Regularity of Free Boundary Minimal Surfaces in Locally Polyhedral Domains

@article{Edelen2022RegularityOF,
  title={Regularity of Free Boundary Minimal Surfaces in Locally Polyhedral Domains},
  author={Nick Edelen and Chaobo Li},
  journal={Communications on Pure and Applied Mathematics},
  year={2022}
}
  • Nick Edelen, Chaobo Li
  • Published 27 June 2020
  • Mathematics
  • Communications on Pure and Applied Mathematics
We prove an Allard-type regularity theorem for free-boundary minimal surfaces in Lipschitz domains locally modelled on convex polyhedra. We show that if such a minimal surface is sufficiently close to an appropriate free-boundary plane, then the surface is $C^{1,\alpha}$ graphical over this plane. We apply our theorem to prove partial regularity results for free-boundary minimizing hypersurfaces, and relative isoperimetric regions. 

Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces

In this note, we establish the dihedral rigidity phenomenon for a collection of parabolic polyhedrons enclosed by horospheres in hyperbolic manifolds, extending Gromov's comparison theory to metrics

References

SHOWING 1-10 OF 43 REFERENCES

Optimal regularity for codimension one minimal surfaces with a free boundary

We consider the problem to minimize n-dimensional area among currents T whose boundary (or part of it) is supposed to lie in a given hypersurface S of ℝn+1. We prove dim (sing T)≤n-7. Thus, optimal

Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law

Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a closed

A polyhedron comparison theorem for 3-manifolds with positive scalar curvature

  • Chaobo Li
  • Mathematics
    Inventiones mathematicae
  • 2019
The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering

On the boundary behavior of minimal surfaces with a free boundary which are not minima of the area

The well known boundary regularity results of H. Lewy and W. Jäger for area minimizing minimal surfaces with a free boundary are shown to be true also for minimal surfaces which are only stationary

Minimal surfaces in a wedge

Abstract. Continuing our papers [4]–[6] we study the global behaviour of minimal surfaces in a wedge which meet the faces of the wedge perpendicularly and are transversal to the edge. We derive a

On a free boundary problem for minimal surfaces

ForC4-embedded manifoldsS ⊂ ℝ3 which are differmorphic to the standard sphere in ℝ3 the existence of non-constant minimal surfaces bounded byS and intersectingS orthogonally along their boundaries is

Minimal surfaces, corners, and wires

Weierstrass representations are given for minimal surfaces that have free boundaries on two planes that meet at an arbitrary dihedral angle. The contact angles of a surface on the planes may be

Minimal surfaces in a wedge I. Asymptotic expansions

In this paper we investigate the boundary behaviour of minimal surfaces in a wedge which are either of class ℐ (Γ, ℱ) or of class ℰ(Γ, ℱ) Here Γ denotes a Jordan curve in ℝ3 whose endpoints lie on a

Dirac and Plateau billiards in domains with corners

Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these,

Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory

The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging