# Almost complex structures on a 4-ball that are not tamed

Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called tamed if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $v$.

Question. Is there an almost complex structure $J$ on a closed ball $B^4$, such that any $C^{\infty}$-small perturbation of $J$ is not tamed by any symplectic form? (I assume that $J$ behaves nicely on the boundary of $B^4$, in particular it is smooth there).

Remark. Clearly, if such $J$ exists on $B^4$, it would exists on any $4$-manifold admitting an almost complex structure.

This question is a follow-up to the following one, where a global obstruction for "tamebility" was found for $\mathbb CP^2$ Almost complex structures on $\mathbb CP^2$ that are not tamed

The following construction provides plenty of examples of non-tamed almost complex structures:

Consider an almost complex structure $J$ on $B^4$ for which the contact hyperplanes of the overtwisted contact structure on $S^3=\partial B^4$ (in the same homotopy class as the standard tight contact structure) become $J$-complex. If there was a taming symplectic form, we would have constructed a so-called weak symplectic filling of an overtwisted contact manifold. However, This is not possible by a result of Eliashberg and Gromov.

Finally, observe that the contact condition is open, and by Gray's stability also a perturbation of $J$ would be tangent to $\partial B^4$ along some tangent hyperplane distribution being an overtwisted contact structure.

• What are you using to construct such a $J$? Certainly if such a $J$ exists it satisfies the conditions of the question, but I don't see where the construction is coming from.
– PVAL
Mar 8, 2017 at 23:22
• Since I am assuming that the contact structure is homotopic to the standard contact structure on $S^3$, the almost complex structure can be taken to be homotopic to the standard integrable complex structure on the ball (we just need to deform the complex tangencies along the boundary). Mar 8, 2017 at 23:26
• In the definition of weak filling one asks that $\omega$ is convex at the boundary. How do you see that any $\omega$ taming $J$ has to be convex at $S^3$? Mar 8, 2017 at 23:29
• The contact condition means that the exterior differential of any contact form (which in this case is equal to the Levi form of $J$) is nondegenerate on the contact planes. Mar 8, 2017 at 23:31
• @aglearner There is also an argument of Eliashberg which shows given that the form is zero in cohomology, that any weak filling of a contact structure can be modified to a strong filling of the same contact structure. This is a short and straightforward argument in the "A few remarks..." paper (prop 4.1 here arxiv.org/pdf/math/0311459.pdf).
– PVAL
Mar 8, 2017 at 23:43