The second variation of nonorientable minimal submanifolds
HTML articles powered by AMS MathViewer
- by Marty Ross PDF
- Trans. Amer. Math. Soc. 349 (1997), 3093-3104 Request permission
Abstract:
Suppose $M$ is a complete nonorientable minimal submanifold of a Riemannian manifold $N$. We derive a second variation formula for the area of $M$ with respect to certain perturbations, giving a sufficient condition for the instability of $M$. Some simple applications are given: we show that the totally geodesic $\mathbb {R} \mathbb {P}^{2}$ is the only stable surface in $\mathbb {R} \mathbb {P}^{3}$, and we show the non-existence of stable nonorientable cones in $\mathbb {R}^{4}$. We reproduce and marginally extend some known results in the truly non-compact setting.References
- F. J. Almgren Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. of Math. (2) 84 (1966), 277–292. MR 200816, DOI 10.2307/1970520
- J. Lucas Barbosa, Manfredo do Carmo, and Jost Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Z. 197 (1988), no. 1, 123–138. MR 917854, DOI 10.1007/BF01161634
- Glen E. Bredon and John W. Wood, Non-orientable surfaces in orientable $3$-manifolds, Invent. Math. 7 (1969), 83–110. MR 246312, DOI 10.1007/BF01389793
- Manfredo P. do Carmo, Stability of minimal submanifolds, Global differential geometry and global analysis (Berlin, 1979) Lecture Notes in Math., vol. 838, Springer, Berlin, 1981, pp. 129–139. MR 636273
- Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207, DOI 10.1007/978-1-4757-2201-7
- Jaigyoung Choe, Index, vision number and stability of complete minimal surfaces, Arch. Rational Mech. Anal. 109 (1990), no. 3, 195–212. MR 1025170, DOI 10.1007/BF00375088
- 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
- D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math. 82 (1985), no. 1, 121–132. MR 808112, DOI 10.1007/BF01394782
- Susana Fornari and Jaime Ripoll, Stability of compact hypersurfaces with constant mean curvature, Indiana Univ. Math. J. 43 (1994), no. 1, 367–381. MR 1275464, DOI 10.1512/iumj.1994.43.43015
- 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
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- Robert Gulliver and H. Blaine Lawson Jr., The structure of stable minimal hypersurfaces near a singularity, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984) Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986, pp. 213–237. MR 840275, DOI 10.1090/pspum/044/840275
- John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
- Rob Kusner, Conformal geometry and complete minimal surfaces, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 2, 291–295. MR 903735, DOI 10.1090/S0273-0979-1987-15564-9
- Rob Kusner, Comparison surfaces for the Willmore problem, Pacific J. Math. 138 (1989), no. 2, 317–345. MR 996204, DOI 10.2140/pjm.1989.138.317
- H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
- H. Blaine Lawson Jr. and James Simons, On stable currents and their application to global problems in real and complex geometry, Ann. of Math. (2) 98 (1973), 427–450. MR 324529, DOI 10.2307/1970913
- I.C. Lima and A.M. da Silveira, Stability of complete nonorientable minimal surfaces in $\mathbf {R}^{3}$, preprint.
- Francisco J. López and Antonio Ros, Complete minimal surfaces with index one and stable constant mean curvature surfaces, Comment. Math. Helv. 64 (1989), no. 1, 34–43. MR 982560, DOI 10.1007/BF02564662
- 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
- Yukihiko Namikawa, A new compactification of the Siegel space and degeneration of polarized Abelian varieties, Sūgaku 28 (1976), no. 3, 214–225 (Japanese). MR 498608
- 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
- Mario J. Micallef, Stable minimal surfaces in Euclidean space, J. Differential Geom. 19 (1984), no. 1, 57–84. MR 739782
- James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR 0464128
- Bennett Palmer, Stability of minimal hypersurfaces, Comment. Math. Helv. 66 (1991), no. 2, 185–188. MR 1107838, DOI 10.1007/BF02566644
- Marty Ross, Complete nonorientable minimal surfaces in $\mathbf R^3$, Comment. Math. Helv. 67 (1992), no. 1, 64–76. MR 1144614, DOI 10.1007/BF02566489
- Marty Ross, Schwarz’ $P$ and $D$ surfaces are stable, Differential Geom. Appl. 2 (1992), no. 2, 179–195. MR 1245555, DOI 10.1016/0926-2245(92)90032-I
- Manuel Ritoré and Antonio Ros, Stable constant mean curvature tori and the isoperimetric problem in three space forms, Comment. Math. Helv. 67 (1992), no. 2, 293–305. MR 1161286, DOI 10.1007/BF02566501
- Marty Ross and Chad Schoen, Stable quotients of periodic minimal surfaces, Comm. Anal. Geom. 2 (1994), no. 3, 451–459. MR 1305713, DOI 10.4310/CAG.1994.v2.n3.a4
- Katsuhiro Shiohama, Total curvatures and minimal areas of complete surfaces, Proc. Amer. Math. Soc. 94 (1985), no. 2, 310–316. MR 784184, DOI 10.1090/S0002-9939-1985-0784184-3
- L. Simon, First and second variation in geometry and topology, University of Melbourne Research Report, 1979.
- Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
- James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
- J.L. Synge, On the connectivity of spaces of positive curvature, Quart. J. Math. (Oxford Series) 7 (1936), 316-320.
- Ilie Burdujan, Une application des systèmes homogènes de K. Yamaguti dans la géométrie différentielle, Bul. Inst. Politehn. Iaşi Secţ. I 25(29) (1979), no. 1-2, 47–49 (French, with Romanian summary). MR 558409
- Shing-Tung Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), no. 1-2, 109–158. MR 896385
- Sahnur Yaprak, Bernstein type theorems for minimal surfaces, Geometry and topology of submanifolds, IV (Leuven, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 30–42. MR 1185714
Additional Information
- Marty Ross
- Affiliation: Department of Mathematics, Melbourne University, Parkville, Victoria, 3052, Australia
- Address at time of publication: Antarctic CRC, Box 252-80, Hobart, Tasmania, Australia
- Email: marty@mundoe.maths.mu.oz.au
- Received by editor(s): July 21, 1994
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 3093-3104
- MSC (1991): Primary 53C45; Secondary 58E12
- DOI: https://doi.org/10.1090/S0002-9947-97-01936-3
- MathSciNet review: 1422909