Abstract. We show existence and uniqueness of classical solutions for the motion of immersed hypersurfaces driven by surface diffusion. If the initial surface is embedded and close to a sphere, we prove that the solution exists globally and converges exponentially fast to a sphere. Furthermore, we provide numerical simulations showing the creation of singularities for immersed curves.

The notion of gradient flows is generalized to a metric space setting without any linear structure. The metric spaces considered are a generalization of Hilbert spaces, and the properties of such metric spaces are used to set up a finite-difference scheme of variational form. The proof of the Crandall-Liggett generation theorem is adapted to show convergence. The resulting flow generates a strongly continuous semigroup of Lipschitz-continuous mappings, is Lipschitz continuous in time for positive time, and decreases the energy functional along a path of steepest descent. In case the underlying metric space is a Hilbert space, the solutions resulting from this new theory coincide with those obtained by classical methods. As an application, the harmonic map flow problem for maps from a manifold into a nonpositively curved metric space is considered, and the existence of a solution to the initial boundary value problem is established.

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