Convex Symplectic Manifolds

Transcription

1 Convex Symplectic Manifolds Jie Min April 16, 2017 Abstract This note for my talk in student symplectic topology seminar in UMN is an gentle introduction to the concept of convex symplectic manifolds. In particular, the notions of Liouville, Weinstein and Stein manifolds are introduced and some properties of them are discussed. At last, I will introduce an h-principle for subcritical Weinstein manifolds based on Gromov s h-principle for subcritical isotropic embeddings. Everything is based on the paper of Eliashberg and Gromov[1] and chapter 11 of the book of Cieliebak and Eliashberg[2]. 1 Liouville manifolds Definition 1. Exact symplectic manifold (V 2n, ω, X), symplectic form ω = dλ, λ is called the Liouville form. Since ω is nondegenerate, there exists a vector field X dual to the Liouville form, i.e. ι X ω = λ. So we have L X ω = ω and L X λ = λ. Denote by X t the time t flow of X, we have (X t ) λ = e t λ. Definition 2. A Liouville manifold (V, ω, X) is an exact symplectic manifold s.t (1)X is complete (2)the manifold is convex: there is an exhaustion of V by compact domains V k V with smooth boundaries along which X is pointing outward, V 1 V 2 V and V k = V. Definition 3. Skeleton V 0 = k=1 t>0 X t (V k ). It is indepedent of the choice of exhaustion {V k }. V is of finite type if V 0 is compact. 1

2 Assume all following is of finite type. Definition 4. A diffeomorphism f : (V, ω V ) (W, ω W ) between two exact symplectic manifolds is called exact symplectomorphism if f λ W λ V is exact. Lemma 5. Any symplectomorphism between finite type Liouville manifolds is diffeotopic to an exact symplectomorphism. Lemma 6. The stable manifolds of X is isotropic. Proof. Let p V be a zero of X, and the stable manifold M = {x V : lim t Xt (x) = p}. Let v T x M, we have e t λ x (v) = ((X t ) λ)(v) = λ X t (x)((x t ) v) 0 as t. So λ(v) = 0, and ω T M = 0. 2 Stein and Weinstein manifolds Definition 7. A Weinstein manifold (W, ω, X, f) is a Liouville manifold (W, ω, X) together with an exhausting Morse function f on W, s.t. X is gradient like for f, i.e. X(f) δ( X 2 + df 2 ). Here exhausting means proper and bounded below. Since the unstable manifolds of any Liouville field is isotropic, thus at most n-dim, then index(f) n. (W, ω, X, f)is called subcritical if index(f) < n. Definition 8. A Stein manifold is a complex manifold, which can be properly embedded into C N. Equivalently, there exists a smooth exhausting J-convex function f : V R. Denote d C f = df J, ω f = dd C f, the associated metric g f (u, v) = ω f (u, Jv). f is called J-convex if g f is a Riemannian metric. So clearly, a Stein structure induces a Weinstein structure. Theorem 1. Let φ, ψ be two exhausting J-convex functions on a Stein manifold M. Then the symplectic structures (M, ω φ ), (M, ω ψ ) are symplectomorphic. 2

3 3 h-principle Consider the monomorphism F : T V T W which covers f : V W, i.e. F is fiberwise injective bundle map. Definition 9. F is a symplectic tangential equivalence (STE) if F ω W = ω V is fiberwise bijective, and f is smooth homotopy equivalence. Definition 10. (f, F 1 ) is quasi-symplectic (QS) if f is diffeomorphism and F t is a homotopy of bundle isomorphisms, s.t. F 0 = df and F 1 is STE. Theorem 2 (Gromov, P334). Still need to think if this is the right h-principle to use. Let ω V be an arbitrary closed C 2-form on V and let F 0 : (T V, ω V ) (T W, ω W ) be a fiberwise injective isometric homomorphism and f0 [ω W ] = [ω V ]. If dimv <dimw, then the map f 0 admits a fine C 0 -approximation by isometric C -immersions f : (V, ω V ) (W, ω W ), and a homotopy of isometric monomorphisms F t, such that F 1 = df. Moreover, for open manifolds, we may have dimv=dimw. f 0 can be homotoped to a C -immersion f which may not be C 0 -close to f 0. (This argument might not be right) From this theorem, one can see that on open manifolds, STE and QS are equivalence notions. If (f, F ) is STE, then f can be homotoped to an immersion g and F t such that F 1 = dg is also isometric. Since g ωw n = ωn V, we have g is local diffeomorphism, so g is a smooth covering map. Since f is homotopic to g, then g is homotopy equivalence. So g is a diffeomorphism and (g, F 1 ) is quasisymplectic. Let V be either subcritical Weinstein manifold where V 0 is skeleton of V or V 0 R 4 where V 0 is Liouville manifold. V 0 is invariant under the flow of Liouville field and V can be recovered by the flow from any neighborhood of V 0. Theorem 3. Let V be as above, W =Liouville, and X, Y be the Liouville fields correspondingly. Then the exact symplectic embeddings V W satisfy h-principle, i.e. the inclusion {f : V W exact symplectic embeddings} {(f, F 1 ) quasi symplectic} is a homotopy equivalence. 3

