Invariant flows

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  • These lectures intend to give a self-contained exposure of some techniques for computing the evolution of plane curves. The motions of interest are the so-called motions by curvature. They mean that, at any instant, each point of the curve moves with a normal velocity equal to a function of the curvature at this point. This kind of evolution is of some interest in differential geometry, for instance in the problem of minimal surfaces.

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  • Inspired by Lorenz’ remarkable chaotic flow, we describe in this paper the structure of all C 1 robust transitive sets with singularities for flows on closed 3-manifolds: they are partially hyperbolic with volume-expanding central direction, and are either attractors or repellers. In particular, any C 1 robust attractor with singularities for flows on closed 3-manifolds always has an invariant foliation whose leaves are forward contracted by the flow, and has positive Lyapunov exponent at every orbit, showing that any C 1 robust attractor resembles a geometric Lorenz attractor. ...

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  • An important problem in conformal geometry is the construction of conformal metrics for which a certain curvature quantity equals a prescribed function, e.g. a constant. In two dimensions, the uniformization theorem assures the existence of a conformal metric with constant Gauss curvature. Moreover, J. Moser [20] proved that for every positive function f on S 2 satisfying f (x) = f (−x) for all x ∈ S 2 there exists a conformal metric on S 2 whose Gauss curvature is equal to f . A natural conformal invariant in dimension four is 1 Q = − (∆R −...

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  • Many classical integrable systems (like the Euler, Lagrange and Kowalewski tops or the Neumann system) as well as finite dimensional reductions of many integrable PDEs share the property of being algebraically completely integrable systems4. This means that they are completely integrable Hamiltonian systems in the usual sense and, moreover, their complexified invariant tori are open subsets of complex Abelian tori on which the complexified flow is linear. To such systems the powerful algebro-geometrical techniques may be applied...

    pdf260p maket1311 16-10-2012 14 1   Download


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