Minimal immersions of compact bordered Riemann surfaces with free boundary

Chen, Jingyi; Fraser, Ailana; Pang, Chao

doi:10.1090/S0002-9947-2014-05990-4

Minimal immersions of compact bordered Riemann surfaces with free boundary
HTML articles powered by AMS MathViewer

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

William Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. (2) 105 (1977), no. 1, 29–44. MR 442293, DOI 10.2307/1971024
L. Bers, Topics in the real analytic theory of Teichmüller space, Mimeographed notes, Univ. of Illinois at Urbana-Champaign.
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, DOI 10.1016/0022-1236(92)90123-Z
H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765, DOI 10.1007/978-1-4612-2034-3
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, DOI 10.1002/cpa.3160330206
Ailana M. Fraser, On the free boundary variational problem for minimal disks, Comm. Pure Appl. Math. 53 (2000), no. 8, 931–971. MR 1755947, DOI 10.1002/1097-0312(200008)53:8<931::AID-CPA1>3.3.CO;2-0
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 362153, DOI 10.2307/2373697
Joseph Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645–A1648 (French). MR 292357
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.
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
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
Ernst Kuwert, A compactness result for loops with an $H^{1/2}$-bound, J. Reine Angew. Math. 505 (1998), 1–22. MR 1662232, DOI 10.1515/crll.1998.117
M. Li, A general existence theorem for embedded minimal surfaces with free boundary, arXiv:1204.2883.
M. Li, Width and rigidity of min-max minimal disks in three-manifolds with boundary, preprint.
M. Li, Rigidity of area-minimizing disks in three manifolds with boundary, preprint.
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, DOI 10.1007/BF01399507
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, DOI 10.1215/00127094-1813410
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, DOI 10.2307/2007026
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, DOI 10.2307/1971088
Charles B. Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807–851. MR 27137, DOI 10.2307/1969401
Antonio Ros, One-sided complete stable minimal surfaces, J. Differential Geom. 74 (2006), no. 1, 69–92. MR 2260928
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, DOI 10.2307/1971131
J. Sacks and K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 639–652. MR 654854, DOI 10.1090/S0002-9947-1982-0654854-8
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, DOI 10.2307/1971247
M. Struwe, On a free boundary problem for minimal surfaces, Invent. Math. 75 (1984), no. 3, 547–560. MR 735340, DOI 10.1007/BF01388643
Shing-Tung Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), no. 1-2, 109–158. MR 896385
Rugang Ye, On the existence of area-minimizing surfaces with free boundary, Math. Z. 206 (1991), no. 3, 321–331. MR 1095757, DOI 10.1007/BF02571346