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Andreas Gastel

The extrinsic polyharmonic map heat flow in the critical dimension

We consider the gradient flow of higher order elliptic functionals of the type ∫_M ² for maps from a compact Riemannian manifold M to ℝⁿ with image contained in another compact manifold, called the extrinsic polyharmonic map heat flow. We prove that smooth initial values can be continued as an eternal solution to the flow if M is of dimension < 2m. In the critical case dim M = 2m, we find a unique eternal weak solution which is smooth except possibly for finitely many times. A singularity can occur only if a “bubble” separates, using up a certain amount of energy.

Advances in Geometry, Walter de Gruyter

Print ISSN: 1615-715X
Volume: 6, 10/2006
Pages: 501 - 521

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