L 2 Geometry of the Symplectomorphism Group

Transcription

1 University of Notre Dame Workshop on Innite Dimensional Geometry, Vienna 2015

2 Outline 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of

3 Outline 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of

4 Setting M is a closed, orientable Symplectic manifold of dimension 2n, with Symplectic form ω, Riemannian metric g and almost complex structure J satisfying J 2 = I g(j, J ) = g(, ) g(, J ) = ω(, )

5 Setting Let D s ω denote the group of all Sobolev Hs dieomorphisms of M preserving the Symplectic form ω. If s > n + 1 then Dω s becomes an innite dimensional manifold whose tangent space at the identity is given by T e D s ω = {J F + h : F H s+1 0 (M), h harmonic}. Using right translations, the L 2 inner product on vector elds (u η, v η) L 2 = g(u, v) η d µ M denes a right-invariant metric on the group, which induces a smooth right invariant Levi-Civita connection and curvature tensor.

6 The Geodesic Equation on D s ω A curve η(t) in D s ω is a geodesic of the L2 metric if and only if the vector eld v = η(t) η 1 (t) satises the Symplectic Euler equations on M t v + v v = ω δ 1 dω ( v v) L v ω = 0, v(0) = v o When M is two dimensional the Symplectic Euler Equations coincide with the 2D Euler equations of incompressible hydrodynamics

7 Global Existence of Geodesics Theorem [Ebin 2012] Solutions to the geodesic equation of the L 2 metric on D s ω(m) are unique and exist for all time. Consequently, the L 2 metric admits an exponential map which is dened on the whole tangent space T e D s ω exp e : T e D s ω D s ω v o exp e (tv o ) = η(t) where η(t) is a geodesic of the L 2 metric issuing from the identity in the direction v o.

8 Outline 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of

9 Denition Let η(t) be a geodesic of the L 2 metric in Dω s with initial velocity v o. A point η(t ) (t > 0) is conjugate to the point η(0) if the linear map D exp(t v o ) fails to be an isomorphism. In contrast with nite dimensional geometry a linear map between innite dimensional spaces, with empty kernel, may not be an isomorphism. Following Grosman [Gro], η(t ) is monoconjugate to η(0) = e if D exp(t v o ) fails to be injective and epiconjugate if D exp(t v o ) fails to be surjective.

10 Singularities of exp e In Ebin-Misiolek-Preston ([E-M-P]), singularities of the exponential map (i.e. conjugate points) were studied in the context of the Euler equations of Hydrodynamics. Their results were Theorem For M 2 a compact two-dimensional manifold without boundary, the exponential map of the L 2 metric on D s µ(m 2 ) is a non-linear Fredholm map of index zero. For M 3 a compact three dimensional manifold, the exponential map of the L 2 metric is NOT a Fredholm map of index zero on D s µ(m 3 ). In three dimensions monoconjugate points accumulate and converge to an epiconjugate point; Preston ([P2]) has shown that this is a typical pathology.

11 The Exponential Map on D s ω(m) Theorem [B, 2014] Let M be a closed, orientable Symplectic manifold of dimension 2n. Then, the L 2 exponential map on D s ω(m) is a non-linear Fredholm map of index zero. Corollary Monoconjugate and epiconjugate points coincide in D s ω(m), are isolated and of nite multiplicity along nite geodesic segments.

12 Proof Sketch The Jacobi equation along η(t) = exp e (tu o ) D s ω is given by D 2 dt 2 J + Rω (J, η) η = 0 (1) with initial conditions J(0) = u o, J (0) = w o, (2) where R ω is the smooth, right-invariant Riemann curvature tensor of the L 2 metric.

13 Proof Sketch If J is the Jacobi eld along η with initial conditions then J(0) = 0, J (0) = w o, (3) Φ(t)w o := D exp e (tu o )tw o = J(t) (4) denes a family of bounded linear operators from T e D s ω to T η(t) D s ω.

14 Proof Sketch Using the Jacobi equation we nd that [ t Φ(t) = Dη(t) Ad η 1 (s)ad η 1 (s) ds 0 t ] Ad η 1 (s)ad η 1 (s) K v o dr η 1 (s)φ(s) ds 0 where Ad η X = Dη X η 1 is the usual push-forward of vector elds, Ad η its formal L2 adjoint, K vo (w) = ad w v o with ad w the formal L2 adjoint of L w - the usual Lie derivative, and dr η 1 (t) the dierential of composition on the right with η 1 (t).

15 Proof Sketch In brief, invertibility of Ω(t) = t Ad 0 η 1 (s)ad η 1 (s) ds on T e Dω s, follows from the estimates C(t) w o L 2 Ω(t)w o L 2 and C(t) w o H s Ω(t)w o H s + K w o H s 1. To see that K vo is compact, we compute it explicitly and nd that for any w T e Dω s K vo w = ad w v o = J 1 g(w, (dg (v o ) ω n 1 )).

16 Outline Existence of Multiplicity of 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of

17 The Isometry Subgroup Existence of Multiplicity of Let M be a closed Symplectic manifold with Symplectic form and Riemannian metric g. The isometry group, denoted by Iso(M), consists of those dieomorphisms satisfying η g = g. Every isometry of M is contained in D s ω. Every Killing vector eld generates a stationary solution to the Symplectic Euler equations.

