A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary

By M. Li. 2012

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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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/08b4f037e344813ecdca1e513c10927cb9d0db2a

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Citations in the corpus

Free boundary minimal surfaces in the unit ball : recent advances and open questions (2019, methodology): “metry and topology (see the ICM proceedings [12] [75] for a more detailed discussion), we have also witnessed rapid development in the free boundary setting. Some recent works related to this include [65] [66] [16] [35] [34] [92] [93]. Acknowledgements. The author is greatly indebted to Prof. Richar”
Min-max for phase transitions and the existence of embedded minimal hypersurfaces (2015, methodology): “We refer the reader to [7, 17, 20, 23, 43, 44] for applications to compact manifolds; [6, 26] for applications to non-compact manifolds; [18] for applications to a free-boundary situation; and to Marques-Neves [22, 21] for the solution of the long-standing Willmore conjecture and the existence of in”
Index characterization for free boundary minimal surfaces (2016, methodology): “Several existence results were obtained by different methods including various minimax constructions, varifold theory, minimizing the Dirichlet functional and maximizing the first Steklov eigenvalue; see [5, 24, 30, 31, 34, 15, 19, 8, 21, 23, 12, 13] and references therein.”
Min-max theory for free boundary minimal hypersurfaces, I: Regularity theory (2016, methodology): “n the sense that the minimal submanifold we seek has to lie completely inside M. This imposes substantial diculty to the problem as seen in (2) above. Moreover, as already remarked in [44] (see also [24]), the minimal submanifold will possibly touch @M along interior portions of arbitrary size. We r”
Min-max theory for minimal hypersurfaces with boundary (2016, methodology): “In passing, since many of the techniques are essentially the same, we also handle the case of free boundaries, which for n = 2 has already been considered by Grüter and Jost in [17] and by Li in [21] (modifying the Simon–Smith approach and reaching stronger conclusions).”
Compactness analysis for free boundary minimal hypersurfaces (2017, result): “In more general settings, we shall mention here the results by Li [25], Li-Zhou [26], De LellisRamic [11] and Maximo-Nunes-Smith [27], the former three works extending the min-max theory to the free boundary setting and the latter being based on a degree-theoretic approach.”