Pseudoholomorphic mappings and Kobayashi hyperbolicity
Let \((M^{2n}, J)\) be an almost complex manifold, that is, \(M\) is a smooth manifold of real dimension \(2n\) and \(J\) a smooth field of automorphisms of its tangent space satisfying \(J^2=-1\). A \textit{pseudo-holomorphic mapping } of the unit disk \({\mathbf D}\) into \((M^{2n}, J)\) is a smooth mapping \(f:{\mathbf D}\to M\) satisfying \(J\circ f_*= f_*\circ J_0\), where \(J_0\) is the almost complex structure on \(\mathbf D\) induced by the standard complex structure. By using such mappings, the authors define a pseudo-metric on \(M\) in much the same way as Kobayashi defined a pseudo-metric (the \textit{Kobayashi} pseudo-metric) on an arbitrary complex manifold. After defining this new pseudo-metric (which they also call the \textit{Kobayashi} pseudo-metric), the authors establish several of its elementary properties. These properties are similar to the elementary properties of the Kobayashi pseudo-metric on complex manifolds. These results include for instance the fact that in the case where \(M\) is closed and hyperbolic (hyperbolic meaning that the pseudo-metric is in fact a metric), then the automorphism group of \((M^{2n}, J)\) is finite. In the general case (\(M\) hyperbolic but not necessarily closed), they show that the dimension of the automorphism group is bounded by \(2n+n^2\), and that equality holds if and only if there is an isomorphism between \((M^{2n}, J)\) and the standard complex ball \((B^{2n}, J_0)\). This kind of results, in the case of a complex manifold, are due basically to Kobayashi. The authors deal then in particular with almost complex structures tamed by a symplectic form, that is, the case where there exists a symplectic form \(\omega\) such that \(\omega(X,JX)>0\) for all nonzero tangent vector \(X\) on \(M\). They use this notion to give sufficient conditions for an almost complex domain in \(\mathbb{C}^n\) to be hyperbolic. The results are illustrated by several examples. The authors make some use of Gromov's techniques of pseudo-holomorphic curves.
scientific article; zbMATH DE number 1454702