Introduction
Green Energy
Discrete and Continuous Green Energy on
Compact Manifolds
Juan G. Criado del Rey
Departamento de Matemáticas, Estadística y Computación
Universidad de Cantabria
Joint work with Carlos Beltrán and Nuria Corral
OPCOP 2017 – Castro Urdiales
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Minimal Energy Points
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Minimal Energy Points
Given an infinite compact set X and a lower semicontinuous
symmetric function K : X × X → R, minimize
X
K (xi , xj ).
(x1 , ..., xN ) 7→
i6=j
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Minimal Energy Points
Given an infinite compact set X and a lower semicontinuous
symmetric function K : X × X → R, minimize
X
K (xi , xj ).
(x1 , ..., xN ) 7→
i6=j
Common examples are logarithmic and Riesz s–energy:
X
log kxi − xj k−1
i6=j
and
X
i6=j
1
kxi − xj ks
for subsets of Rn .
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Minimal Energy Points
A particularly interesting example is that of elliptic Fekete
points: points minimizing the discrete logarithmic energy for the
sphere S2
X
log kxi − xj k−1 , X = (x1 , ..., xN ) ∈ (S2 )N .
Elog (X ) =
i6=j
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Minimal Energy Points
A particularly interesting example is that of elliptic Fekete
points: points minimizing the discrete logarithmic energy for the
sphere S2
X
log kxi − xj k−1 , X = (x1 , ..., xN ) ∈ (S2 )N .
Elog (X ) =
i6=j
Problem (Smale’s 7th Problem)
Let us denote Elog (N) = minX Elog (X ). Can one find X s.t.
Elog (X ) − Elog (N) ≤ c log N,
c a universal constant?
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Minimal Energy Points
How can we define minimal logarithmic energy points, or, in
general, minimal energy points in an arbitrary compact
manifold?
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Minimal Energy Points
How can we define minimal logarithmic energy points, or, in
general, minimal energy points in an arbitrary compact
manifold?
If we want to use logarithmic or Riesz energy, we need an
embedding into some Rn .
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Minimal Energy Points
How can we define minimal logarithmic energy points, or, in
general, minimal energy points in an arbitrary compact
manifold?
If we want to use logarithmic or Riesz energy, we need an
embedding into some Rn .
The analog definitions using intrinsic distance
X
X
log d(xi , xj )−1 or
d(xi , xj )−s
i6=j
i6=j
are not smooth functions in general.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
A Facility Location Problem
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
A Facility Location Problem
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
A Facility Location Problem
In which position should we place the sources of heat in order
to maximize the average temperature?
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
A Facility Location Problem
In which position should we place the sources of heat in order
to maximize the average temperature?
Maximise
Z
(x1 , ..., xN ) 7→
N
X
x∈S2 i=1
Juan G. Criado del Rey
log kx − xi k−1 dx.
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
A Facility Location Problem
In which position should we place the sources of heat in order
to maximize the average temperature?
Maximise
Z
(x1 , ..., xN ) 7→
N
X
x∈S2 i=1
log kx − xi k−1 dx.
However, this function is obviously constant by symmetry.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
A Facility Location Problem
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
A Facility Location Problem
Maximize
N
X
Z
(x1 , ..., xN ) 7→
S2 \
S
j
log kx − xi k−1 dx.
B(xj ,ε) i=1
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
A Facility Location Problem
Maximize
N
X
Z
(x1 , ..., xN ) 7→
S2 \
S
j
log kx − xi k−1 dx.
B(xj ,ε) i=1
Theorem (Beltrán, ’14)
The points x1 , ..., xN solve this facility location problem if and
only if they minimize the logarithmic energy.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
A Facility Location Problem
The logarithmic kernel is the Green function for the Laplacian in
the sphere S2 :
1
−1
∆S2 x 7→
log kx − y k
= δy (x) − vol(S2 )−1 ,
2π
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
A Facility Location Problem
The logarithmic kernel is the Green function for the Laplacian in
the sphere S2 :
1
−1
∆S2 x 7→
log kx − y k
= δy (x) − vol(S2 )−1 ,
2π
R
i.e. for every f ∈ C 2 (S) with f = 0,
Z
1
f (y ) =
log kx − y k−1 ∆f (x)dx.
2π x∈S2
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Green Energy
Inspired by this fact, we consider the Green energy.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Green Energy
Inspired by this fact, we consider the Green energy.
Theorem
Let M be a compact Riemannian manifold. There is a smooth
function G(x, y ) defined in M × M minus the diagonal s.t.
∆x G(x, y ) = δy (x) − V −1 ,
where V = vol(M).
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Green Energy
Inspired by this fact, we consider the Green energy.
Theorem
Let M be a compact Riemannian manifold. There is a smooth
function G(x, y ) defined in M × M minus the diagonal s.t.
∆x G(x, y ) = δy (x) − V −1 ,
where V = vol(M).
Define
EG (x1 , ..., xN ) =
X
G(xi , xj ).
i6=j
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Green Energy
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
For M = S2 ,
G(x, y ) =
1
d(x, y )
1
log sin
+C =
log kx − y k−1 + C.
2π
2
2π
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
For M = S2 ,
G(x, y ) =
1
d(x, y )
1
log sin
+C =
log kx − y k−1 + C.