18 Existence of Multiplicity of Examples of Conjugate points in D s ω Let M be the complex projective plane CP 2 with the Fubini-Study metric: h i j = h( i, j ) = (1 + z 2 )δ i j z i z j (1 + z 2 ) 2 where z = (z 1, z 2, z 3 ) is a point in CP 2, z 2 = z z z 2 3. The isometry group of CP 2 is given by PU(3), the projective unitary group. PU(3) is given by the quotient of the unitary group, U(3), by it's center, U(1), embedded as scalars. Thus, in terms of matrices, PU(3) consists of complex 3 3 matrices whose center consists of elements of the form e iθ I. Elements of PU(3) correspond to equivalence classes of unitary matrices, where two matrices A and B are equivalent if A = e iθ I B and we write A B..

19 Existence of Multiplicity of Examples of conjugate points in D s ω Consider the following 2-parameter variation of isometries acting on CP 2 γ(s, t) = i i cos s sin s 0 sin s i cos s i cos t sin t 0 sin t i cos t i i i cos s sin s 0 sin s i cos s Notice that γ(s,0) = ii I since ii corresponds to a rotation by 90 in each coordinate, i.e. is an element of U(1) embedded as scalars.

20 Existence of Multiplicity of Examples of conjugate points in D s ω Compute γ(s, t) (γ(s, t)) 1 = 0 i cos s sin s i cos s 0 0 sin s 0 0 = V (s, t) and V (s, t) satises the Symplectic Euler equation for each s. The variation eld of the family of geodesics γ(s, t) is Y (t) = d ds (γ(s, t)) s=0 = 0 0 i sin t cos t i sin t cos t 1 0 which clearly vanishes for t = 0 and t = 2π. That is, γ(2π) is conjugate to e = γ(0).

21 Existence of Multiplicity of on D s ω Theorem [B, 2014] Conjugate points exist on D s ω(cp n ) for all n 2.

22 Outline Existence of Multiplicity of 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of

23 Existence of Multiplicity of on D s ω Theorem [B, 2014] Every geodesic of the L 2 metric which lies in Iso(M) and which is of length greater than πr (for some positive constant r ) has conjugate points, all of which have even multiplicity.

24 Proof Idea Existence of Multiplicity of Restrict the L 2 metric to the nite dimensional subgroup Iso(M) and its nite dimensional Lie (sub) algebra T e Iso(M) T e D s ω. The L 2 metric becomes bi-invariant and the Riemann curvature tensor becomes R(w,u)v = 1 4 ad v ad u w u, v,w T e Iso(M) (5) Using (5) we can show that the sectional curvature of the plane σ spanned by any two unit vectors in T e Iso(M) is positive and bounded away from zero. The rst statement then follows from the classical Theorem of Bonnet and Myers.

25 Proof Idea Existence of Multiplicity of When η(t) consists of isometries, Ad η(t) = Ad η 1 (t) Jacobi equation reduces to and the t (Ad η 1 (t) y(t)) = K v o (Ad η 1 (t) y(t)) + w o where y = J η 1, J(0) = 0, J (0) = w o. The operator K vo is skew self-adjoint in the L 2 metric and compact. Thus we have a complete orthonormal basis of T e D 0 ω consisting of eigenvectors of K vo

26 Proof Idea Existence of Multiplicity of Letting u(t) = Ad η 1 (t) y(t), and using the orthonormal basis of K vo our equation becomes or with solution j t u j (t)ϕ j = j since u j (0) = y j (0) = 0. λ j u j (t)ϕ j + woϕ j j j t u j (t) = λ j u j (t) + w j o u j (t) = 1 eλ j t wo j λ j

27 Proof Idea Existence of Multiplicity of Now 0 = u(t ) = Ad η 1 (t ) y(t ) if and only if y(t ) = 0. So, η(t ) is conjugate to η(0) if and only if 0 = u i (t ) = 1 eλ i t λ i w i o, i.e. if and only if K vo has an eigenvector with eigenvalue λ i = 2πi t k, k Z/{0}. Since complex eigenvalues come in conjugate pairs the multiplicity of every conjugate point is even: u 1 (t) = 1 eλ 1t Re(λ 1 ) w 1 o u 2 (t) = 1 e λ1t Re(λ 1 ) w 2 o

28 Existence of Multiplicity of Arnold, V. I., Khesin, B. A., Topological Methods in Hydrodynamics, Springer-Verlag (1998) D. G. Ebin, Geodesics on the Symplectomorphism Group, GAFA, Published Online (2012) Ebin, D., Marsden, J., Groups of Dieomorphisms and the Motion of an Incompressible Fluid, Ann. of Math. 92 (1970) Ebin, D., Misiolek, G., Preston, S., Singularities of the Exponential map on the Volume-Preserving Dieomorphism Group, GAFA, Vol.16 (2006), Grosman, N., Hilbert Manifolds without Epiconjugate Points, Proc. Amer. Math. Soc. 16 (1965), Khesin, B., Dynamics of Symplectic Fluids and Point Vortices, GAFA, to appear.

29 Existence of Multiplicity of Khesin, B., Wendt, R., The Geometry of Innite-Dimensional Groups, Springer-Verlag (2009) Misiolek, G., Stability of Flows of Ideal Fluids and the Geometry of the Groups of Dieomorphisms, Ind. Uni. Math. Jour., Vol. 42, No. 1 (1993) Misiolek, G., Preston S., Fredholm Properties of Riemannian exponential maps on dieomorphism groups, Invent. Math, 2010, Preston, S., On the Volumorphism Group, the First Conjugate Point is the Hardest, Comm. Math. Phys. 276 (2006)