Symplectic embeddings of polydisks

Abstract

We show that the polydisk $P(1,2)$, the product of disks of areas $1$ and $2$, can be symplectically embedded in a ball $B(R)$ of capacity $R$ if and only if $R \ge 3$. Hence, the inclusion map gives the optimal embedding and neither the Embedded Contact Homology nor Ekeland–Hofer capacities give sharp obstructions in this situation. Our proof applies the theory of pseudoholomorphic curves in manifolds with cylindrical ends, and in particular finite energy foliations.

Appendix A: Morse–Bott Riemann–Roch

First, we state the index formula for a real-linear Cauchy–Riemann-type operator acting on sections of a complex line bundle over a punctured Riemann surface $(\dot{S}, j)$ (where $\dot{S} = S{\setminus }\Gamma$, with $\Gamma$ a finite collection of punctures). We define $W^{1,p,\delta }(\dot{S}, E)$ to be the weighted space of sections with exponential decay (for $\delta > 0$) at the punctures. (See the discussion preceding Theorem 4.2 for a review of the definition of an asymptotic operator. We also refer the reader to [1, Sections 2.1–2.3] for a nice exposition of the relevant index computations and discussion of the asymptotic operator.)

Then, we reformulate the index theorem from [19]:

Theorem 7.1

A real Cauchy–Riemann-type operator $T{:} W^{1,p,\delta }(\dot{S}, E) \rightarrow L^{p,\delta } (\dot{S}, \Omega ^{0,1}(E))$ is Fredholm if and only if all of its $\delta$–perturbed asymptotic operators are non-degenerate.

For any choice of cylindrical asymptotic trivialization $\Phi$, the index is given by:

$$\begin{aligned} \mathrm {index}= n \chi ( \dot{S}) + \sum \limits _{z \in \Gamma _{+}} \mathrm{CZ }^{\Phi }( A_{z} + \delta _{z} ) - \sum \limits _{z \in \Gamma _{-}} \mathrm{CZ }^{\Phi }(A_{z} - \delta _{z}) + 2 c^{\Phi }_{1}( E ), \end{aligned}$$

where $n$ denotes the (complex) rank of $E$, $\chi (\dot{S})$ is the Euler characteristic of the punctured surface, $\mathrm{CZ }^\Phi (A)$ is the Conley–Zehnder index of the asymptotic operator $A$, and $c_{1}^{\Phi }(E)$ is the first Chern number of the bundle $E$ relative to the asymptotic trivialization data $\Phi$.

In particular then, for a punctured holomorphic curve $u$ in an almost complex symplectic $4$-manifold $(V, J)$ with symplectization ends, considered modulo domain reparametrizations, the deformation index is given by

$$\begin{aligned} \mathrm {index}(u) = - \chi (\dot{S}) + \sum \limits _{z \in \Gamma _{+}} \mathrm{CZ }^{\Phi }( A_{z} + \delta _{z} ) - \sum \limits _{z \in \Gamma _{-}} \mathrm{CZ }^{\Phi }(A_{z} - \delta _{z}) + 2 c^{\Phi }_{1}( u^*TV ) \end{aligned}$$

where, now, the $\Phi$ are given by trivializations of the contact structure over the asymptotic limits of $u$ and $c^{\Phi }_{1}( u^*TV )$ is, as before, the first Chern number of the bundle relative to the asymptotic trivialization given by $\Phi$.

The data of the asymptotic trivialization gives a preferred class of sections that are asymptotically constant. The first Chern number of the bundle relative to an asymptotic trivialization is then defined to be the signed count of zeros of a generic asymptotically constant section.

