We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface into IRn, n = 3, 4, if there is one conformal immersion with Willmore energy smaller than a certain bound Wn,p depending on codimension and genus p of the Riemann surface. For tori in codimension 1, we know W3,1 = 8 .

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].

Let X CPN be a smooth subvariety. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X which attempts to deform the given embedding into a balanced one. If L X is an ample line bundle, considering embeddings via H0(Lk) gives a sequence of balancing flows. We prove that, provided these flows are started at appropriate points, they converge to Calabi flow for as long as it exists. This result is the parabolic analogue of Donaldsons theorem relating balanced embeddings to metrics with constant scalar curvature [12]. In our proof we combine Donaldson乫s techniques with an asymptotic result of Liu and Ma [17] which, as we explain, describes the asymptotic behavior of the derivative of the map FS . Hilb whose fixed points are balanced metrics.

Let U(n) be the unitary group, and u(n)¤ the dual of its Lie algebra, equipped with the Kirillov Poisson structure. In their 1983 paper, Guillemin-Sternberg introduced a densely defined Hamiltonian action of a torus of dimension (n−1)n/2 on u(n)¤, with moment map given by the Gelfand-Zeitlin coordinates. A few years later, Flaschka-Ratiu described a similar, ‘multiplicative’ GelfandZeitlin system for the Poisson Lie group U(n)¤. By the Ginzburg-Weinstein theorem, U(n)¤ is isomorphic to u(n)¤ as a Poisson manifold. Flaschka-Ratiu conjectured that one can choose the Ginzburg-Weinstein diffeomorphism in such a way that it intertwines the linear and nonlinear Gelfand-Zeitlin systems. Our main result gives a proof of this conjecture, and produces a canonical Ginzburg-Weinstein diffeomorphism.

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