Dumitru Vulcanov - Numerical simulations with Ricci flow, an overview and cosmological applications

Numerical simulations withNumerical simulations with
Ricci flow,Ricci flow,
an overview and cosmologicalan overview and cosmological
applicationsapplications..
DumitruDumitru VulcanovVulcanov
The West University of TimisoaraThe West University of Timisoara
20102010

In differential geometry, the Ricci
flow is a process which deforms
the metric of a Riemannian
manifold in a manner formally
analogous to the diffusion of heat,
smoothing out irregularities in the
metric.
It plays in important role in the
apparent proof of the Poincare
conjecture, one of the seven
Millennium Prize Problems for
which the Clay Mathematics
Institute offers a 1,000,000 USD
prize for a correct solution.
R. S. Hamilton
G. Perelman
Introduction

Poincare conjecture says that :
Every simply connected, closed 3-manifold is
homeomorphic to the 3-sphere.
G. Perelman
Introduction
In particular, Perelman proved
Thurston's geometrization
conjecture. This solved in the
affirmative the Poincaré conjecture,
posed in 1904, which before its
solution was viewed as one of
the most important and difficult
open problems in topology.

)1(2 ij
ij
R
t
g
−=
∂
∂
Given a Riemannian manifold with metric
tensor gij the Ricci flow may be defined
by the geometric evolution equation :
Where Rij is the Ricci tensor and t the “time”
The normalised version of (1) is
)2(
2
2 ijij
ij
rg
n
R
t
g
+−=
∂
∂
Introduction

- Euclidean space, or more generally (Ricci) flat; Ricci
flow has no effect.
- Sphere ; Ricci flow collapses the manifold to a point
in finite time.
- Einstein manifolds (Ricci = constant×metric) ; Ricci
flow will collapse it to a point if it has positive
curvature, leave it invariant if it has zero curvature,
and expand it if it has negative curvature.
In particular, this shows that in general the Ricci
flow cannot be continued for all time, but will
produce singularities. For a 3 dimensional manifold,
Perelman showed how to continue past the singu-
larities using surgery on the manifold.
Examples (trivial ones)
Introduction

Introduction
Examples (non-trivial ones)
- Cigar soliton or Witten Black-Hole having a
metric
222
2
222
22
22
tanh
11
θ
θ
dsds
r
drdr
yx
dydx
g +=
+
+
=
++
+
=
Where : )11log(arcsinh 2
rrs ++==
-This is a stationary solution to the Ricci flow
(steady soliton)
- Has positive curvature
- Asymptotic to a cylinder
- Curvature decays exponentially
- Is rotationally symmestric

Many versions of the Ricci flow have also
been studied:
• Various curvature flows defined using either an
extrinsic curvature, which describes how a curve or
surface is embedded in a higher dimensional flat
space, or an intrinsic curvature, which describes
the internal geometry of some Riemannian
manifold,
• Various flows which extremalize some quantity
mathematically analogous to an energy or entropy,
• Various flows controlled by a p.d.e. which is a
higher order analog of a nonlinear diffusion
equation….
Introduction

The problem of numerical simulations and
visualizations with Ricci flow of 2 or 3 dimensional
manifolds comes naturally as Ricci flow is
geometric by nature.
Ricci flow acts directly on the metric of the surface,
tending not to preserve the embeddedness. A
number of interesting results have been obtained
by restricting to classes of metrics of revolution,
since such symmetries are preserved under Ricci
flow and the metric depends on considerably
fewer parameters in such cases.
These surfaces tend to remain embedded in RRRR3
making direct visualization possible.
Numerical simulations

Since the Ricci flow equations (1) or (2) are only
weekly parabolic and since numerical evolutions
appear to be very unstable, we need some solutions
to avoid this. Mainly there are in the literature two
solutions (see bellow), namely :
- filtering and reparamatrization method (inspired
by spectral methods)
- using of other flow equations derived from (1), the
deTurck flow is such an example.
However an explicit finite-difference scheme for (1)
is still usable with very small time-steps and high
number digits precision (several hundred) -
removing short wavelength instabilities.