If we consider the case of a punctured holomorphic curve $u$ in an almost complex manifold, asymptotic at a given puncture to a closed Reeb orbit $\gamma$ that comes in a Morse–Bott family, the exponential decay condition imposed on sections of $\xi$ corresponds to requiring that the deformations of this curve be among curves that converge to the same orbit. In order to consider the moduli space of curves where the asymptotic limit is allowed to move in the Morse–Bott family, one must consider sections of $E$ that converge exponentially fast to sections of $\gamma ^*E$ that correspond to deformations in the Morse–Bott family. By an argument of elliptic regularity, this is the same as considering the linear operator acting on sections with small exponential growth. Note that the nonlinear Cauchy–Riemann operator does not make sense on this space, even though the linearized one does. We refer the reader to [23, Section 2] for further development of this theory.

To apply Theorem 7.1 to obtain the formulas in Sect. 3.3, we must compute the relevant Conley–Zehnder indices and Chern numbers relative to appropriate trivializations along Reeb orbits in $\Sigma = T^3$. The results can be substituted directly into Theorem 7.1 to obtain Propositions 3.1, 3.2 and 3.3. We start with the Conley–Zehnder indices.

The key tool we will consider is the alternative description of Conley–Zehnder indices in terms of winding numbers of eigenvectors of the corresponding asymptotic operators [8, Section 3]. (Further exposition is also in [7, Theorem 3.9].) The main result is that the asymptotic operator $A$, which is a self-adjoint unbounded operator on $L^2(\gamma ^*E)$, has spectrum given by eigenvalues, each eigenvalue has multiplicity at most 2, and each eigenvector is nowhere vanishing. After choice of a trivialization $\Phi$ of $\gamma ^*E$, an eigenvector has a well-defined winding number. Furthermore, each eigenvector associated with the same eigenvalue has the same winding number, which we denote $w(\lambda )$. By a perturbation theoretic argument (see [8] or references in [7]), each integer is attained by $w(\lambda )$ and each winding number is attained twice—either by an eigenvalue with multiplicity $2$, or by two simple eigenvalues. Furthermore, the winding number $w(\lambda )$ is monotone non-decreasing in $\lambda$. Non-degeneracy of $A$ is equivalent to saying that $0$ is not an eigenvalue of $A$. For a non-degenerate operator $A$, we consider the largest negative eigenvalue $\lambda ^- < 0$ and the smallest positive eigenvalue $0 < \lambda ^+$. Then,

$$\begin{aligned} \mathrm{CZ }^\Phi (A) = w^\Phi (\lambda ^+) + w^\Phi (\lambda ^-). \end{aligned}$$

(4)

In $T^3$, the contact structure is globally trivial. Given explicitly, writing $T^3 = (\mathbb R/ 2\pi \mathbb Z)^3$ with circle valued coordinates $\theta , q_1, q_2$, we have $\alpha = \cos (\theta ) dq_1 + \sin (\theta ) dq_2$ and a trivialization of $\xi$ by $\sin (\theta ) \partial _{q_1} - \cos (\theta ) \partial _{q_2}$ and $\partial _\theta$. With respect to this trivialization, the asymptotic operator at a $T$-periodic orbit is

$$\begin{aligned} - J_0 \frac{\mathrm{d}}{\mathrm{d}t} - T \left( \begin{array}{ll} 0 &{}\quad 0 \\ 0 &{}\quad 1 \end{array}\right) \end{aligned}$$

We note that the spectrum includes $0$ and $-T$. Each of these eigenvalues has multiplicity $1$ and has eigenvectors of winding $0$ with respect to our trivialization.

The perturbed operator corresponding to a negative puncture allowed to move in the Morse–Bott family is the one coming from allowing asymptotic exponential growth. At the level of the asymptotic operator, this corresponds to considering the operator $A+\delta$, thus pushing the entirety of the spectrum to the right. Thus, the eigenvalues of the $\delta$-perturbed operator, on either side of $0$ are $-T+\delta < 0 < 0 + \delta$. These both have winding number $0$, so by Eq. (4), we have Conley–Zehnder index $0$.

