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Let $f$ be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of $f$ under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is isotopic to a constant map.

This work uncovers the tropical analogue, for measured laminations, of the convex hull construction in decorated Teichm¨uller theory; namely, it is a study in coordinates of geometric degeneration to a point of Thurston’s boundary for Teichm¨uller space. This may offer a paradigm for the extension of the basic cell decomposition of Riemann’s moduli space to other contexts for general moduli spaces of flat connections on a surface. In any case, this discussion drastically simplifies aspects of previous related studies as is explained. Furthermore, a new class of measured laminations relative to an ideal cell decomposition of a surface is discovered in the limit. Finally, the tropical analogue of the convex hull construction inMinkowski space is formulated as an explicit algorithm that serially simplifies a triangulation with respect to a fixed lamination and has its own independent interest.

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In this paper, we sketch the proof of the extension of the stability theorem of the Minkowski space in General Relativity done explicitly in [6] and [7]. We discuss solutions of the Einstein vacuum (EV) equations (obtained in the author’s Ph.D. thesis ([6]) in 2007). We solve the Cauchy problem for more general, asymptotically flat initial data than in the pioneering work of D. Christodoulou and S. Klainerman [21] or than in any other work. Moreover, we describe precisely the asymptotic behavior. Our relaxed assumptions on the initial data yield a spacetime curvature which is not bounded in L1(M). As a major result, we encounter in our work borderline cases, which we discuss in this paper as well. The fact that certain of our estimates are borderline in view of decay indicates that the conditions in our main theorem are sharp in so far as the assumptions on the decay at infinity on the initial data are concerned. Thus, the borderline cases are a consequence of our relaxed assumptions on the data, [6, 7]. They are not present in the other works, as all of them place stronger assumptions on their data. We work with an invariant formulation of the EV equations. Our main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime. This result generalizes the work of D. Christodoulou and S. Klainerman [21].