Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] William Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. (2) 105 (1977), no. 1, 29-44. MR 0442293 (56 #679)
  • [2] L. Bers, Topics in the real analytic theory of Teichmüller space, Mimeographed notes, Univ. of Illinois at Urbana-Champaign.
  • [3] A. El Soufi and S. Ilias, Majoration de la seconde valeur propre d'un opérateur de Schrödinger sur une variété compacte et applications, J. Funct. Anal. 103 (1992), no. 2, 294-316 (French, with English summary). MR 1151550 (93g:58150), https://doi.org/10.1016/0022-1236(92)90123-Z
  • [4] H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765 (93a:30047)
  • [5] Doris Fischer-Colbrie and Richard Schoen, The structure of complete stable minimal surfaces in $ 3$-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199-211. MR 562550 (81i:53044), https://doi.org/10.1002/cpa.3160330206
  • [6] Ailana M. Fraser, On the free boundary variational problem for minimal disks, Comm. Pure Appl. Math. 53 (2000), no. 8, 931-971. MR 1755947 (2001g:58026), https://doi.org/10.1002/1097-0312(200008)53:8$ \langle $931::AID-CPA1$ \rangle $3.3.CO;2-0
  • [7] R. D. Gulliver II, R. Osserman, and H. L. Royden, A theory of branched immersions of surfaces, Amer. J. Math. 95 (1973), 750-812. MR 0362153 (50 #14595)
  • [8] Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645-A1648 (French). MR 0292357 (45 #1444)
  • [9] J. Jost, Existence results for embedded minimal surfaces of controlled topological type, Ann. Sc. Norm. Sup. Pisa, Part I, 13 (1986), 15-50; Part II, 13 (1986), 401-426; Part III, 14 (1987), 165-167.
  • [10] Jürgen Jost, On the existence of embedded minimal surfaces of higher genus with free boundaries in Riemannian manifolds, Variational methods for free surface interfaces (Menlo Park, Calif., 1985), Springer, New York, 1987, pp. 65-75. MR 872889 (89b:58051)
  • [11] Jürgen Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons Ltd., Chichester, 1991. A Wiley-Interscience Publication. MR 1100926 (92h:58045)
  • [12] Ernst Kuwert, A compactness result for loops with an $ H^{1/2}$-bound, J. Reine Angew. Math. 505 (1998), 1-22. MR 1662232 (2000a:58036), https://doi.org/10.1515/crll.1998.117
  • [13] M. Li, A general existence theorem for embedded minimal surfaces with free boundary, arXiv:1204.2883.
  • [14] M. Li, Width and rigidity of min-max minimal disks in three-manifolds with boundary, preprint.
  • [15] M. Li, Rigidity of area-minimizing disks in three manifolds with boundary, preprint.
  • [16] Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269-291. MR 674407 (84f:53049), https://doi.org/10.1007/BF01399507
  • [17] Fernando C. Marques and André Neves, Rigidity of min-max minimal spheres in three-manifolds, Duke Math. J. 161 (2012), no. 14, 2725-2752. MR 2993139, https://doi.org/10.1215/00127094-1813410
  • [18] William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621-659. MR 678484 (84f:53053), https://doi.org/10.2307/2007026
  • [19] William H. Meeks III and Shing Tung Yau, Topology of three-dimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. (2) 112 (1980), no. 3, 441-484. MR 595203 (83d:53045), https://doi.org/10.2307/1971088
  • [20] Charles B. Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807-851. MR 0027137 (10,259f)
  • [21] Antonio Ros, One-sided complete stable minimal surfaces, J. Differential Geom. 74 (2006), no. 1, 69-92. MR 2260928 (2007g:53008)
  • [22] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $ 2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1-24. MR 604040 (82f:58035), https://doi.org/10.2307/1971131
  • [23] J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 639-652. MR 654854 (83i:58030), https://doi.org/10.2307/1998902
  • [24] R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127-142. MR 541332 (81k:58029), https://doi.org/10.2307/1971247
  • [25] M. Struwe, On a free boundary problem for minimal surfaces, Invent. Math. 75 (1984), no. 3, 547-560. MR 735340 (85a:58019), https://doi.org/10.1007/BF01388643
  • [26] Shing-Tung Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), no. 1-2, 109-158. MR 896385 (88g:58003)
  • [27] Rugang Ye, On the existence of area-minimizing surfaces with free boundary, Math. Z. 206 (1991), no. 3, 321-331. MR 1095757 (92g:58023), https://doi.org/10.1007/BF02571346

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 58E12, 53C21, 53C43

Retrieve articles in all journals with MSC (2010): 58E12, 53C21, 53C43


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.

American Mathematical Society