Mike Usher

Lagrangian submanifolds of $\mathbb CP^n$

Usher mostly describes constructions of Lagrangians. (Joint work with Joel Oakley.)

Some nice examples:

• Chekanov torus in $\mathbb C^2$ (comes in 1-parametric family by "radius"). Chekanov-Eliashberg-Polterovich: these are Lagrangian isotopic to the standard "Clifford" tori, but not Hamiltonian isotopic. (In particular, same Maslov and area classes.)
• Both of these can then be shoved in $\mathbb{C}P^2$ and in $S^2 \times S^2$, and are monotone for specific values of the parameter $r$. (Non-displaceable)
• Polterovich torus in $T^*S^2$. Albers and Frauenfelder: monotone & non-displaceable.
• Higher dimensional version $P^r_{k,m}$ $k \ge 0$, $m \ge 1$.  Take $S^k \times \{0\}$ and $\{ 0 \} \times S^m$ in $S^{k+m+1}$. Look at the union of lifts of geodesics of speed $r$ passing through these two sets. This is monotone, non-displaceable Lagrangian diffeo to $S^1 \times S^k \times S^{m}/ (-id, -id, -id)$ (Polterovich torus corresponds to taking $k=0$ and $m=1$.)
• Passing to a quotient, can get these to live in $T^* \mathbb{R}P^{k+m+1}$. Embedding these in $S^2 \times S^2$ or in $\mathbb CP^n$, get shove these in here.
• Biran construction: $\mathbb{C}P^n = \mathbb{R}P^n \cup (\text{disk bundle over }Q_n)$. 

The main theorem is an alternative construction of Chekanov in dimension 4 that generalizes differently. In (much?) higher dimensions, however, this construction generalizes to a surprising displaceable monotone Lagrangian.

In dimension 6, get Lagrangian torus and Lagrangian $S^1 \times S^2$ whose displaceability are unknown, but that have vanishing Floer homology over $\mathbb Z$.

Pedro Salamão

(joint work with Umberto Hryniewicz and Joan Licata)

Question: determine the topology of a co-oriented contact manifold from the Reeb flow of defining contact form.

Motivating example: characterisation of the tight 3-sphere. If $\alpha$ is dynamically convex & non-degenerate and admits a closed Reeb orbit $P$ so that $P$ is unknotted, has self-linking number $-1$ and the CZ index is $3$. Then $M$ is the tight 3-sphere. (Converse also true.)

(If, $P$ as above, then is binding of an open book decomposition of $M$ with disk-like pages. This theorem is proved by constructing this open book.)

$L(p,q)$ with standard contact structure $\xi_{\mathrm{std}}$.

Let $K$ be a knot. Let $p \ge 1$. Then a $p$-disk for $K$ is a disk whose interior embeds in $M \setminus K$ and $u|_{\partial D}$ is a $p$-cover of $K$.

If $K$ admits a $p$-disk, it is $p$-unknotted and is an order $p$ rational unknot.

The image of the Hopf fibres project to $p$-unknots $K_0$ and $K_1$ inside $L(p,q)$ with self-linking number $-p$.

If you look at the $p$-disk with boundary on $K_0$, and go around the boundary annulus once, you come back to a sheet that is shifted over by $-q$. (Call this the monodromy.)

For $K_1$ the monodromy is $q'$ where $qq' = 1 (mod p)$.

Theorem:  (Hryniewicz, Licata, Salamão)
 $(M, \xi)$ with $c_1(\xi) = 0$ on $\pi_2(M)$. Let $p \in \mathbb N^*$

Then, contactomorphic to $L(p,q), \xi_{\mathrm{std}}$ if and only if $\xi$ is the kernel of a non-degen contact form $\alpha$ admitting a prime closed Reeb orbit $K$ such that

• $K$ is p-unknotted, $CZ( K^p ) \ge 3$, self-linking $K^p = -p$ and monodromy is $-q$
• every contractible Reeb orbit $P \subset M \setminus K$ with CZ index 2 is not contractible in $M \setminus K$.

Strategy of proof: make $p$ disk for $K$ become a fast pseudoholomorphic plane with $\pi du \ne 0$. Then, foliate complement with these planes. Technical condition on index $2$ orbits gives rational open book and then result.