For a positive puncture allowed to move in the Morse–Bott family, we again consider a $\delta$-perturbed operator, but now consider the shifted operator $A-\delta$. The spectrum is then shifted to the left, so we now have that $-T-\delta < 0$ and $0-\delta < 0$ are both in the spectrum of $A - \delta$, and have winding number $0$. Thus, the smallest positive eigenvalue must have winding $1$. It follows then that for a positive puncture allowed to move in the Morse–Bott family, the Conley–Zehnder index is $1$.

Now we move to the Chern numbers. Here we need a separate analysis for each of the possible ranges $X{\setminus }L$, $T^* L$ and ${\mathbb R}\times \Sigma$ as described in Sect. 3.1.

For a curve contained in $\mathbb R\times \Sigma$, it is clear that the Chern number of $u^*(T(\mathbb R\times \Sigma ))$ with respect to this asymptotic trivialization is $0$, since this trivialization of the contact structure is global.

For a curve contained in $T^*T^2$, we claim that the Chern number of $u^*T(T^*T^2)$ relative to this trivialization is also $0$. Consider $\Sigma$ as the boundary of the unit disk bundle in $u^*T(T^*T^2)$. Let $p_1, q_1, p_2, q_2$ be canonical coordinates on $T^*T^2$, where $q_1, q_2$ are $T^2$ valued coordinates as before and $p_1, p_2$ the corresponding cotangent coordinates. The trivialization of $T(\mathbb R\times \Sigma )$ induced by the trivialization of $\xi$ corresponds to the trivialization of $T(T^*T^2)|_\Sigma$ by $p_1 \partial _{q_1} + p_2 \partial _{q_2}$, $p_1 \partial _{q_1} + p_2 \partial _{q_2}$, $p_2 \partial _{q_1} - p_1 \partial _{q_2}$ and $p_1 \partial _{p_2} - p_2 \partial _{p_1}$. This defines a map $\Sigma \rightarrow U(2)$ by, at each point, considering the matrix to write this frame with respect to the standard trivialization $\partial _{p_1}, \partial _{q_1}, \partial _{p_2}, \partial _{q_2}$. The matrix has constant determinant $1 \in U(1)$ and thus when restricted to any Reeb orbit represents a contractible loop in $U(2)$. From this, we obtain that any asymptotically constant section of $u^*T(T^*T^2)$ over a curve can be deformed to a nowhere vanishing section, establishing the claim.

Finally, we need to compute $c_1^\Phi (u^*TX)$ for a punctured sphere in $X{\setminus }L$, of degree $d$ and of intersection $e$ with the exceptional divisor $E$, and asymptotic ends on orbits in classes $(k_i, l_i)$. First, observe that this is additive with respect to connect sums, so we only need to compute this for a curve with $d=e=0$, since the first Chern number of $TX|_{{\mathbb C}P^1(\infty )}$ is $3$ and the first Chern number of $E$ is $1$ (and a closed curve of degree $d$ and intersection $e$ with $E$ represents the class $d[{\mathbb C}P^1] - e[E]$). Any such curve can now be deformed and decomposed to a collection of $s$ disks of degree $0$ and $e=0$, each with boundary on a geodesic of class $(k_i, l_i)$, $i=1, \ldots , s$, and thus, these may be taken to be model disks contained in $\mathbb C^2$ with boundary on $L$. Now, observe that the relative first Chern number for a rank $k$ complex vector bundle $V$, $c_1^\Phi (V)$, can be computed by computing the relative first Chern number for the determinant bundle $\Lambda _\mathbb C^k(V)$. Our trivialization of $\xi$ induces a trivialization of $T(T^*T^2)|_{\Sigma }$ and thus of $\Lambda ^2_\mathbb C(T(T^*T^2))|_{\Sigma }$. A straightforward computation shows that this trivialization has winding number $k_i+l_i$ with respect to the obvious trivialization of $\Lambda ^2 (T\mathbb C^2)$, and thus, the relative Chern number is $k_i+l_i$.