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MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
71/2017

Asymptotic analysis and qualitative behavior at the free boundary for Sacks-Uhlenbeck $\alpha$-harmonic maps

Jürgen Jost, Lei Liu and Miaomiao Zhu

Abstract

We investigate the possible blow-up behavior of sequences of Sacks-Uhlenbeck $\alpha$-harmonic maps from a compact Riemann surface with boundary to a compact Riemannian manifold $N$ with a free boundary on a closed submanifold $K\subset N$. We discover and explore a new phenomenon, that the connection between bubbles, instead of being a geodesic joining them, can be a more general curve that involves the geometry of both $N$ and $K$. In technical terms, by comparing the blow-up radius with the distance between the blow-up position and the boundary, we define a new quantity, based on which we show a generalized energy identity for the blow-up sequence and give new length formulas for the necks in the case that there is only one bubble occurring at a boundary blow-up point.

Received:
24.10.17
Published:
25.10.17
Keywords:
harmonic map, $\alpha$-harmonic map, free boundary, blow-up, energy identity, neck analysis

Related publications

inJournal
2022 Repository Open Access
Jürgen Jost, Lei Liu and Miaomiao Zhu

Asymptotic analysis and qualitative behavior at the free boundary for Sacks-Uhlenbeck \(a\)-harmonic maps

In: Advances in mathematics, 396 (2022), p. 108105