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A convergence result for the Gradient Flow of ∫ |A|2in Riemannian Manifolds

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We study the gradient flow of the L2−norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L2−concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result for the flow and the subconvergence to a critical immersion.
  • Institut für Angewandte Mathematik, Westfälische–Wilhelms Universität Münster,
    Einsteinstrasse 62, 48149 Münster, Germany
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