Cartan uniqueness theorem on nonopen sets

Cartan's uniqueness theorem does not hold in general for CR mappings, but it does hold under certain conditions guaranteeing extendibility of CR functions to a fixed neighborhood. These conditions can be defined naturally for a wide class of sets such as local real-analytic subvarieties or subanalytic sets, not just submanifolds. Suppose that $V$ is a locally connected and locally closed subset of ${\mathbb{C}}^n$ such that the hull constructed by contracting analytic discs close to arbitrarily small neighborhoods of a point always contains the point in the interior. Then restrictions of holomorphic functions uniquely extend to a fixed neighborhood of the point. Using this extension, we obtain a version of Cartan's uniqueness theorem for such sets. When $V$ is a real-analytic subvariety, we can generalize the concept of infinitesimal CR automorphism and also prove an analogue of the theorem. As an application of these two results we show that, for circular subvarieties, the only automorphisms, CR or infinitesimal, are linear.Comment: 10 pages, updated proof of Lemma 2.2, expanded section 3, and added reference [9

Aquaporin water channels in the nervous system.

The aquaporins (AQPs) are plasma membrane water-transporting proteins. AQP4 is the principal member of this protein family in the CNS, where it is expressed in astrocytes and is involved in water movement, cell migration and neuroexcitation. AQP1 is expressed in the choroid plexus, where it facilitates cerebrospinal fluid secretion, and in dorsal root ganglion neurons, where it tunes pain perception. The AQPs are potential drug targets for several neurological conditions. Astrocytoma cells strongly express AQP4, which may facilitate their infiltration into the brain, and the neuroinflammatory disease neuromyelitis optica is caused by AQP4-specific autoantibodies that produce complement-mediated astrocytic damage