Great idea: $C^0$ perturb the disk we start with, to obtain the following picture:

• characteristic foliation has only one nicely elliptic singular point $e$ and all leaves reach the boundary transversely (recall the boundary is a transverse knot)
• transverse to $R$ near the boundary (this uses the fact the CZ index is at least 3 for the $p$-fold cover of the knot)
• the disk does not contain any periodic orbits of the Reeb flow (i.e. only one is the boundary)
• Local model near $e$ with nice Darboux coordinates.

Ah, idea now is to consider the Bishop disks near singularity. One boundary component is the constant disk in $e$. Other one is a building. Chasing through what happens, must break on orbits with index at least $2$. This index $2$ orbit condition allows us to exclude most of what can happen. From this, we conclude that the other boundary component of the building is the plane we want.

Additional applications:

Geodesic flows and restricted planar 3 body problem: both live on $\mathbb RP^3 = L(2,1)$.

Surface of revolution: double cover of meridians have CZ index 3 or 4. Linking argument means hypotheses satisfied for other orbits.

Finsler metrics with pinched flag curvature (pinching based on reversibility). Paternain says is dynamically convex and Rademacher says there is a simple closed geodesic. Some work says this satisfies hypotheses of the theorem.

planar restricted 3 body problem with circular earth-moon orbit.
Albers, Fish, Frauenfelder, Hofer, Paternain, von Koert: subsets of these in various papers show that low energy gives two components each of which regularizes to a Reeb flow on $L(2,1)$. Obtain disk like surfaces of section for component near the moon (by working on $S^3$). Also show dynamical convexity but not geometric convexity near the first critical value for rotating Kepler problem. Retrograde orbit has index $3$ but the direct orbit has higher index.

Possible application of this rational open book stuff: work with prescribed orbit with high index, or for perturbations that destroy dynamical convexity.

Miguel Abreu

(work with Macarini)

Problem: existence of elliptic orbits (i.e. all eigenvalues of return map on unit circle)

Elliptic closed orbits give many dynamical consequences (e.g. KAM tori; transverse homoclinic connections?; positivity of topological entropy)

Conjecture of Ekeland: every convex hypersurface in $\mathbb{R}^{2n}$ carries an elliptic closed orbit.

In general, still open.

Thm (Dall Antionio, D'Oriofrio & Ekeland '95)
If $M \subset \mathbb R^{2n}$ convex and invariant by antipodal map, then carries an elliptic closed orbit.

Goal: understand and generalize this using contact homology.

Thm (Abreu-Macarini)

Consider a prequantization manifold $(M, \xi = \ker \beta)$ where $\beta$ generates a free $S^1$ action.

Let $\alpha$ be some other contact form. Let $G \subset S^1$ be a finite subgroup. Suppose $\alpha$ is invariant under the $G$ action.

Let $a$ be the free homotopy class of the fibre of $M$.

Assume one of the following holds:

• $M/S^1$ admits a Morse function with only even indices
• $a^j \ne 0$ for all $j \ge 1$.

Then for every $G$-invariant positively [negatively] $a$-dynamically convex form $\alpha$ has an elliptic orbit representing the class $a$ with $CZ(\gamma) = k_-$ [respectively $k_+$].

Definition: Fix a free homotopy class of loops $a$. Associate to $a$ the minimal degree $k_-$ and maximal degree $k_+$ in which contact homology is non-zero for this class.

We say positively $a$-dynamically convex if all closed orbits in the free class $a$ have at least CZ index $k_-$. (Negatively, at most $k_+$) [ok, modulo the grading shift or whatever]

Applications:
Suppose have metric $g$ on $S^2$ with $\frac{1}{4} \le \kappa \le 1$. Then there is an elliptic closed geodesic.

Proof: Harris-Paternain proved that this curvature condition means the geodesic flow lifts to dynamically convex on $S^3$ + perturbation.

