• DOI:10.4310/CAG.1998.V6.N2.A1
• Corpus ID: 2368343

Gradient flows on nonpositively curved metric spaces and harmonic maps

@article{Mayer1998GradientFO,
  title={Gradient flows on nonpositively curved metric spaces and harmonic maps},
  author={Uwe F. Mayer},
  journal={Communications in Analysis and Geometry},
  year={1998},
  volume={6},
  pages={199-253}
}

• U. Mayer
• Published 1998
• Mathematics
• Communications in Analysis and Geometry

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.