Extrinsic geometric flows - On joint work with Vladimir Rovenski from

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Extrinsic geometric flows
On joint work with Vladimir Rovenski from Haifa
Paweł Walczak
Uniwersytet Łódzki
CRM, Bellaterra, July 16, 2010
Paweł Walczak Uniwersytet Łódzki
Setting
Throughout this talk:
(M, F, g0 ) is a (compact, complete, any) foliated, Riemannian
manifold,
dim M = n + 1
codim F = 1,
both M and F are oriented,
b is the 2nd fundamental form of F
A = −∇N (N ⊥ F, kNk = 1) is the Weingarten operator,
σk is the k-th mean curvature of F
τk = the sum of k-th powers of the principal curvatures of F
~τ = (τ1 , . . . , τn ), ~σ = (σ1 , . . . , σn ) etc.
The flow
Fix N the unit normal and consider the one parameter family (gt )
of Riemannian structures on M varying along F subject to the
equation
n−1
X
dgt
= ht :=
fj (~τ )bj
(1)
dt
j=0
where
fj ∈C∞ (Rn ) are given a priori
and bj = g (Aj (·), ·). The family (gt ) satisfying (1) is called
extrinsic geometric flow (EGF).
Geometric flows
Our motivation comes from:
Ricci flow: dgt /dt = Rict
(famous !)
Mean curvature flow: dFt /dt = Ht
(well known !)
Other flows, see
H.-D. Cao, S. T. Yau, Geometric flows, Surveys in Diff.
Geom., 2008
Examples
Extrinsic Ricci flow (by Gauss equation)
Ricex (g ) = τ1 b1 − b2 ,
Extrinsic Newton transformation flows:
Ti (g ) = σi g − σi−1 b1 + . . . + (−1)i bi
Remark
Newton transformations were used in the variational calculus
for σj ’s and recently for a generalization of Asimov and
Brito-Langevin-Rosenberg integral formulae, see
K. Andrzejewski, P. W., Ann. Global. Anal. Geom. 2010.
The strategy
To prove existence/uniqueness results for (1) we shall:
1. derive and the corresponding equation for τ
2. substitute the solution into fj ’s of (1) to get the equation
n−1
X
dgt
hj (·)bj
=
dt
j=0
with hj = fj (~τ ) ∈C∞ (M).
3. solve (2) locally, in bifoliated coordinates and show that this
solution satisfies (1).
(2)
Variational formulae
Subject to (1) we have:
gt (Πt (X , Y ), Z ) =
1 t
(∇X ht )(Y , Z )+(∇tY ht )(X , Z )−(∇tZ ht )(X , Y )
2
where
Πt = d∇t /dt.
and ht is the RHS of (1). Consequently,
d(At )/dt = −
1 Xn−1
[N(fm (~τ ))Am
τ )∇tN Am
t + fm (~
t ]...
m=0
2
Equations for ~τ (1)
... and the corresponding power sums τi (i > 0) of principal
curvatures satisfy the infinite quasilinear system
dτi /dt +
n−1
X mfm (~
i
τ , t)
τi−1 N(f0 (~τ , t)) +
N(τi+m−1 )
2
i+m−1
m=1
+ τi+m−1 N(fm (~τ , t))
= 0...
(3)
Equations for ~τ (2)
... which (due to algebraic relations between τj ’s) reduces to the
following finite system of quasilinear PDE’s:
∂t ~τ + A(s, t, ~τ )∂s τ = 0,
(4)
where s is the parameter along an N-trajectory and A = B + C is
the n × n matrix given by
Cij = (i/2)
X
m
τi+m−1 fm,τj ,
B=
X
(m/2)fm · B̃ m−1
m
with B̃ being the generalized companion matrix to the
characteristic polynomial of At .
Companion matrices (1)
Let Pn = λn − p1 λn−1 − . . . − pn−1 λ − pn be a polynomial over R
and λ1 ¬ λ2 ¬ . . . ¬ λn be the roots of Pn . Hence,
pi = (−1)i−1 σi , where σi are elementary symmetric functions of
the roots λi . The generalized companion matrices of Pn are
defined by


