Gradient flows on nonpositively curved metric spaces and harmonic maps

Comm. Anal. Geom., 6, no. 2, pp. 199-253. 1998
Gradient flows on nonpositively curved metric spaces and harmonic maps
eBay Authors

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. 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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