dξ
ξ − zdz ∧ dz = Z
∂Ω
Z
∂Ω
ξ − z
ξ − zdz dξ ≤ L2. 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 W1,2([0, 2π]) satisfying
R2π
0 f (t)dt = 0 then Z 2π
0 |f (t)|2dt ≤ Z 2π
0 |f0(t)|2dt , (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 groupHn 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 ⊂ R2. Denote by x = (x1, x2) the position vector. By translating the regionΩ which preserves area and perimeter, we may assume thatR
∂Ωxds = 0. The divergence theorem and Wirtinger’s inequality applied to x1, x2then gives
2A = Z
Ωdiv(x) dA = Z
∂Ω< x,~n > ds ≤ Z
∂Ω|x|ds ≤p L
µZ
∂Ω|x|2ds
¶12
≤p L
·µ L 2π
¶2Z
∂Ω
¯
¯
¯
¯ dx ds
¯
¯
¯
¯
2
ds
¸
1 2
≤L2 2π. Equality holds if and only ifΩ is a disc.
First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg groupHn 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 ofR3was 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 groupHn 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 ofR3was 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 groupHn 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 ofR3was 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 groupHn 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 ofR3was 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 groupHn November 20, 2010 14 / 44
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 ofR3was 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 groupHn 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 ofR3was 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 groupHn November 20, 2010 14 / 44
The Heisenberg groupHnand relevant concept/quantities.
The Heisenberg groupHnis a Lie group onR2n+1with the following group law:
(x, y, t) ◦ (x0, y0, t0) = (x + x0, y + y0, t + t0+1
2(x0· y − y0· x)) . where x = (x1, ..., xn), y = (y1, ..., yn), 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)
Xi= ∂
∂xi−yi 2
∂
∂t, Yi= ∂
∂yi+xi 2
∂
∂t, T = [Xi, Yi] = ∂
∂t, together with an inner product <, > with respect to which these 2n + 1 vector fields form an orthonormal system.Hnequiped with <, > is then a Riemannian manifold.
measure and therefore a Haar measure onHn. For sets E ⊂ Hn, 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 groupHn November 20, 2010 15 / 44
The Heisenberg groupHnand relevant concept/quantities.
The Heisenberg groupHnis a Lie group onR2n+1with the following group law:
(x, y, t) ◦ (x0, y0, t0) = (x + x0, y + y0, t + t0+1
2(x0· y − y0· x)) . where x = (x1, ..., xn), y = (y1, ..., yn), 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)
Xi= ∂
∂xi−yi 2
∂
∂t, Yi= ∂
∂yi+xi 2
∂
∂t, T = [Xi, Yi] = ∂
∂t, together with an inner product <, > with respect to which these 2n + 1 vector fields form an orthonormal system.Hnequiped with <, > is then a Riemannian manifold.
The Lebesgue measure onR2n+1is both left and right translation measure and therefore a Haar measure onHn. For sets E ⊂ Hn, 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 groupHn November 20, 2010 15 / 44
The Heisenberg groupHnis a Lie group onR2n+1with the following group law:
(x, y, t) ◦ (x0, y0, t0) = (x + x0, y + y0, t + t0+1
2(x0· y − y0· x)) . where x = (x1, ..., xn), y = (y1, ..., yn), 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)
Xi= ∂
∂xi−yi 2
∂
∂t, Yi= ∂
∂yi+xi 2
∂
∂t, T = [Xi, Yi] = ∂
∂t, together with an inner product <, > with respect to which these 2n + 1 vector fields form an orthonormal system.Hnequiped with <, > is then a Riemannian manifold.
The Lebesgue measure onR2n+1is both left and right translation measure and therefore a Haar measure onHn. For sets E ⊂ Hn, 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 groupHn November 20, 2010 15 / 44
The Heisenberg groupHnand relevant concept/quantities.