Rmk different proof exists (Ballmann, Thorbergsson & Ziller) and furthermore maybe better result due to Contrera and Oliveira (http://arxiv.org/abs/math/0312005)

Also an application to magnetic flows. $T^*N$ with $\omega_0 + \pi^* \Omega$ for a closed 2-form $\Omega$. Take Hamiltonian from geodesic flow.

Benedetti: If $N$ is a surface of genus $g \ne 1$, and $\Omega$ is symplectic on $N$, then small energy levels are contact-type. If $g=0$, lifts to $S^3$ and is dynamically convex.

If $g > 1$, there exists a $|\chi_N|$ covering $\tilde M \to H^{-1}(c)$ such that $\tilde M$ is prequant
and the lift to $\tilde M$ is negatively dynamically convex.

Toric contact manifolds:
$(M^{2n-1}, \xi)$ has $\mathbb T^{n}$ action. (equivalently: symplectization is toric symplectic cone)

"Good" toric contact manifolds: $\mathbb{C}^d // K$ for some $K \subset \mathbb T^d$ linear sub-torus.

Miguel made an interesting remark about being able to construct examples in higher dimensions with finitely many periodic orbits. This is related to a question from Michael's talk. He wondered then if the correct generalization of Michael's conjecture is to suggest that the only manifolds with finitely many distinct orbits are these toric contact manifolds.

Marco Mazzucchelli

Isometry invariant geodesics

(a question from Grove?)

Key idea: Marco is interested in geometrically distinct geodesic arcs. The key tool is to consider the local homology of a critical point/set. The idea is to look at the local homology of iterates.

(For instance, to find infinitely many geodesics from p to q, the variational problem doesn't have a $\mathbb{Z}/m$ symmetry anymore, so cannot use the symmetry anymore)

Suppose I is an isometry. Want the existence of $\tau$ so $I( \gamma(t) ) = \gamma( t + \tau)$.

If $I$ is the identity, then just looking for closed.

Question: when do we have infinitely many invariant geodesics?

Thm (Grove): If there exists a non-closed $I$ invariant geodesic then the closure of $\gamma(\mathbb R)$ contains uncountably infinitely many $I$-invariant geodesics.

It therefore remains to study periodic $I$ invariant geodesics.

Variational setting:
$$ \Lambda( M, I )
=
\{
\gamma \colon \mathbb R \to M
\, | \,
I ( \gamma(t) )
= \gamma (t+1)
\quad \forall t \in \mathbb R
\}
$$

Has $\mathbb R$ action by time shift.

Energy $E( \gamma ) = \int_0^1 | \dot \gamma |^2 dt$

The critical points of $E$ are $I$-invariant geodesics. The critical orbit of $\gamma$ is $\mathbb R$ if not closed, or $S^1$ if closed.

Warning: every such $\gamma$ comes with infinitely many critical points of $E$.

$\tau \in \mathbb R$. $\gamma^\tau(t) = \gamma( \tau t)$.
For each $\gamma$ that is a periodic critical point of $E$ of period $p \ge 1$, also have $\gamma^{mp + 1}$ for any $m \in \mathbb{N}$.

Study Morse-theoretic properties of this set $\{ \gamma^{mp+1} \}$.

Grove-Tanaka: show that $m \mapsto nul( \gamma^{mp+1})$ is bounded,
$m \mapsto index( \gamma^{mp+1} )$ groes linearly and finally $m \mapsto rank L_*( \gamma^{mp+1})$ is bounded.

Theorem: Assume $\pi_1(M) =0$ and $I$ is homotopic to the identity and $H^*(M, \mathbb Q)$ has more than 1 generator.

Then, there exist infinitely many $I$-invariant geodesics.

(RMK: I homotopic to the identity allows us to identify the homology of $\Lambda(M, I)$ with the loop space homology.)

(When $I$ is the identity, this recovers Gromoll-Meyer theorem.)

Theorem (Mazzucchelli):
Assume:

•  $M = M_1 \times M_2$ (as manifolds... do not require a product metric).
•  $rank H_1(M, \mathbb Z) > 0$ and $dim M_2 \ge 2$.
•  $I$ is homotopic to the identity.

Then, there exist infinitely many $I$-invariant geodesics.


