Transcript Gary Lawlor

I wouldn’t give a fig for the simplicity
on this side of complexity. But I would
give my right arm for the simplicity
on the far side of complexity.
-- Oliver Wendell Holmes
The double bubble theorem in 3-space
Every standard double bubble in R3
has the least surface area required
to separately enclose two volumes.
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Morgan
Foisy
Hass, Schlafly
Hutchings
Wichiramala
Ritore, Ros
Reichardt
A unified isoperimetric inequality
Objects in Rn having volume (or area) V and
surface area (or perimeter) S satisfy the
sharp inequality
where r is the inradius of the minimizer
in any of the following classes of objects:
Classes for which this inequality holds include:
• All bodies (with minimizer the round ball)
• All rectangular boxes (with minimizer the cube)
• All triangles (minimizer is equilateral)
• Cylindrical cans (popular calculus problem)
Classes for which this inequality holds include:
• All bodies (with minimizer the round ball)
• All rectangular boxes (with minimizer the cube)
• All triangles (minimizer is equilateral)
• Cylindrical cans (popular calculus problem)
• Double and triple bubbles in the plane
• Double bubbles in 3-space
• Conjecturally:
• All multiple bubbles in the plane
• Triple to quintuple bubbles in 3-space
• (n+1)-fold bubbles in Rn
Calibration is a great way to prove
minimization.
• Find a progress monitor, in the form of a
differential form or vector field
• Using a form of Stokes’ theorem to
orchestrate the process,
• Make a (fully) local comparison
between area and the integral of the
monitor.
• The total monitor integral is the
same for all competitors.
• Conclude the global comparison
between competitors.
Metacalibration can be described as
calibration combined with slicing, and
enhanced by emulation.
Slicing makes possible new variable types,
and can average out a pointwise inequality
requirement over a curve or sub-surface.
Emulation guides and simplifies the
statement and calculations.
Benefits of metacalibration are centered
around the concepts of:
• Partial reduction
• Reallocation
• Emulation
• Differentiation of a measuring stick
Emulation
1. Start with two objects to compare:
• An ideal object I
• A competing object C
2. Match some aspect of C and I
3. Measure some aspect of I, based on step 2
4. Use that quantity to help measure C
To illustrate emulation, we offer the following
isoperimetric proof, a-la-Schmidt:
Theorem: For any body of volume V in Rn,
the surface area S satisfies
where r is the radius of the round ball
of volume V.
Theorem: For any body C of volume V in Rn,
the surface area S satisfies
Proof: The theorem is true if n=1, in which
case V = L = 2r and S = 2.
Now take any n>1 and assume
the theorem true for n-1.
Let C be a body of volume V in Rn.
Slice C with horizontal planes
Pt: {xn=t}.
Let Amax be the largest cross-sectional area.
Let B be the round ball whose largest
horizontal slice has area Amax as well.
Let V be the volume of B.
B
C
Now as the slicing plane Pt passes
upward through C, for every t find
a plane Qt slicing through B so as
to match the cross-sectional area A(t).
Let z(t) be the z-coordinate of the plane
Qt, with z=0 at the center of B.
B
C
Define
G(t) =
Then
G’ =
2V(t) + V(t) - z(t) A(t)
r
2A + Az’ - z’A - z A’
r
B
C
Define
G(t) =
Then
(n-1)V(t) + V(t) - z(t) A(t)
r
G’ = (n-1)A + Az’ - z’A - z A’
r
=
(n-1)A - z A’
r
Define
G(t) =
Then
(n-1)V(t) + V(t) - z(t) A(t)
r
G’ = (n-1)A + Az’ - z’A - z A’
r
(n-1)A - z A’
r
 P - z A’

r
by the induction hypothesis, where  is the
radius of the current slice of B.
=
Define
G(t) =
Then
(n-1)V(t) + V(t) - z(t) A(t)
r
G’ = (n-1)A + Az’ - z’A - z A’
r
(n-1)A - z A’
r
 P - z A’
G’ 
r
=
by induction, where  is the radius of the current slice of B.
G’ 
1
r
( - z) (P, A’)
Define
G(t) =
Then
(n-1)V(t) + V(t) - z(t) A(t)
r
G’ = (n-1)A + Az’ - z’A - z A’
r
(n-1)A - z A’
r
 P - z A’
G’ 
r
=
by induction, where  is the radius of the current slice of B.
G’ 
1
r
( - z) (P, A’)  S’
Define
G(t) =
(n-1)V(t) + V(t) - z(t) A(t)
r
G’  S’
G  S
In the end, G = [(n-1)V + V ] / r
= V+ V + … + V + V
r r
r r
which, by the AM-GM inequality, is minimized when V = V
and thus r = r. So
G  S
In the end, G = [(n-1)V + V ] / r
= V+ V + … + V + V
r r
r r
which, by the AM-GM inequality, is minimized when V = V
and thus r = r. So
completing the proof by induction.
Comparison of methods for
proving geometric minimization
Deformation
• Variational methods
• Symmetrization
Reduction
• Symmetrization
• Mod out by symmetry
• Directed slicing
• Calibration
• Equivalent problems
• Paired calibration
Slicing
Deformation
• Variational methods
• Symmetrization
Reduction
• Symmetrization
• Mod out by symmetry
• Directed slicing
• Calibration
• Equivalent problems
• Paired calibration
Paired vector fields
Deformation
• Variational methods
• Symmetrization
Reduction
• Symmetrization
• Mod out by symmetry
• Directed slicing
• Calibration
• Equivalent problems
• Paired calibration
Metacalibration brings all these
methods into one framework.
Deformation
• Variational methods more localized
• Symmetrization more versatile
Reduction
• Symmetrization
• Mod out by symmetry
• Directed slicing more flexible
• Calibration applicable to more types
• Equivalent problems a central feature
• Paired calibration less rigid
Metacalibration brings all these
methods into one framework.
The double bubble theorem in 3-space
2
3
Metacalibration
What is the question to which
this piece is the answer?
What is the question to which
this piece is the answer?
Answer (i.e., question):
Least “capillary surface area” for
the given, fixed volumes
Divide and conquer
 Partition
into pieces…
 Solve planar problems via
Hutchings
 Coordinate
these results
over all slices
• Slice competitor with horizontal planes
• Slice standard model with slanted planes,
matching both volumes:
h1
h2
Proof.
• Slice competitor with horizontal planes
• Slice standard model, matching both volumes:
h1
h2
•Prove that such slicing planes exist and are unique
• Prove that S’ ≥ G’, where G is the calibration
h1
• Proof that S’ ≥ G’
uses
• variations
• equivalent problems
• calibration
• spherical inversion
• escorting
• Michael Hutchings’ planar method
h2
h1
h2