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A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary
Author(s) -
Li Martin Manchun
Publication year - 2015
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21513
Subject(s) - mathematics , convexity , minimal surface , boundary (topology) , disjoint sets , genus , pure mathematics , manifold (fluid mechanics) , invariant (physics) , riemannian manifold , surface (topology) , combinatorics , mathematical analysis , geometry , mathematical physics , economics , biology , mechanical engineering , botany , financial economics , engineering
In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3‐manifold M with boundary ∂ M . These minimal surfaces are either disjoint from ∂ M or meet ∂ M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ∂ M . We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus , for compact 3‐manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M . Our proof employs a variant of the min‐max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.© 2014 Wiley Periodicals, Inc.