Extrinsic geometric flows

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Extrinsic geometric flows

On joint work with Vladimir Rovenski from Haifa

Paweł Walczak Uniwersytet Łódzki

CRM, Bellaterra, July 16, 2010

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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, σ_k is the k-th mean curvature of F

τ_k = the sum of k-th powers of the principal curvatures of F

~τ = (τ1, . . . , τn), ~σ = (σ1, . . . , σn) etc.

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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

dg_t

dt = h_t:=

n−1

X

j =0

f_j(~τ )b_j (1)

where

fj ∈C^∞(Rⁿ) are given a priori

and b_j = g (A^j(·), ·). The family (gt) satisfying (1) is called extrinsic geometric flow (EGF).

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Geometric flows

Our motivation comes from:

Ricci flow: dgt/dt = Rict (famous !)

Mean curvature flow: dF_t/dt = H_t (well known !) Other flows, see

H.-D. Cao, S. T. Yau, Geometric flows, Surveys in Diff.

Geom., 2008

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Examples

Extrinsic Ricci flow (by Gauss equation) Ric^ex(g ) = τ₁b₁− b₂, Extrinsic Newton transformation flows:

T_i(g ) = σ_ig − σ_{i −1}b1+ . . . + (−1)ⁱb_i

Remark

Newton transformations were used in the variational calculus for σ_j’s and recently for a generalization of Asimov and Brito-Langevin-Rosenberg integral formulae, see

K. Andrzejewski, P. W., Ann. Global. Anal. Geom. 2010.

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The strategy

To prove existence/uniqueness results for (1) we shall:

1. derive and the corresponding equation for τ

2. substitute the solution into f_j’s of (1) to get the equation dg_t

dt =

n−1

X

j =0

h_j(·)b_j (2)

with hj = fj(~τ ) ∈C^∞(M).

3. solve (2) locally, in bifoliated coordinates and show that this solution satisfies (1).

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Variational formulae

Subject to (1) we have:

g_t(Π_t(X , Y ), Z ) = 1 2

(∇^t_Xh_t)(Y , Z )+(∇^t_Yh_t)(X , Z )−(∇^t_Zh_t)(X , Y ) where

Π_t = d ∇_t/dt.

and h_t is the RHS of (1). Consequently, d (A_t)/dt = −1

2 Xn−1

m=0[N(f_m(~τ ))A^m_t + f_m(~τ )∇^t_NA^m_t ]...

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Equations for ~ τ (1)

... and the corresponding power sums τ_i (i > 0) of principal curvatures satisfy the infinite quasilinear system

d τ_i/dt + i 2

τ_{i −1}N(f₀(~τ , t)) +

n−1

X

m=1

mf_m(~τ , t)

i +m−1 N(τ_{i +m−1})

+ τi +m−1N(fm(~τ , t))= 0... (3)

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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 )∂sτ = 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−1fm,τj, B =^X

m

(m/2)fm· ˜B^m−1

with ˜B being thegeneralized companion matrix to the characteristic polynomial of A_t.

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Companion matrices (1)

Let P_n = λⁿ− p₁λⁿ⁻¹− . . . − p_n−1λ − p_n be a polynomial over R and λ1 ¬ λ₂¬ . . . ¬ λ_n be the roots of Pn. Hence,

pi = (−1)^{i −1}σi, where σi are elementary symmetric functions of the roots λ_i. Thegeneralized companion matrices of P_n are defined by

B_~_c =







0 ^cn−1_cn 0 ··· 0

0 0 ^cn−2

cn−1 ··· 0

··· ··· ··· ··· ···

0 0 ··· 0 ^c1_c2

cnpn cn−1pn−1 ... c2p2c1p1







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where c₁ = 1 and c_i 6= 0 (i > 1) are arbitrary numbers.

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Companion matrices (2)

Our matrix ˜B coincides with B_~_c, where c_i = n

n + 1 − i

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Existence/uniqueness for ~ τ

From the theory of quasi-linear PDE’s:

Theorem

If the matrix A in (4) ishyperbolic(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}.

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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.

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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}.

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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 g_t-totally umbilical for any t.

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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”:

U(F ) = Z

M

X

i <j

|k_j− k_i|ⁿ· Ω. (6)

and have shown that all the foliations of compact Riemannian manifolds of negative Ricci curvature are far from being umbilical.

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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 , g_t) → 0 as t → T )?

Perhaps, one should consider rather ”normalized EGF’s”, that is the flows satisfying

dgt/dt = ht−ρt

n gˆt with ρt = R

MTrace Ahd volt

vol(M, gt) . (7) A_h being a (1,1)-tensor dual to h.

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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:

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An example - continuation

If ht = kt· g_t along F (kt = the curvature of the leaves), then the Gaussian curvature K_t (t > 0) of (M, g_t) becomes:

negative in a nbhood of the line x = 0 positive in a nbhood of the lines x = ±1

(More detailed study of (M, g_t) should be performed with the use of Maple.)

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Solitons

Example

If f_j(0) = 0 for all j⁰s in (1) and F is totally geodesic for t = 0, then (trivially) gt= g0 for all t.

Definition

A solution to (1) is calleda (EG) soliton, when

g_t= σ_t· ψ^∗_tg₀ (8)

for some σt ∈ R and ψt, diffeo’s preserving F . Differentiating (8), we get

˙σ(0) g₀+ σ(0)L_{X (0)}g₀ = h₀ (9)

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Solitons - continuation

Depending on X in (9), one may distinguish betweentangent (X ∈ T F ) andnormal(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:

L_Xg = (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.

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Bibliography

V. Rovenski, P. W., Extrinsic geometric flows on foliated manifolds, arXiv:1003.1607.

Extrinsic geometric flows