Uniqueness in Rough Almost Complex Structures, and Differential Inequalities

Rosay, Jean-Pierre

Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2261-2273.

L’étude des applications $J$-holomorphes conduit à l’étude des inéquations $|\frac{\partial u}{\partial \overline{z}}|\le C|u|$, et $|\frac{\partial u}{\partial \overline{z}}|\le ϵ|\frac{\partial u}{\partial z}|$. La première inéquation est facile à utiliser. La seconde, qui intervient naturellement dans les structures non lisses, est plus difficile. De façon intéressante, le cas d’applications vectorielles $u$ est différent du cas scalaire. Les questions étudiées ont trait à l’unicité de prolongement et aux zéros isolés. Parmi les résultats, il est démontré que, pour les structures presque complexes de classe Hölderienne $\frac{1}{2}$, toute courbe $J$-holomorphe constante sur un ouvert non vide, est constante. Ceci est en contraste avec des exemples immédiats de non-unicité.

The study of $J$-holomorphic maps leads to the consideration of the inequations $|\frac{\partial u}{\partial \overline{z}}|\le C|u|$, and $|\frac{\partial u}{\partial \overline{z}}|\le ϵ|\frac{\partial u}{\partial z}|$. The first inequation is fairly easy to use. The second one, that is relevant to the case of rough structures, is more delicate. The case of $u$ vector valued is strikingly different from the scalar valued case. Unique continuation and isolated zeroes are the main topics under study. One of the results is that, in almost complex structures of Hölder class $\frac{1}{2}$, any $J$-holomorphic curve that is constant on a non-empty open set, is constant. This is in contrast with immediate examples of non-uniqueness.

Classification : 32Q65, 35R45, 35A02
Keywords: $J$-holomorphic curves, differential inequalities, uniqueness
Mot clés : courbes $J$-holomophes, inégalités différentielles, unicité

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