4 Proof. In particular, we will only prove the following: For all (f, F 1 ), there exists an isotopy of embeddings f t : V W, s.t. f 0 = f, f 1 is an exact symplectic embedding. (This argument needs to be refined) First, the h-principle holds for a very small neighborhood of V 0 by Gromov s h-principle, because V 0 has positive codimension. So from (f, F 1 ) we can construct an exact symplectic embedding g 0 : U W of a neighborhood U of V 0, homotopic to (f, F 1 ) in the space of quasi-symplectic embeddings. Actually can just take U = X 1 (V 0 ). Then we extend the embedding to the whole V by induction. Consider an exhaustion of V by V = V k, such that V k is invariant under the contraction flow and X 1 (V k ) V k+1. Let Y 1 be a Liouville field on W, which is equal to dg 0 (X) on g 0 (V 2 U) and equal to Y out side a compact set. Such a field exists because field Y corresponds to Liouville form β. f exact, i.e. f β α = dh on U V 2. Then define ξ be a cutoff function being 1 on U V 2 and define β 1 = β + d((ξh) f 1 ). Then dβ 1 = dβ, β 1 is a Liouville form and gives Y 1. Now extend g 0 to g 1 (x) = Y1 1 (g 0 (X 1 )), defined on V 1 U. Note that for g 1 U = g 0, because dg 0 (X) = Y 1 on f(v 2 U). Inductively, suppose we have g i : V i U W. Let Y i+1 be equal to dg i (X) on g i (V i+2 (V i U)) and equal to Y outside a compact set. Then we can extend g i to g i+1 (x) = Yi+1(g 1 i (X 1 )). Alternative argument for extension:alternatively, we may recognize V = U Σ [0, ), where the product is the symplectization of Σ with symplectic form ω = d(e t λ Σ ). Take a Liouville field Ỹ s.t. Ỹ = dg 0 (X) on g 0 (U) and Ỹ = Y at infinity. We may define g = g 0 on U and g(y, t) = Ỹ t ((g 0 Σ )(y)). Finally, note that two quasisymplectic embeddings are isotopic if their germs along V 0 are isotopic. Theorem 4. Suppose V, W are subcritical Weinstein manifolds, and there exists a STE (f, F ). then V, M are symplectomorphic. Proof. STE (f, F ) implies a quasisymplectic diffeomorphism f : V W. By above h-principle, there exists exact symplectic embeddings f : V W and g : W V. So we have an sequence of embeddings V W V M. 4

5 Let M be the direct limit of this system. Then by Mazur s trick, we have V = M = W. Corollary 11. Let V and W be open manifolds, then T V and T W are symplectomorphic iff V and W are STE. Proof. Open n-manifold has a Morse function with only critical points of index n 1. Then use this function we can construct a Morse function on cotangent bundle with only critical points of index n 1. So cotangent bundles are subcritical Weinstein manifolds. Theorem 5. V and W are STE Weinstein manifolds, then V R 2 W R 2 are symplectomorphic. and Corollary 12. If V R and W R are STE, then T (V R) and T (W R) are symplectomorphic. References [1] Eliashberg & Gromov, Convex symplectic manifolds. [2] Cieliebak & Eliashberg, From Stein to Weinstein and back. [3] Gromov, Partial Differential Relations. 5