Abstract
Given a unit vector \({\textbf {v}}\in {\mathbb {R}}^3\) and \(\lambda \in {\mathbb {R}}\), a translating \(\lambda \)-soliton is a surface in \({\mathbb {R}}^3\) whose mean curvature H satisfies \(H=\langle N,{\textbf {v}}\rangle +\lambda \), where N is the Gauss map of the surface. In this paper, we extend the phenomenon of instability of Plateau–Rayleigh for translating \(\lambda \)-solitons of cylindrical type, proving that long pieces of these surfaces are unstable. Specifically, we will provide explicit bounds on the length of these unstable surfaces in terms of \(\lambda \) and the amplitude of the generating curve. It will be also proved that a graphical translating \(\lambda \)-soliton is a minimizer of the weighted area in a suitable class of surfaces with the same boundary and the same weighted volume.
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Funding
Antonio Bueno has been partially supported by the Project P18-FR-4049 and CARM, Programa Regional de Fomento de la Investigación, Fundación Séneca- Agencia de Ciencia y Tecnología Región de Murcia, reference 21937/PI/22. Rafael López is a member of the Institute of Mathematics of the University of Granada and he has been partially supported by the Projects PID2020-117868GB-I00 and MCIN/AEI/10.13039/501100011033. Irene Ortiz is partially supported by the grant PID2021-124157NB-I00 funded by MCIN/AEI/10.13039/501100011033/ "ERDF A way of making Europe", Spain, and she is also supported by Comunidad Autónoma de la Región de Murcia, Spain, within the framework of the Regional Programme in Promotion of the Scientific and Technical Research (Action Plan 2022), by Fundación Séneca, Regional Agency of Science and Technology, REF. 21899/PI/22.
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Bueno, A., López, R. & Ortiz, I. The Plateau–Rayleigh Instability of Translating \(\lambda \)-Solitons. Results Math 79, 58 (2024). https://doi.org/10.1007/s00025-023-02091-2
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DOI: https://doi.org/10.1007/s00025-023-02091-2