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Minimal immersions of compact bordered Riemann surfaces with free boundary


Authors: Jingyi Chen, Ailana Fraser and Chao Pang
Journal: Trans. Amer. Math. Soc. 367 (2015), 2487-2507
MSC (2010): Primary 58E12; Secondary 53C21, 53C43
DOI: https://doi.org/10.1090/S0002-9947-2014-05990-4
Published electronically: November 24, 2014
MathSciNet review: 3301871
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Abstract: Let $ N$ be a complete, homogeneously regular Riemannian manifold of dim$ N \geq 3$ and let $ M$ be a compact submanifold of $ N$. Let $ \Sigma $ be a compact orientable surface with boundary. We show that for any continuous $ f: \left ( \Sigma , \partial \Sigma \right ) \rightarrow \left ( N, M \right )$ for which the induced homomorphism $ f_{*}$ on certain fundamental groups is injective, there exists a branched minimal immersion of $ \Sigma $ solving the free boundary problem $ \left ( \Sigma , \partial \Sigma \right ) \rightarrow \left ( N, M \right )$, and minimizing area among all maps which induce the same action on the fundamental groups as $ f$. Furthermore, under certain nonnegativity assumptions on the curvature of a $ 3$-manifold $ N$ and convexity assumptions on the boundary $ M=\partial N$, we derive bounds on the genus, number of boundary components and area of any compact two-sided minimal surface solving the free boundary problem with low index.


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Additional Information

Jingyi Chen
Affiliation: Department of Mathematics, The University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada
Email: jychen@math.ubc.ca

Ailana Fraser
Affiliation: Department of Mathematics, The University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada
Email: afraser@math.ubc.ca

Chao Pang
Affiliation: Department of Mathematics, The University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada
Email: ottokk@math.ubc.ca

DOI: https://doi.org/10.1090/S0002-9947-2014-05990-4
Received by editor(s): September 5, 2012
Published electronically: November 24, 2014
Additional Notes: This work was partially supported by NSERC
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.