Abstract
In this paper we prove existence and uniqueness of the Björling problem for the class of immersed surfaces in \(\mathbb {R}^3\) whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a Möbius strip for an arbitrary large class of prescribed functions. In particular, we use the Björling problem to construct the first known examples of self-translating solitons of the mean curvature flow with the topology of a Möbius strip in \(\mathbb {R}^3\).
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Acknowledgements
The author was partially supported by MICINN-FEDER Grant No. MTM2016-80313-P, Junta de Andalucía Grant No. FQM325 and FPI-MINECO Grant No. BES-2014-067663. The author wants to express his gratitude to his Ph.D. advisor Pablo Mira, for fruitful conversations about this topic.
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Bueno, A. The Björling problem for prescribed mean curvature surfaces in \(\mathbb {R}^3\). Ann Glob Anal Geom 56, 87–96 (2019). https://doi.org/10.1007/s10455-019-09657-w
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DOI: https://doi.org/10.1007/s10455-019-09657-w