The main goal of numerical simulations with Ricci
flow is to exhibit the formation of singularities
and/or neck pinching phenomenon, which occurs
naturally for metrics of revolution.
Numerical simulations in Ricci flow are very similar
to those in Numerical relativity, thus some methods
and experience can be… imported here – even
codes.
There are not very manny results in this direction
reported till now. We can mention only two articles,
namely :
1) Garfinkle & Isenberg - math.DG/0306129 (GI)
2) Rubinstein & Sinclair - math.DH/0406189 (RS)

2-surfaces
A two-dimensional surface of revolution of genus
zero embedded in R3 can be defined in a polar repre
sentation with coordinates [2,0[, 21
πθρ ∈== xx






=
)(0
0)(
][
ρ
ρ
µν
m
h
g
)( ρm
by a metric of the form :
where has the direct physical inter-
pretation as the radius from the axis of rotation.
For a closed surface, we have
0)( =polem ρ
Choosing ρρρρ =0 at the North Pole and ρρρρ =ππππ at the South
Pole

2-surfaces
To avoid the numerical instabilities in an implicit
scheme for (1) and inspired by spectral methods,
RS introduced a filter which consists of transfor-
ming to Fourier space (DFT), dropping shorter
wavelength terms, and then transforming back.
Rubnstein and Sinclair used their own code,
called Ricci_rot (in C) for numerical simulations
and special codes for vizualisations (in OpenGL)
No convergence and stability analysis was done
– at least not reported, for the explicit direct
finite differencing schema used. These produced
several troubles, especialy in the 3D case.

2-surfaces
The Ricci flow forces some parts of a surface to
contract while others are inflating creating further
sources of numerical instability. The solution to
this problem is to reparametrize the metric, the
most pleasant one let h(ρ) be a constant.
Here are some results reported by RS
Dumbbell shape 2-surface
under Ricci flow –
Reproduced from [1]
Courtesy of H. Rubinstein

Numerical simulations with Maple
Why using Maple for numerical simulations in Ricci flow ?
Because Maple is and integrated platform wich can do symbolic
computations, numerical computing and vizualisations in the
same .. worksheet !
Thus we done all the necessary steps for numerical simulations
in Ricci flow using only Maple, namely :
1) calculations of the differential equations in several cases we
investigated
2) finite differencing of the PDE obtained in the previous step
3) stability and convergence analysis on the FD schema used in
order to establish a Courant factor for stable numerical
evolution
4) analysis of initial data
5) the numerical evolution of the FD equations in question
6) vizualisation of results – even movies !!

Advantages :
-Storing the analytical and numerical results in separate Maple
libraries, for later use. Example : step (1) above produce the main
equations stored in libraries loaded later in separate Maple
worksheets (bisurface_eqs.m is for 2 surfaces of revolution – se
below)
- as a result steps (2) and (3) or (4) above can be done separately in
spcial workshhets
-Separately doing the numerical evolution and vizualitation after
analysis steps
-Everything done under the same language and environment !
- Easy to use results for anybody familiar with Maple and not only !
A special mention for the step (1) . Here we used the GrtensorII
package for algebraic computing of the main equations and
geometrical objects (as the Ricci tensor and scalar). GrTensorII is
specially designed for differential geometry calculations in
riemannian geometry

For the 2-surface described earlier (the metric (3)) the Ricci tensor
has only two non-vanishing components, and the Ricci flow
corresponding equations are :
0
),(),(
),(),(
2
1
),(
),(
)),((
)),((
2
1
),(
2
2
2
2
=
∂
∂
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
tmth
tmth
tm
tm
tm
tm
th
t ρρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
0
),(
),(),(
2
1
),(
),(
)),(),((
)),((
2
1
),( 2
2
2
2
=
∂
∂
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
th
tmth
th
tm
tmth
tm
tm
t ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρρ
ρ
ρ
ρ
These and other geometrical objects are then stored in a
Maple library (*.m extension) for later use and load. For the 2
surfaces these are the main equations we used for numerical
simulations.

