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We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, we give a unified vanishing theorem of mean curvature for these entire graphs of constant mean curvature., 11 pages, no figure, minor revision

The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean three-space, weakly complete improper affine spheres in the affine three-space and weakly complete flat surfaces in the hyperbolic three-space. For this purpose, we give an effective curvature bound for a specified conformal metric on an open Riemann surface., 13 pages, to appear in Mathematische Zeitschrift

Acetylcholinesterase (AChE) at the neuromuscular junction (NMJ) is anchored to the synaptic basal lamina via a triple helical collagen Q (ColQ) in the form of asymmetric AChE (AChE/ColQ). We exploited the proprietary NMJ-targeting signals of ColQ to treat congenital myasthenia and to explore the mechanisms of autoimmune myasthenia gravis (MG). Mutations in COLQ cause congenital endplate AChE deficiency (CEAD). First, a single intravenous administration of adeno-associated virus serotype 8 (AAV8)-COLQ to Colq−/− mice normalized motor functions, synaptic transmission, and partly the NMJ ultrastructure. Additionally, injection of purified recombinant AChE/ColQ protein complex into gluteus maximus accumulated AChE in non-injected forelimbs. Second, MuSK antibody-positive MG accounts for 5–15 % of MG. In vitro overlay of AChE/ColQ to muscle sections of Colq−/− mice, as well as in vitro plate-binding of MuSK to ColQ, revealed that MuSK-IgG blocks binding of ColQ to MuSK in a dose-dependent manner. Passive transfer of MuSK-IgG to wild-type mice markedly reduced the size and intensity of ColQ signals at NMJs. MuSK-IgG thus interferes with binding of ColQ to MuSK. Elucidation of molecular mechanisms of specific binding of ColQ to NMJ enabled us to ameliorate devastating myasthenic symptoms of Colq−/− mice and also to reveal underlying mechanisms of anti-MuSK-MG.