## Minimal immersions of compact bordered Riemann surfaces with free boundary

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- by Jingyi Chen, Ailana Fraser and Chao Pang PDF
- Trans. Amer. Math. Soc.
**367**(2015), 2487-2507 Request permission

## 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.## References

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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
- MR Author ID: 662533
- 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
- Received by editor(s): September 5, 2012
- Published electronically: November 24, 2014
- Additional Notes: This work was partially supported by NSERC
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3301871