Gromov hyperbolicity of strongly pseudoconvex almost complex manifolds. (English) Zbl 1327.32040
Consider an almost-complex manifold \((M,J)\), and a relatively compact domain \(D\subset M\) with smooth strictly \(J\)-pseudoconvex boundary. This means that \(D=\{\rho<0\}\) for some smooth defining function \(\rho\) (whose gradient never vanishes on \(\partial D\)), which is strictly \(J\)-plurisubharmonic in a neighborhood of \(\overline{D}\). As in the integrable case, one can use \(J\)-holomorphic discs to define a Kobayashi distance \(d\) on \(D\). The main theorem of the paper is that \((D,d)\) is a Gromov-hyperbolic metric space. Furthermore, they show that \(\partial D\) is connected.
Reviewer: Valentino Tosatti (Evanston)
MSC:
| 32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |
| 32Q60 | Almost complex manifolds |
| 32T15 | Strongly pseudoconvex domains |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
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