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      Min-max theory for capillary surfaces

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          Abstract

          We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces with any given constant mean curvature \(c\), and with smooth boundary contacting at any given constant angle \(\theta\). Moreover, if \(c\) is nonzero and \(\theta\) is not \(\frac{\pi}{2}\), then our min-max solution always has multiplicity one. We also establish a stable Bernstein theorem for minimal hypersurfaces with certain contact angles in higher dimensions.

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          Author and article information

          Journal
          18 November 2021
          Article
          2111.09924
          61565a25-3623-47bc-8da3-78bde561965d

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          53A10, 49J35, 53C42
          42 pages, 4 figures; comments welcome!
          math.DG math.AP

          Analysis,Geometry & Topology
          Analysis, Geometry & Topology

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