Extrinsic geometric flows

Transcription

1 On joint work with Vladimir Rovenski from Haifa Paweł Walczak Uniwersytet Łódzki CRM, Bellaterra, July 16, 2010

2 Setting Throughout this talk: (M, F, g 0 ) 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, N = 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 = (τ 1,..., τ n ), σ = (σ 1,..., σ n ) etc.

3 The flow Fix N the unit normal and consider the one parameter family (g t ) of Riemannian structures on M varying along F subject to the equation n 1 dg t dt = h t := f j ( τ)b j (1) where f j C (R n ) are given a priori j=0 and b j = g(a j ( ), ). The family (g t ) satisfying (1) is called extrinsic geometric flow (EGF).

4 Geometric flows Our motivation comes from: Ricci flow: dg t /dt = Ric t (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

5 Examples Extrinsic Ricci flow (by Gauss equation) Ric ex (g) = τ 1 b 1 b 2, Extrinsic Newton transformation flows: T i (g) = σ i g σ i 1 b ( 1) i b 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

6 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 n 1 dg t dt = h j ( )b j (2) with h j = f j ( τ) C (M). 3. solve (2) locally, in bifoliated coordinates and show that this solution satisfies (1). j=0

7 Variational formulae Subject to (1) we have: g t (Π t (X, Y ), Z) = 1 [ ( t 2 X h t )(Y, Z)+( t Y h t )(X, Z) ( t Z h t )(X, Y ) ] where Π t = d t /dt. and h t is the RHS of (1). Consequently, d(a t )/dt = 1 2 n 1 m=0 [N(f m( τ))a m t + f m ( τ) t NA m t ]...

8 Equations for τ (1)... and the corresponding power sums τ i (i > 0) of principal curvatures satisfy the infinite quasilinear system n 1 dτ i /dt + 2{ i [mf m ( τ, t) τi 1 N(f 0 ( τ, t)) + i+m 1 N(τ i+m 1) m=1 + τ i+m 1 N(f m ( τ, t)) ]} = 0... (3)

9 Equations for τ (2)... which (due to algebraic relations between τ j 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 C ij = (i/2) m τ i+m 1 f m,τj, B = (m/2)f m B m 1 m with B being the generalized companion matrix to the characteristic polynomial of A t.

10 Companion matrices (1) Let P n = λ n p 1 λ n 1... p n 1 λ p n be a polynomial over R and λ 1 λ 2... λ n be the roots of P n. Hence, p i = ( 1) i 1 σ i, where σ i are elementary symmetric functions of the roots λ i. The generalized companion matrices of P n are defined by B c = c 0 n cn c 0 0 n 2 0 c n 1 c c 2 c np n c n 1 p n 1... c 2 p 2 c 1 p 1 where c 1 = 1 and c i 0 (i > 1) are arbitrary numbers. (5)

11 Companion matrices (2) Our matrix B coincides with B c, where c i = n n + 1 i

12 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 R n ) 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}.

13 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 ( ɛ, ɛ) for some ɛ > 0.

14 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}.

15 Umbilicity Ricci flow maps Einstein to Einstein our EGFlows map umbilical to umbilical: Proposition Let (M, g 0 ) be a Riemannian manifold endowed with a codimension-1 totally umbilical foliation F. If g t (0 t < ɛ) provide an EGFlow on (M, F), then F is g t -totally umbilical for any t.

16 Umbilicity - continuation In R. Langevin and P. Walczak. Conformal geometry of foliations, Geom. Dedicata 132 (2008), p we defined a measure of non-umbilicity : U(F) = k j k i 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.

17 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, g 0 ), the members (g t ) 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 dg t /dt = h t ρ t n ĝt M with ρ t = Trace A h d vol t. (7) vol(m, g t ) A h being a (1,1)-tensor dual to h.

18 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.

19 An example - continuation If h t = k t g t along F (k t = 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.)

20 Solitons Example If f j (0) = 0 for all j s in (1) and F is totally geodesic for t = 0, then (trivially) g t = g 0 for all t. Definition A solution to (1) is called a (EG) soliton, when g t = σ t ψ t g 0 (8) for some σ t R and ψ t, diffeo s preserving F. Differentiating (8), we get σ(0) g 0 + σ(0)l X (0) g 0 = h 0 (9) and may call solitons also Riemannian structures g 0 satisfying (9).

21 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: L X g = (ψ(λ) ɛ) g along F, where ψ(λ)g 0 = h 0. If F is g-totally geodesic, then X is the infinitesimal homothety along leaves with the factor f 0 (0) ɛ. If f 0 (0) = ɛ, then X is a leaf-wise Killing field, for ex., when M is a surface of revolution foliated by parallels.

22 More problems Problem Describe possible types of singularities for EGFlows as t T, the largest value of time parameter for which the regular solution g t exists. Problem Describe the behaviour of geometry (sectional, Ricci, scalar, principal, mean curvatures and so on) of (M, F, g t ) as t T... Problem and much more, so we need to find young people to deal with...

23 Bibliography V. Rovenski, P. W., on foliated manifolds, arxiv:

24 Finis coronat opus Thanks! Merci! Obrigado! Gracias! Cπaciδa! Danke schön! Arigato! Dziękuję!