Given an oriented C2embedded hypersurfaceS ⊂ Hn(and after an orientation is chosen) we let N be the Riemannian unit normal toS and we write N =Pn
i=1(piXi+ qiYi) + ωT
The projection of N onto the horizontal plane span{Xi, Yi| i = 1, .., n} at each point g ∈ S is called the horizontal normal and is denoted by
NH=Pn
i=1piXi+ qiYi.
The set
ΣSdef
= {g ∈ S | NH(g) = 0}
is called the characteristic set (singular set by some authors) of the hypersurfaceS . For our purpose, it suffices to know that for any C2 hypersurfaceS we have σ(ΣS) = 0 where dσ is the Riemannian volume onS .
For points g ∉ ΣSwe define the horizontal Gauss map (the horizontal unit normal) by setting
νH(g) = NH(g)
|NH(g)|= Xn i=1
piXi+ qiYi.
The vector fieldνHplays an important role in the analysis of hypersurfaces in this setting.
First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg groupHn November 20, 2010 16 / 44
The Heisenberg groupHnand relevant concept/quantities.
Given an oriented C2embedded hypersurfaceS ⊂ Hn(and after an orientation is chosen) we let N be the Riemannian unit normal toS and we write N =Pn
i=1(piXi+ qiYi) + ωT
The projection of N onto the horizontal plane span{Xi, Yi| i = 1, .., n} at each point g ∈ S is called the horizontal normal and is denoted by
NH=Pn
i=1piXi+ qiYi. The set
ΣSdef
= {g ∈ S | NH(g) = 0}
is called the characteristic set (singular set by some authors) of the hypersurfaceS . For our purpose, it suffices to know that for any C2 hypersurfaceS we have σ(ΣS) = 0 where dσ is the Riemannian volume onS .
normal) by setting
νH(g) = NH(g)
|NH(g)|= Xn i=1
piXi+ qiYi.
The vector fieldνHplays an important role in the analysis of hypersurfaces in this setting.
First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg groupHn November 20, 2010 16 / 44
The Heisenberg groupHnand relevant concept/quantities.
Given an oriented C2embedded hypersurfaceS ⊂ Hn(and after an orientation is chosen) we let N be the Riemannian unit normal toS and we write N =Pn
i=1(piXi+ qiYi) + ωT
The projection of N onto the horizontal plane span{Xi, Yi| i = 1, .., n} at each point g ∈ S is called the horizontal normal and is denoted by
NH=Pn
i=1piXi+ qiYi. The set
ΣSdef
= {g ∈ S | NH(g) = 0}
is called the characteristic set (singular set by some authors) of the hypersurfaceS . For our purpose, it suffices to know that for any C2 hypersurfaceS we have σ(ΣS) = 0 where dσ is the Riemannian volume onS .
For points g ∉ ΣSwe define the horizontal Gauss map (the horizontal unit normal) by setting
νH(g) = NH(g)
|NH(g)|= Xn i=1
piXi+ qiYi.
The vector fieldνHplays an important role in the analysis of hypersurfaces in this setting.
First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg groupHn November 20, 2010 16 / 44
Given an oriented C2embedded hypersurfaceS ⊂ Hn(and after an orientation is chosen) we let N be the Riemannian unit normal toS and we write N =Pn
i=1(piXi+ qiYi) + ωT
The projection of N onto the horizontal plane span{Xi, Yi| i = 1, .., n} at each point g ∈ S is called the horizontal normal and is denoted by
NH=Pn
i=1piXi+ qiYi. The set
ΣSdef
= {g ∈ S | NH(g) = 0}
is called the characteristic set (singular set by some authors) of the hypersurfaceS . For our purpose, it suffices to know that for any C2 hypersurfaceS we have σ(ΣS) = 0 where dσ is the Riemannian volume onS .
For points g ∉ ΣSwe define the horizontal Gauss map (the horizontal unit normal) by setting
νH(g) = NH(g)
|NH(g)|= Xn i=1
piXi+ qiYi.
The vector fieldνHplays an important role in the analysis of hypersurfaces in this setting.
First Taiwan Geometry Symposium, NCTS South ()The Isoperimetric Problem in the Heisenberg groupHn November 20, 2010 16 / 44