Theorem (M):

in a general setting: if exist $\gamma$ closed $I$ invariant and $d \ge 2$ such that $L_*( \gamma^{mp+1})$ is non-trivial for infinitely many $m$ then there are infinitely many $I$ invariant geodesics.


Marco remarks that the ideas are closely related to Umberto's discussion of symplectically degenerate maxima.

IMPA conference: Michael Hutchings

Questions:

(1) Can we improve lower bound on number of closed Reeb orbits?
(2) Can we get existence with a priori upper bound on length?
(3) Can we replace Reeb vector fields with something else?

Theorem for (1): Hutchings & Taubes. 2008. $M^3$ contact, compact, oriented. At least two distinct orbits for non-degenerate. If manifold is not $S^3$ or lens space, then have at least 3 (again, assume non-degenerate).

Theorem (Cristofaro-Gardiner & Gripp-Ramos 2012)
Every possibly degenerate contact form on a closed oriented $M^3$ has two embedded Reeb orbits.

Conjecture (Ekeland-Hofer): every star-shaped hypersurface in $\mathbb{R}^{2n}$ has at least $n$ closed Reeb orbits.

If contact form non-degenerate, conjecture follows by an argument from CH algebra. (Jean Gutt explained this to me earlier. [INSERT EXPLANATION HERE])

Colin-Honda 2008: for many contact structures $\xi$ on closed oriented $Y^3$, every contact form $\ker \alpha = \xi$ has infinitely many distinct orbits. (Uses linearized CH.)

Conjecture: $Y^3$ closed oriented connected different from $S^3$ or a lens space, then every contact form on $Y$ has infinitely many embedded Reeb orbits.

Definition: volume of $(Y, \alpha)$ is $\int \alpha \wedge d\alpha$.

Question: Does there exist $c(Y, \xi)$ so $A(\gamma)^2 \le c(Y, \xi) vol(Y, \alpha)$ for some closed $\gamma$?

Stronger version: $c(Y, \xi) \le 1$.

$\newcommand{\vol}{\mathrm{vol}}$

Examples: Take a prequantization bundle with Euler class $e$ and $\omega = 2\pi e$. Take some positive function $f \colon \Sigma \to \mathbb{R}$. Then look at $f \alpha_0$.

Conjecture holds for this example, since $ vol( Y, f \alpha_0) = 2 \pi \int_{\Sigma} f^2 \omega
\ge 4 \pi^2 e \min(f^2)$. Action of critical fibres are $2\pi f(x)$ (where $x$ is a critical point of $f$.)

Another example: relationship to systolic inequalities.
Take $g$ a Riemannian metric on $S^2$ with area $4 \pi$. Then, action of a Reeb orbit agrees with the length of the corresponding geodesic.

Go to double cover $S^3$ and take pull-back contact form. This gives volume $16 \pi^2$. Reeb orbits are even geoedeics (geods which determine plane curves with odd rotation number). The conjecture then says there is an even geodesic of length $\le 4 \pi$.

This is sharp for the round metric (double cover of a great circle on $S^2$).

Calabi-Croke example: two flat equilateral triangles glued together to give metric of area $4 \pi$ and no closed geodesics of length $\le 2 \pi$. However! exists an even geodesic of length $\le 4 \pi$.

Sketch of proof : suppose $Y$ has only two Reeb orbits. Then, some trickery and work shows that you can construct a foliation from $u$ map. (Foliation by cylinders) This gives a genus 1 Heegaard splitting.

Some consequences of isomorphism to SW Floer homology:
non-canonical $\mathbb{Z}$ grading when $c_1(\xi) + 2 PD(\Gamma)$ is torsion.

For sufficiently negative grading, $ECH$ vanishes. For sufficiently large grading, $U : ECH_* \to ECH_{*-2}$ is isomorphism. Infinitely many $*$ so $ECH_*$ non-zero.

Filtered ECH: look at subcomplex filtered by action. This depends heavily on the contact form.

Michael with Daniel & Vinicius (2012) : consider sequence of classes so that $U \sigma_{k+1} = \sigma_k$. Then they show $\lim_k c_{\sigma_k}(Y, \alpha)^2/ k = 2 \vol(Y, \alpha)$.

