×

The extrinsic polyharmonic map heat flow in the critical dimension. (English) Zbl 1136.58010

Let \(u:M\to N\) be a \(C^\infty\)-map between compact Riemannian manifolds, where \(N\) is isometrically embedded into some \(\mathbb R^n\). The critical points of the energy \(E(u)\) of \(u \) which is defined by using the covariant derivative of order \(m\) (resp. the \(m\)-th total derivative) are called intrinsic (resp. extrinsic) polyharmonic maps. (Biharmonic maps are examples for intrinsic case, with \(m = 2\)).
In the extrinsic case, concerning the gradient flow of higher order elliptic functionals, by generalizing the results of T. Lamm [Ann. Global Anal. Geom. 26, No. 4, 369–384 (2004; Zbl 1080.58017)] and M. Struwe [Comment. Math. Helv. 60, 558–581 (1985; Zbl 0595.58013)], the author proves here that smooth initial values can be continued as an eternal solution to the flow if \(M\) is of dimension \(< 2m\) and in the critical case \(\dim M = 2m\), a unique eternal weak solution is found, which is smooth except possibly for finitely many items. A singularity can occur only if a “bubble” separates, using up a certain amount of energy.
The proof benefits of interpolation inequalities in higher order heat flows, pointed out by E. Kuwert and R. Schätzle [J. Differ. Geom. 57, No. 3, 409–441 (2001; Zbl 1035.53092)] in their work on the Willmore flow.

MSC:

58E20 Harmonic maps, etc.
35K30 Initial value problems for higher-order parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kuwert E., J. Di{\currency}erential Geom. 57 pp 409– (2001)
[2] DOI: 10.1023/B:AGAG.0000047526.21237.04 · Zbl 1080.58017 · doi:10.1023/B:AGAG.0000047526.21237.04
[3] DOI: 10.1007/s00526-004-0283-8 · Zbl 1070.58017 · doi:10.1007/s00526-004-0283-8
[4] DOI: 10.1007/s005260050072 · Zbl 0946.35017 · doi:10.1007/s005260050072
[5] DOI: 10.1007/BF02567432 · Zbl 0595.58013 · doi:10.1007/BF02567432
[6] DOI: 10.1007/BF01191339 · Zbl 0807.58010 · doi:10.1007/BF01191339
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.