2π
2
2π
Easy expression for the Green function?
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
For M = S2 ,
G(x, y ) =
1
d(x, y )
1
log sin
+C =
log kx − y k−1 + C.
2π
2
2π
Easy expression for the Green function?
Maybe G(x, y ) = φ(d(x, y ))?
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
For M = S2 ,
G(x, y ) =
1
d(x, y )
1
log sin
+C =
log kx − y k−1 + C.
2π
2
2π
Easy expression for the Green function?
Maybe G(x, y ) = φ(d(x, y ))?
Only for locally harmonic Blaschke manifolds.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
A manifold is locally harmonic if the mean curvature of the
geodesic spheres is constant along each geodesic sphere.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
A manifold is Blaschke if it is compact and inj(M) = diam(M).
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
The sphere Sn is an example of locally harmonic Blaschke
manifold. Other examples are the projective spaces RPn , CPn ,
HPn , and the Cayley plane OP2 .
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
The sphere Sn is an example of locally harmonic Blaschke
manifold. Other examples are the projective spaces RPn , CPn ,
HPn , and the Cayley plane OP2 .
Conjecture (Lichnerowicz)
The only locally harmonic Blaschke manifolds are the Compact
Rank One Symmetric Spaces (CROSS) listed above.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
The sphere Sn is an example of locally harmonic Blaschke
manifold. Other examples are the projective spaces RPn , CPn ,
HPn , and the Cayley plane OP2 .
Conjecture (Lichnerowicz)
The only locally harmonic Blaschke manifolds are the Compact
Rank One Symmetric Spaces (CROSS) listed above.
For the CROSS we have that
G(x, y ) = φ(d(x, y ))
with a very simple expression for φ.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
Denote by D = diam(M) and by v (r ) = vol(S(x, r )) the volume
of the geodesic sphere for some point x. Then
0
φ (r ) = −
V −1
RD
v (t)dt
,
v (r )
r
and φ(r ) is some primitive of φ0 (r ).
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
Denote by D = diam(M) and by v (r ) = vol(S(x, r )) the volume
of the geodesic sphere for some point x. Then
0
φ (r ) = −
V −1
RD
v (t)dt
,
v (r )
r
and φ(r ) is some primitive of φ0 (r ). Alternatively,
0
φ (r ) = −
V −1
RD
n−1 Ω(t)dt
r t
,
r n−1 Ω(r )
where Ω(r ) is volume density.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
0
φ (r ) = −
V −1
RD
n−1 Ω(t)dt
r t
r n−1 Ω(r )
M
r n−1 Ω(r )
Sn
sinn−1 r
RPn
2n−1 sinn−1 r
CPn
22n−1 sin2n−1 r cos r
HPn
24n−1 sin4n−1 r cos3 r
OP2
215 sin15 r cos7 r
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
In the case of S2 ,
G(x, y ) =
1
log kx − y k−1 + C
2π
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
In the case of S2 ,
G(x, y ) =
1
log kx − y k−1 + C
2π
and in the case of Sn , G(x, y ) = ...
4 −2
4
t − 2 log t + C
2
3π
3π
for S4
8 −4
12
12
t + 3 t −2 − 3 log t + C
for S6
8
5π
5π
5π
192 −4
48
48
128 −6
t +
t + 4 t −2 − 4 log t + C
for S8
35π 4
35π 4
7π
7π
···
where t = kx − y k.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
For CP3 ,
1
G(x, y ) =
24vol(CP3 )
1
2
+ 2 − 4 log sin r
4
sin r
sin r
+ C,
and for CP4 ,
1
G(x, y ) =
96vol(CP4 )
2
2
6
+ 4 + 2 − 12 log sin r +C,
sin6 r
sin r
sin r
where r = d(x, y ).
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Examples
Theorem (Beltrán, Corral, J.G.C. 16’)
Let x1 , ..., xN ∈ Sn be distinct points. The following are
equivalent:
1
The points x1 , ..., xN minimize the discrete Green energy.
2
For any ε > 0 such that d(xi , xj ) > 2ε, i 6= j, the points
x1 , ..., xN maximize the function
Z
(x1 , ..., xN ) 7→
N
X
S
Sn \ j B(xj ,ε) i=1
Juan G. Criado del Rey
G(x, xi )dvol(x).
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Asymptotically uniform distribution
Minimal logarithmic energy points for S2 are known to be
asymptotically uniformly distributed.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Asymptotically uniform distribution
Minimal logarithmic energy points for S2 are known to be
asymptotically uniformly distributed. Moreover,
Theorem (Brauchart, ’08)
For a sequence {XN }N of minimizers for the discrete logarithmic
energy on S2 , the spherical cap discrepancy satisfies
DC (XN ) = O(N −1/4 )
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Asymptotically uniform distribution
Theorem (Beltrán, Corral, J.G.C., ’17)
Let M be a compact Riemannian manifold and let {XN }N be a
sequence of minimizers for the discrete Green energy. Then
{XN }N is asymptotically uniformly distributed, i.e.
1 X
∗
δx * V −1 dvol
N
x∈XN
as N → ∞.
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds
Introduction
Green Energy
Thank you!
Juan G. Criado del Rey
Discrete and Continuous Green Energy on Compact Manifolds