In our investigations we used as initial the function m(ρ(ρ(ρ(ρ)))) as
with appropiate values for the coefficients c3 and c5 controlling
the shape of the surface - for example for c3 = c5 = 0 the surface
is a sphere.
For initial value of the h(ρ(ρ(ρ(ρ)))) functions we also used h0(ρ)(ρ)(ρ)(ρ) = 1.
Before using these initial data, we investigate the behaviour and
shape of these functions and their derivatives in order to reveal
some tricky points, if any.
2
53
2
53
0
)531(
))5sin()3sin()(sin(
)(
cc
cc
m
++
++
=
ρρρ
ρ

Thus for c3 = 0.766 and c5 =-
0.0091 we have the shape of
the m0() function (left panel)
and of the first main equation
above with m(ρρρρ,,,, t) replaced
with m(0) (right panel).
The first and the second
spatial derivatives of the m0
function are plotted in
revealing some turning points
where we need special care
with the future numerical
simulations.
0.4
0
0.3
0.1
0.2
rho
2.52 31.50.50 1
20
0
-20
-40
rho
32.521.510.5
40
1
0
-1
-0.5
rho
32.521.50.50 1
0.5
0.50
1
2
0
-2
-1
rho
32.521.51

With these established, we composed a Maple program for finite
differencing the above equations.
First we denoted the two unknown functions m(ρρρρ,,,, t) and h(ρρρρ,,,, t) whith
two matrices, mm[i,j] and hh[i,j] respectively.
We also denoted the spatial interval between the points on the ρρρρ axis
with δδδδ =∆ρ∆ρ∆ρ∆ρ and the time step with dete =∆∆∆∆ t variables.
Separately we denoted also the parts from the two above equations
containing only spatial derivatives with two matrices, namely eno1[i,j]
and eno2[i,j] .
For the time derivatives we used the forward Euler method (it has the
advantage that one is able to calculate quantities at timestep j + 1 in
terms of only quantities known at timestep j) and for the spatial
derivatives we can use a second-order representation still using only
quantities known at timestep j.

Thus the “eno” matrices have the shape as :
2
j)δj)hh(i,8mm(i,
j))1,mm(ij)1,j))(mm(i1,hh(ij)1,(hh(i
2
j)δmm(i,
j)1,mm(ij)2mm(i,j)1,mm(i
2
δ
2
j)8mm(i,
2
j))1,mm(ij)1,(mm(i
ji,
eno1
−−+−−+
−+−+
+
−−+
−=
2
δ
2
j)8hh(i,
j))1,mm(ij)1,j))(mm(i1,hh(ij)1,(hh(i
2
j)δhh(i,
j)1,mm(ij)2mm(i,j)1,mm(i
2
j)δj)hh(i,8mm(i,
2
j))1,mm(ij)1,(mm(i
ji,
eno2
−−+−−+
−+−+
+
−−+
−=
Now the time integration of the two finite-differenced equations
is straitforward as :
j)eno2(i,*detej)mm(i,1)jmm(i,
j)eno1(i,*detej)hh(i,1)jhh(i,
+=+
+=+
for any time j > 0 if the initial values for hh[i.0] and mm[i,0]
are provided.

With the above established theoretical steps we proceeded to com-
pose a special Maple program (worksheet) for accomplishing the
numerical tasks.
First the program starts with the necessary loadings, namely the
library for the bisurface case equations (bisurface eq.m) previously
saved and then the finite-differences discretization of the main
equations :
> restart;
> read("bisurface_eqs.m");
> eno1[i,j]:= ((subs(m(rho,t)=mm[i,j],subs(h(rho,t)=hh[i,j],
subs(diff(m(rho,t),rho)=(mm[i+1,j]-mm[i1,j])/2/delta,
subs(diff(h(rho,t),rho)=(hh[i+1,j]-hh[i-1,j])/2/delta,
(subs(diff(m(rho,t),rho,rho)=(mm[i+1,j]-2*mm[i,j]+
mm[i-1,j])/delta/delta,eco1)))))))):
And similarly for the eno2 object.

The next lines simply introduce of the initial data (as the function
m0(ρ(ρ(ρ(ρ)) and some checkings of it’s values in different points of the
grid. Then comes the establishing of the grid values, the number of
the points on the axis (n), the values of the values for
the c3, c5 constants and of the spatial interval on the axis (delta):
> m0(rho):=((sin(rho)+c3*sin(3*rho)+c5*sin(5*rho))/
(1+3*c3+5*c5))**2;
> plot(subs(c3=0.766,c5=-0.091,m0(rho)),rho=0..10);
> n:=50;dete:=0.0018;
> c3:=0.766;c5:=-0.091;delta:=evalf(Pi/n);
> courfac:=dete/delta/delta/2;
After a series of commands establishing the initial values of main
matrices, comes the most important part of the program, were we
integrate the two equations :
> tmax:=80 ; for v from 1 to tmax do; for k from 1 to n-v-1 do
> hhini:=eval(subs(i=k,j=v-1,hh[i,j])); mmini:=eval(subs(i=k,j=v-1,mm[i,j]));
> coco1:=eval(subs(i=k,j=v-1,eno1[i,j]));coco2:=eval(subs(i=k,j=v-1,eno2[i,j]));
> hh[k,v]:=eval(hhini+dete*coco1);
mm[k,v]:=eval(mmini+dete*coco2);end do:end do:
and this is all, as for the numerical calculations !!!! Well, almost !

