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Register a new userHigher systolic inequalities for 3-dimensional contact manifolds
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We prove that Besse contact forms on closed connected 3-manifolds, that is, contact forms with a periodic Reeb flow, are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.
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Let $X subset mathbb{R}^4$ be a convex domain with smooth boundary $Y$. We use a relation between the extrinsic curvature of $Y$ and the Ruelle invariant $text{Ru}(Y)$ of the natural Reeb flow on $Y$ to prove that there exist constants $C > c > 0$ independent of $Y$ such that [c < frac{text{Ru}(Y)^2}{text{vol}(X)} cdot text{sys}(Y) < C] Here $text{sys}(Y)$ is the systolic ratio, i.e. the square of the minimal period of a closed Reeb orbit of $Y$ divided by twice the volume of $X$. We then construct dynamically convex contact forms on $S^3$ that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salom~{a}o. These are the first examples of dynamically convex contact $3$-spheres that are not strictly contactomorphic to a convex boundary $Y$.
We study homotopically non-trivial spheres of Legendrians in the standard contact R3 and S3. We prove that there is a homotopy injection of the contactomorphism group of S3 into some connected components of the space of Legendrians induced by the natural action. We also provide examples of loops of Legendrians that are non-trivial in the space of formal Legendrians, and thus non-trivial as loops of Legendrians, but which are trivial as loops of smooth embeddings for all the smooth knot types.
We provide examples of contact manifolds of any odd dimension $geq 5$ which are not diffeomorphic but have exact symplectomorphic symplectizations.
A Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal S^1-orbibundles over integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomological condition this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry. We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result without referring to additional structures. More precisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizer groups. By introducing all relevant orbifold notions in this equivariant way we avoid patching constructions with orbifold charts. As an application, and building on work by Cristofaro-Gardiner--Mazzucchelli, we deduce a complete classification of closed Besse contact 3-manifolds up to strict contactomorphism.
This is an exposition of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. The original work appeared in [1].
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