A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary

Abstract

In this paper, we develop a general existence theory for properly embedded minimal surfaces with free boundary in any compact Riemannian 3-manifold $M$ with boundary $\partial M$. The main feature of our result is that no convexity assumption is required on $\partial M$. Our proof uses a variant of the min-max construction first considered by Almgren and Pitts. Recently, Colding-De Lellis gave a simplified proof of the interior regularity and here, we prove the boundary regularity of the limiting embedded minimal surfaces at their free boundaries. In addition, 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 $M$.

arXiv comment40 pages, 2 figures; published in Communications on Pure and Applied Mathematics (there is an error in Lemma 3.5, the issue has been addressed in arXiv:1611.02612 in a joint work with Xin Zhou)

Cited by in corpus (2)

• Minimal immersions of compact bordered Riemann surfaces with free boundary
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