Numerical vizalisations with Maple
The next lines of the program are just a series of plotting
commands in order to visualise the shape and values ofthe
unknown functions, at all the times processed or atdifferent
times separately.
> for j from 0 to tmax do
> bebe[j]:=plot([[r[p],mm[p,j]]$p=0..n],t) od;
> display(seq(bebe[m],m=0..tmax));
> plot([[r[p],mm[p,0]]$p=0..n],t);
> plot([[r[p],mm[p,1]]$p=0..n],t);.....
plot([[r[p],mm[p,tmax]]$p=0..n],t);
Here we need plots package and of course the program contains
similar lines for plotting the values for h(ρ(ρ(ρ(ρ;;;; t)
The rest of the program is mainly dedicated to vizualisations of
the results, several plottings were done, including some series
of pictures for animation movies !

Numerical vizalisations with Maple
For more striking wiew of the results we used a separate sequence of
commands, where we plotted in 3D figures the time evolution of the
two functions m(ρρρρ:t) and h(ρρρρ:t), namely
> ccc:=[seq([ seq([p,k, evalf(subs(i=p,j=k,mm[i,j]))],
p=tmax+1..n-tmax-1)], k=0..tmax)]: surfdata(ccc,axes=BOXED);
> ddd:=[seq([ seq([p,k, evalf(subs(i=p,j=k,hh[i,j]))],
p=tmax+1..n-tmax-1)], k=0..tmax)]: surfdata(ddd,axes=BOXED);
The time evolution of both functions are plotted as a surface in a
figure having the time as one of the axes. Some examples will be
plotted in the next slides.
For the stability analysis, we used, in a separate Maple worksheet the
von Neumann analysis – other methods are also possible. For our case
of a 2-surface of revolution we established the Courant factor as :
2533029590.01
)(2 2
usedweactually
t
≤
∆
∆
ρ
For both finite differences equations !
Special care also we had for the boundary values where we used
specific method in every case – this is to be reported elsewhere !

Numerical vizualisations with Maple
Time evolution of the
functions m(t) (left)
and h(t) (right)
–1
–0.5
0
0.5
1
z
0
5
10
15
20
t
0
0.1
0.2
0.3
0.4
R
–1
–0.5
0
0.5
1
rho
0
5
10
15
20
t
–120
–100
–80
–60
–40
–20
0
20
R
Time evolution of the
radius m(t)1/2 (left) and
of the Ricci scalar R(t)
(right)
C3=0,766 C5= -0.091
N=50 ∆t=0.002
δ = 0.06283185308
–1
–0.5
0
0.5
1
z
0
20
40
60
80
t
0
0.05
0.1
0.15
0.2
m
–1
–0.5
0
0.5
1
z
0
20
40
60
80
t
1
2
3
4
5
6
7
h

The sphere
03 =c
05 =c
Numerical vizualisations with Maple - Movies

Two folded surface
766.03 =c
091.05 −=c

Three folded surface
021.03 =c
598.05 =c

3-surfaces
For the 3-dimensional surface of revolution we investigated (as RS)
the metric as
where ρ ≡ x1 plays the role of a latitude and θ ≡ x2 the role of a
latitude on the abstract Riemannian surface of revolution and K2 is
a consant.










=
)(cos)(00
0)(0
00)(
][
2
2
θρ
ρ
ρ
µν
Km
m
h
g
The pinching behavior of this surface under unnormalized Ricci flow (1)
was studied by Rubinstein and Sinclair starting at t = 0. Also this time an
explicit finite-differences schema was used. The instabilities forced to
restrict to fairly large time steps and they were forced to run their code
only to a restricted (even not at equal times) number of steps. These
results were called by the authors as “qualittively correct” !!!

