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Index estimates for minimal surfaces and \(k\)-convexity. (English) Zbl 1127.58007

Author’s abstract: We prove Morse index estimates for the area functional for minimal surfaces that are solutions to the free boundary problem in \(k\)-convex domains in manifolds of nonnegative complex sectional curvature.

MSC:

58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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[1] Theodore Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165 – 174. · Zbl 0107.39002
[2] T. Frankel, On the fundamental group of a compact minimal submanifold, Ann. of Math. (2) 83 (1966), 68 – 73. · Zbl 0189.22401 · doi:10.2307/1970471
[3] Ailana M. Fraser, On the free boundary variational problem for minimal disks, Comm. Pure Appl. Math. 53 (2000), no. 8, 931 – 971. , https://doi.org/10.1002/1097-0312(200008)53:83.3.CO;2-0 · Zbl 1039.58013
[4] Ailana M. Fraser, Minimal disks and two-convex hypersurfaces, Amer. J. Math. 124 (2002), no. 3, 483 – 493. · Zbl 1043.53050
[5] Ailana M. Fraser, Fundamental groups of manifolds with positive isotropic curvature, Ann. of Math. (2) 158 (2003), no. 1, 345 – 354. · Zbl 1044.53023 · doi:10.4007/annals.2003.158.345
[6] Ailana Fraser and Jon Wolfson, The fundamental group of manifolds of positive isotropic curvature and surface groups, Duke Math. J. 133 (2006), no. 2, 325 – 334. · Zbl 1110.53027 · doi:10.1215/S0012-7094-06-13325-2
[7] G. Huisken, C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Preprint. · Zbl 1170.53042
[8] Francesco Mercuri and Maria Helena Noronha, Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2711 – 2724. · Zbl 0862.53003
[9] Jin-ichi Itoh, \?-convex domains in \?\(^{n}\), Geometry of manifolds (Matsumoto, 1988) Perspect. Math., vol. 8, Academic Press, Boston, MA, 1989, pp. 275 – 279. · doi:10.1080/0907676X.2000.9961396
[10] Jürgen Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. A Wiley-Interscience Publication. · Zbl 0729.49001
[11] H. Blaine Lawson Jr., The unknottedness of minimal embeddings, Invent. Math. 11 (1970), 183 – 187. · Zbl 0205.52002 · doi:10.1007/BF01404649
[12] Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. · Zbl 1066.53137
[13] Mario J. Micallef and John Douglas Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199 – 227. · Zbl 0661.53027 · doi:10.2307/1971420
[14] John Douglas Moore and Thomas Schulte, Minimal disks and compact hypersurfaces in Euclidean space, Proc. Amer. Math. Soc. 94 (1985), no. 2, 321 – 328. · Zbl 0574.53038
[15] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1 – 24. · Zbl 0462.58014 · doi:10.2307/1971131
[16] Richard Schoen and Jon Wolfson, Theorems of Barth-Lefschetz type and Morse theory on the space of paths, Math. Z. 229 (1998), no. 1, 77 – 89. · Zbl 0939.58017 · doi:10.1007/PL00004651
[17] Ji-Ping Sha, \?-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437 – 447. · Zbl 0563.53032 · doi:10.1007/BF01394417
[18] Ji-Ping Sha, Handlebodies and \?-convexity, J. Differential Geom. 25 (1987), no. 3, 353 – 361. · Zbl 0661.53028
[19] H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525 – 548. · Zbl 0639.53050 · doi:10.1512/iumj.1987.36.36029
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