Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Index estimates for minimal surfaces and $ k$-convexity


Author: Ailana Fraser
Journal: Proc. Amer. Math. Soc. 135 (2007), 3733-3744
MSC (2000): Primary 58E12; Secondary 53C21
DOI: https://doi.org/10.1090/S0002-9939-07-08894-6
Published electronically: August 2, 2007
MathSciNet review: 2336590
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


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

  • [Fr1] T. Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165-174. MR 0123272 (23:A600)
  • [Fr2] T. Frankel, On the fundamental group of a compact minimal submanifold, Ann. of Math. (2) 83 (1966), 68-73. MR 0187183 (32:4637)
  • [F1] A. Fraser, On the free boundary variational problem for minimal disks, Comm. Pure Appl. Math. 53 (2000), no. 8 931-971. MR 1755947 (2001g:58026)
  • [F2] A. Fraser, Minimal disks and two-convex hypersurfaces, Amer. J. Math. 124 (2002), no. 3 483-493. MR 1902886 (2003d:53104)
  • [F3] A. Fraser, Fundamental groups of manifolds of positive isotropic curvature, Ann. of Math. (2) 158 (2003), no. 1 345-354. MR 1999925 (2004j:53050)
  • [FW] A. Fraser, J. Wolfson, The fundamental group of manifolds of positive isotropic curvature and surface groups, Duke Math. J. 133 (2006), no. 2 325-334. MR 2225695
  • [HS] G. Huisken, C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Preprint.
  • [MN] F. Mercuri, M. H. Noronha, Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature, Trans. Amer. Math. Soc. 348 (1996), no. 7 2711-2724. MR 1348153 (96j:53049)
  • [I] J. Itoh, $ p$-convex domains in $ \mathbb{R}^n$, Geometry of manifolds (Matsumoto, 1988), 275-279, Perspect. Math., 8, Academic Press, Boston, 1989. MR 1040529 (91e:53008)
  • [J] J. Jost, Two Dimensional Geometric Variational Problems, John Wiley and Sons, Ltd., Chichester, (1991). MR 1100926 (92h:58045)
  • [L] H. B. Lawson, The unknottedness of minimal embeddings, Invent. Math. 11 (1970), 183-187. MR 0287447 (44:4651)
  • [McS] D. McDuff, D. Salamon, Introduction to symplectic topology, Second edition, Oxford University Press, New York (1998). MR 1698616 (2000g:53098)
  • [MM] M. Micallef, J. D. 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. MR 924677 (89e:53088)
  • [MS] J. D. Moore, T. Schulte, Minimal disks and compact hypersurfaces in Euclidean space, Proc. Amer. Math. Soc. 94 (1985), no. 2 321-328. MR 784186 (86h:53057)
  • [SU] J. Sacks, K. Uhlenbeck, The existence of minimal immersions of two-spheres, Ann. of Math. (2) 113 (1981), 1-24. MR 604040 (82f:58035)
  • [SW] R. Schoen, J. Wolfson, Theorems of Barth-Lefschetz type and Morse theory on the space of paths, Math. Z. 229 (1998), no.1 77-89. MR 1649314 (2000i:58021)
  • [Sh1] J. Sha, $ p$-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3 437-447. MR 827362 (87e:53071)
  • [Sh2] J. Sha, Handlebodies and $ p$-convexity, J. Differential Geom. 25 (1987), no. 3 353-361. MR 882828 (88f:53079)
  • [Wu] H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3 525-548. MR 905609 (88k:53068)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58E12, 53C21

Retrieve articles in all journals with MSC (2000): 58E12, 53C21


Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-07-08894-6
Received by editor(s): July 26, 2006
Published electronically: August 2, 2007
Additional Notes: The author was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC)
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society