3-surfaces in Maple
Using our programs in Maple (as described before) for this type of
3surfaces we were able, after a von Neumann stability analysis to
establish a Courant factor and to run properly a number of iterations
(20..30) using the intinial data as :
11)()
40
9
(sin
10000
1
)( 2
2
==+= Khm ρ
πρ
ρ
The m(ρ) function (left)
and the Ricci scalar (right)
for a 3-surface of revo-
lution under Ricci flow

Corseted sphere surfaces
In their study on a critical behaviour of 3-surfaces under Ricci flow,
Garfinkle and Isenberg (GI) used a so called corseted sphere
geometry having :
]sin[sin( 22222222
φθθψψ ddedeeg WWX
++= −
Here (ψ, θ, φ) are standard angular coordinates on the three sphere.
The metric functions X and W are functions only of ψ. W = X and X is
choosed so that :
2/1cos2sinsin4 2224
≥= ψψψ fore X
2/1cos2cos42sinsin4 22224
≤+= ψψλψψ fore X
Here λ is a constant, which parameterizes the degree of corseting
for these geometries.

For their numerical investigations, GI used the DeTurck flow, where
the PDE equations are strong parabolic and a finite differences
schema for the time evolution and centered finite differences for the
spatial derivatives. The numerical investigations pointed out a
critical value for the λ parameter (λ = 0.1639) dividing the behavior
under the flow into two regimes :
- a subcritical one - for larger λ the flow converges to a round sphere
- a supercritical one - for smaller λ the flow goes to a S2 neck
pinching singularity
0
0.5
1
1.5
2
2.5
3
3.5
ψ
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
t
2
4
6
8
10
12
14
16
18
0
0.5
1
1.5
2
2.5
3
3.5
ψ 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
t
0
10
20
30
40
50
60
70Subcritical (left) and supercritical
(right) behaviour of RS
2 under
Ricci flow for a corseted sphere
(courtesy of David Garfinkle)
At the critical value for λ the flow approaches a “javelin” geometry,
marked by curvature singularities at the poles, with roughly uniform
curvature between the poles. This javelin geometry corresponds to the
“type 3” singularity described by Hamilton and discussed by Chow.

The work of Garfinkle and Isenberg is a typical example of how the
experience in numerical relativity can be applied to Ricci flow
simulations. The DeTurk “trick” used to make the PDE strongly
parabolic is similar to the ADM-BSSN version of numerical relativty.
In spite of this, we still think that the use of initial Ricci flow
equations (normalised or not) can be used, as the explicit finite
differences schema can be done stable – a Courant factor in certain
conditions is possible to exhibit for this schema – as we proved using
our Maple programs. This is similar to the “pure” ADM method in NR.
We will developp this idea in our next investigations.
But this is an open question from now one, and the results will be
reported elsewhere… I hope !

Selected references
- R.S. Hamilton, J. Diff. Geom., 17, 255 (1982)
- D. Garfinkle, J. Isenberg, math.DG/0306129 (2003)
- J.H. Rubinstein, R. Sinclair, Exp.Math., 14, nr.3 (2005),
math.DG/0506189
-B. Chow, A Survey of Hamilton's Program for the Ricci
flow on 3-Manifolds, math.DG/0211266 (2002)
-D. DeTurck, J. Diff. Geom., 18, 157 (1983)
-http://www.cactuscode.org for numerical relativity
-http://grtensor.org the site for grTensorII package

Selected references
D.N. Vulcanov - Numerical simulations with Ricci flow on
surfaces- A review and some recent results, Annals od
Univ. of Craiova, vol. 18, 120-129, 2008
- D.N. Vulcanov -Errors and Convergence in Numerical
Simulations of Ricci Flow on 2-Dimensional Surfaces,
PROCEEDINGS OF THE PHYSICS CONFERENCE TIM-08,
Book Series: AIP Conference Proceedings Volume: 1131
Pages: 143-148 Published: 2009

Acknowledgements
This is the … very end of the show !!!
Special thanks go to H. Rubinstein and D. Garfinkle for their permission
to reproduce the pictures from their articles I included here !
Manny worm thanks to prof.dr. Goran Djordjevic and
his team for the kind hospitality and support
during my visit in Nis under the auspices of
SEENET-MTP network

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