0
 0

B~c =  ···

cn−1
cn
0
···
0
0
cn pn cn−1 pn−1
0
cn−2
cn−1
···
···
...
···
0
···
0 

···
··· 
c1 
0
c2
c2 p2 c1 p1
where c1 = 1 and ci 6= 0 (i > 1) are arbitrary numbers.
(5)
Companion matrices (2)
Our matrix B̃ coincides with B~c , where
ci =
n
n+1−i
Existence/uniqueness for ~τ
From the theory of quasi-linear PDE’s:
Theorem
If the matrix A in (4) is hyperbolic (that is if its eigenvectors are
real and span Rn ) at (0, 0), then (4) has unique solution in a
neighbourhood of (0, 0). If M is compact and A is hyperbolic on
M × {0}, then (4) has unique solution in a neighbourhood of
M × {0}.
Existence/uniqueness for (2)
Calculations in bifoliated coordinates (adapted to F and N) show
that (2) reduces to a quasilinear system of PDE’s with the diagonal
(hence, hyperbolic) matrix of coefficients. This implies directly
Theorem
The equation (2) has always a unique local (in space and time)
solution; if M is compact, then it has a solution on M × (−, ) for
some > 0.
Existence/uniqueness for (1)
Combinimg Theorems 1 and 2 one gets directly
existence/uniqueness results for the original problem.
Theorem
If the matrix A in (4) is hyperbolic at (0, 0), then (1) has unique
solution in a neighbourhood of (0, 0). If M is compact and A is
hyperbolic on M × {0}, then (1) has unique solution in a
neighbourhood of M × {0}.
Umbilicity
Ricci flow maps Einstein to Einstein
our EGFlows map umbilical to umbilical:
Proposition
Let (M, g0 ) be a Riemannian manifold endowed with a
codimension-1 totally umbilical foliation F. If gt (0 ¬ t < )
provide an EGFlow on (M, F), then F is gt -totally umbilical for
any t.
Umbilicity - continuation
In
R. Langevin and P. Walczak. Conformal geometry of foliations,
Geom. Dedicata 132 (2008), p. 135–178.
we defined a ”measure of non-umbilicity”:
Z
U(F) =
X
|kj − ki |n · Ω.
(6)
M i<j
and have shown that all the foliations of compact Riemannian
manifolds of negative Ricci curvature are far from being umbilical.
Umbilicity - a problem
It is known that Ricci flow on some compact 3-manifolds converges
to a metric of constant sectional curvature.
Problem
Under what conditions on (M, F, g0 ), the members (gt ) of the
corresponding EGFlow converge to one for which F is totally
umbilical ( say, U(F, gt ) → 0 as t → T )?
Perhaps, one should consider rather ”normalized EGF’s”, that is
the flows satisfying
ρt
dgt /dt = ht −
ĝt
n
R
with
Ah being a (1,1)-tensor dual to h.
ρt =
M
Trace Ah d volt
.
vol(M, gt )
(7)
An example
Consider the strip M = [−1, 1] × R equipped with the 1-dim Reeb
foliation obtained from a vector field X making the angle α with
the first factor, α changing linearly form −π/2 to π/2:
Rysunek: Harvest foliation.
An example - continuation
If ht = kt · gt along F (kt = the curvature of the leaves), then the
Gaussian curvature Kt (t > 0) of (M, gt ) becomes:
negative in a nbhood of the line x = 0
positive in a nbhood of the lines x = ±1
(More detailed study of (M, gt ) should be performed with the use
of Maple.)
Solitons
Example
If fj (0) = 0 for all j 0 s in (1) and F is totally geodesic for t = 0,
then (trivially) gt = g0 for all t.
Definition
A solution to (1) is called a (EG) soliton, when
gt = σt · ψt∗ g0
(8)
for some σt ∈ R and ψt , diffeo’s preserving F. Differentiating (8),
we get
σ̇(0) g0 + σ(0)LX (0) g0 = h0
(9)
and may call solitons also Riemannian structures g0 satisfying (9).
Solitons - continuation
Depending on X in (9), one may distinguish between tangent
(X ∈ T F) and normal (X ⊥ F) solitons.
Existence an properties of all such solitons would be of great (we
hope) interest.
Example (EG soliton with conformal Killing X )
If F is totally umbilical with normal curvature λ. then a soliton
X becomes a leaf-wise conformal Killing field:
LX g = (ψ(λ) − ) g along F, where ψ(λ)g0 = h0 . If F is
g -totally geodesic, then X is the infinitesimal homothety along
leaves with the factor f0 (0) − . If f0 (0) = , then X is a leaf-wise
Killing field, for ex., when M is a surface of revolution foliated
by parallels.
More problems
Problem
Describe possible types of singularities for EGFlows as t → T , the
largest value of time parameter for which the regular solution gt
exists.
Problem
Describe the behaviour of geometry (sectional, Ricci, scalar,
principal, mean curvatures and so on) of (M, F, gt ) as t → T ...
Problem
and much more, so we need to find young people to deal with ...
Bibliography
V. Rovenski, P. W., Extrinsic geometric flows on foliated
manifolds, arXiv:1003.1607.
Finis coronat opus
Thanks !
Merci !
Obrigado !
Gracias !
Cπaciδa !
Danke schön !
Arigato !
Dziękuję !

Extrinsic geometric flows - On joint work with Vladimir Rovenski from

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