Min-max construction of minimal surfaces with a fixed angle at the boundary

We prove the existence of minimal surfaces in a bounded convex subset of R M intersecting the boundary of M with a fixed contact angle. The proof is based on a min-max construction in the spirit of Almgren-Pitts for the capillarity functional.

[1]  L. Bers.  Isolated Singularities of Minimal Surfaces , 1951 .

[2]  F. C. Marques,et al.  Density of minimal hypersurfaces for generic metrics. , 2017 .

[3]  A. Song,et al.  Existence of infinitely many minimal hypersurfaces in closed manifolds , 2018, Annals of Mathematics.

[5]  Camillo De Lellis,et al.  Min-max theory for minimal hypersurfaces with boundary , 2016, 1611.00926.

[6]  Leon Simon,et al.  Lectures on Geometric Measure Theory , 1984 .

[7]  A. Friedman,et al.  A free boundary problem for quasilinear elliptic equations , 1984 .

[8]  F. Maggi.  Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory , 2012 .

[9]  Y. Tonegawa,et al.  A fixed contact angle condition for varifolds , 2016, 1606.00164.

[10]  Han Hong,et al.  Capillary surfaces: stability, index and curvature estimates , 2021, 2105.12662.

[11]  Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law , 2014, 1402.0549.

[12]  M. Li.  A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary , 2012, 1204.2883.

[13]  Jonathan J. Zhu,et al.  Min–max theory for constant mean curvature hypersurfaces , 2017, Inventiones mathematicae.

[14]  Antonio Ros,et al.  On stability of capillary surfaces in a ball , 1997 .

[15]  Chao Li,et al.  Min-max theory for capillary surfaces , 2021 .

[16]  Camillo De Lellis,et al.  The min--max construction of minimal surfaces , 2003, math/0303305.

[17]  F. Maggi,et al.  Dimensional estimates for singular sets in geometric variational problems with free boundaries , 2014, 1407.4834.

[18]  Daniela De Silva,et al.  Free boundary regularity for a problem with right hand side , 2009, 0912.2057.

[19]  Fernando C. Marques,et al.  Existence of infinitely many minimal hypersurfaces in positive Ricci curvature , 2013, Inventiones mathematicae.

[20]  Rectifiability of the free boundary for varifolds , 2021, Indiana University Mathematics Journal.

[21]  Camillo De Lellis,et al.  The existence of embedded minimal hypersurfaces , 2009, 0905.4192.

[22]  W. Meeks.  The structure of stable minimal surfaces near a singularity , 2007 .

[23]  T. Willmore.  EXISTENCE AND REGULARITY OF MINIMAL SURFACES ON RIEMANNIAN MANIFOLDS , 1982 .