Abstract
We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy and show the so-called generalized energy identity. We find conditions that are both necessary and sufficient for the compactness in W 1,2 and C 0 modulo bubbles of sequences of such maps.
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Acknowledgments
This paper is developed from a portion of the author’s Ph.D. thesis [40]. He would like to thank his advisor, Prof. Jürgen Jost, for encouragement and inspiration. He would also like to thank Prof. Michael Struwe for conversations and advice, and Brian Clarke for discussions and help.
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M. Zhu was supported by IMPRS “Mathematics in the Sciences” and the Klaus Tschira Foundation.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Zhu, M. Harmonic maps from degenerating Riemann surfaces. Math. Z. 264, 63–85 (2010). https://doi.org/10.1007/s00209-008-0452-0
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DOI: https://doi.org/10.1007/s00209-008-0452-0