Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

Abstract

We give a uniform lower bound for the polynomial complexity of all Reeb flows on the spherization (S * M, ξ) over a closed manifold. Our measure for the dynamical complexity of Reeb flows is slow volume growth, a polynomial version of topological entropy, and our uniform bound is in terms of the polynomial growth of the homology of the based loops space of M . As an application, we extend the Bott-Samelson theorem from geodesic flows to Reeb flows: If (S * M, ξ) admits a periodic Reeb flow, or, more generally, if there exists a positive Legendrian loop of a fibre S * q M , then M is a circle or the fundamental group of M is finite and the integral cohomology ring of the universal cover of M is the one of a compact rank one symmetric space.Proof of (i). We will see in the proof of (ii) that if π 1 (M) has subexponential growth, then M belongs to the list in (ii), and π 1 (M) has polynomial growth of order 0, 1, 3, or 4.Proof of (ii). A main ingredient of the proof is the following Lemma 3.3. Consider a closed orientable 3-manifold M. If π 1 (M) has subexponential growth, then M admits a geometric structure modeled on one of the four geometries Ë 3 , Ë 2 × Ê, 3 , Nil .Proof. The proof can be extracted from [5], and is repeated here for the readers convenience. We distinguish several cases. Case 1: M is not prime. This means that M can be written as a connected sum M = M 1 #M 2 with both π 1 (M 1 ) and π 1 (M 2 ) non-trivial. By the Seifert-Van Kampen Theorem, π 1 (M) is the free product π 1 (M 1 ) * π 1 (M 2 ). It follows from the existence of normal forms for free products that π 1 (M 1 ) * π 1 (M 2 ) contains a free subgroup of rank 2 unless π 1 (M 1 ) = π 1 (M 2 ) = 2 , see Exercise-with-hints 19 in Sec. 4.1 of [56]. Our hypothesis on π 1 (M) thus implies π 1 (M 1 ) = π 1 (M 2 ) = 2 , and so M = ÊP 3 # ÊP 3 . This manifold has a geometric structure modeled on the geometry Ë 2 × Ê, see [79, p. 457].Case 2: M is prime, but not irreducible. Then M = S 2 × S 1 , see [42, Proposition 1.4] or [45, Lemma 3.13]. In particular, M has a geometric structure modeled on Ë 2 × Ê.Case 3: M is irreducible. We distinguish two subcases:

Interval topology in contact geometry

“…A globally hyperbolic spacetime is strongly causal, so the Alexandrov topology coincides with the manifold topology by Proposition 4.1.Remark 4.7. The Legendrian isotopy class of skies or, equivalently, the Legendrian isotopy class of the fibre of ST * M for a Cauchy surface M ⊂ X is orderable if the universal cover M is non-compact by[9, Remark 8.2] or the integral cohomology ring of M is not isomorphic to that of a compact rank one symmetric space by[16, Theorem 1.2] combined with[10, Proposition 4.7]. In the remaining cases, one can use the fact that this Legendrian isotopy class is always universally orderable by [10, Theorem 1.1] and obtain a substitute for Corollary 4.6 by considering the map x → S x from the (finite) universal cover X of the spacetime X to the universal cover of the Legendrian isotopy class of skies in N X , cf.…”

The Bott–Samelson theorem for positive Legendrian isotopies

“…In this paper we prove the following addition to Theorem 1, which was conjectured in [6]. Theorem 2.…”

“…Frauenfelder, Labrousse and Schlenk [6] proved versions of Theorem 1 and 2 for autonomous Reeb flows, using Rabinowitz-Floer homology. They also proved Theorem 1 using Rabinowitz-Floer homology for positive Legendrian isotopies as stated above.…”

“…The puzzle piece missing in [6] to generalize Theorem 2 from autonomous Reeb flows to positive Legendrian isotopies is the fact that the action functional in the construction is Morse-Bott. We provide this in Lemma 3.2, and thus complete the proof in [6]. The key ingredient is the choice of Hamiltonian, which is elaborated in Lemma 2.1.…”

Contact forms with large systolic ratio in dimension three

“…[HMS15], and the same holds for some statements about the topological entropy of geodesic flows, see e.g. [MS11,FLS15,Alv16]. On the other hand, techniques from contact geometry have been recently found to be useful to address systolic questions in Riemannian and Finlser geometry, such as the local systolic maximality of Zoll metrics, see [APB14,ABHS17a,ABHS17b].…”