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Description:
The subject of this thesis is the long time behavior of the spinor flow in two situations. The spinor flow is a geometric flow which arises as the negative gradient flow of the functional which associates to a Riemannian metric and a unit spinor field the spinorial energy. The geometric interpretation of this energy depends on the dimension of the manifold. If the manifold has dimension at least three, critical points of this functional are Ricci-flat special holonomy metrics. In dimension two, a pair of a metric and a unit spinor field can be interpreted as a generalized isometric immersion and the spinorial energy as a generalized Willmore energy. The first theme is the stability of the spinor flow in dimension three and up. Given a critical point of the spinorial energy it is found that the spinor flow with initial condition close enough to the critical point exists for all times and converges to a critical point at an exponential rate. The critical points of the spinorial energy restricted to metrics of constant volume are also geometrically very interesting. A volume constrained critical point is shown to be stable in the above sense if it is a minimizer, the critical set at that point satisfies some regularity constraint and the metric has a discrete isometry group. The rate of convergence depends on the regularity of the critical set. The second theme is the behavior of the spinor flow on closed surfaces of positive genus. Given a solution of the spinor flow on a finite interval, one may ask under what conditions the flow can be continued beyond the length of the interval. It is shown that it suffices to assume bounds on the injectivity radius and a certain integral of the second derivative of the spinor field to continue the flow. From this a pointwise criterium can be derived. The proofs are based on a new compactness theorem for families of metrics on closed surfaces of positive genus. ; Gegenstand dieser Arbeit ist das Langzeitverhalten des Spinorflusses in zwei verschiedenen Situationen. Der ...