Another theorem with the usual suspects: Suppose $(Y, \alpha)$ closed connected 3-man. Then, either exists a short Reeb orbit OR at least 3 embedded Reeb orbits.

Idea uses some trickery with the $U$ map. I guess the key idea is that having 2 orbits means that the picture looks an awful lot like the ellipsoid.

IMPA conference: Mihai Damien

Mihai discussed some results about the topology of Lagrangian submanifolds and pointed out the difference between monotone (rigid) and the flexible results of Yasha, Tobias, Ivan and Emmy. His key method: Lifted Floer homology.

Morse Theory

Want to do Morse theory on the universal cover of a manifold. Have an infinite dimensional complex. Define $d \tilde x = \sum \tilde y$, where the sum is over lifts of gradient lines. If you fix one lift $\tilde x$ in each class, then you can reinterpret this as a complex generated by critical points but now over $R[ \pi_1(L) ]$ ($R$ is the base ring.)

Floer theory

Look at image of $u(s,0)$ in differential to tell us how to interpret going between two different lifts of a given intersection point of $L, L'$. Problem: if $N_L =2$, $d^2 \ne 0$ in this case. Can bubble disks. The issue is then that
(Thm, Oh) $$ d^2 x = \sum (disks) x$$, where we consider the count of Maslov 2 disks passing through $x$.

*Prop* (Oh): L monotone, $N_L = 2$, and $L, L'$ Hamiltonian isotopic. Then, the number passing through $x$ on $L$ and the number on $L'$ passing through $x$ have the same parity.

Therefore, at least $mod 2$, we have $d^2 = 0$, at least in classical Floer homology.

 In this lifted picture on the universal cover, we have some trouble:
If $g \in \pi_1(L, x)$ has $g = [w( \partial D]$, then this disk on $L$ counts as $g$ and a similar disk on $L'$ counts as 1. Thus, $$d^2(\tilde x) = \sum_{g \in \pi_1(L) } (1+g) \#_2^g(x).$$
 And where $\#_2^g(x)$ is the [mod 2] number of Maslov 2 disks whose boundaries are in homotopy class $g$.
 In particular then, have well-defined if $N_L \ge 3$.

 If not defined: Fix some $g$ so $\#_2^g = 1$. Now, take $\gamma \in \pi_1(L, x)$. We now construct some kind of cobordism to go from $g$ to $\gamma g \gamma^{-1}$. (Idea is to move the marked point in the evaluation map around the loop $\gamma$. It is important that we consider the based fundamental group and not the free one.)
 Thus, the set $\{ \gamma g \gamma^{-1} \, | \, \gamma \in \pi_1(L, x) \}$ is finite. (Gromov compactness + rigidity)
Thus, $Z(g)$ is of finite index.

UPSHOT: defined/not-defined is a key tool in Mihai's work. This is a really cool variation of the usual pain we have to say something is well-defined.

Dan Popovici (Toulouse) - “Deformation Limits of Compact Kaehler Manifolds.”

The speaker considered the following situation: suppose $\pi \colon X \to D$ is a proper holomorphic submersion. Let $n$ be the complex dimension of the fibre, and suppose fibres compact.

Example of Hironaka '62:

explicit example where fibres $X_t$, $t \ne 0$ are projective but $X_0$ is not Kähler.

Conjecture:

If $X_t$ Kähler, then $X_0$ should be a class $\mathcal C$ manifold.

Definition:

$X$ is class $\mathcal C$ if exists $\mu \colon \tilde X \to X$ proper, holomorphic, bimeromorphic and $\tilde X$ is Kähler.

i.e. should think of $X$ as blown up Kähler.

Theorem

(Demailly, Pali 2004??) Suppose $X$ compact, cplx. $X$ is of class $\mathcal C$ if and only if there is a Kähler current $T$. (i.e. $T$ is a $(1,1)$ current, $dT = 0$ and $T \ge \epsilon \omega$ where $\omega$ is a $C^\infty$ $(1,1)$ form on $X$.)
The goal is then to produce a Kähler current on $X_0$. Dan then showed the problem reduced to studying a conjecture of Demailly on "transcendental Morse inequalities".