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Transactions of the American Mathematical Society

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



The second variation of
nonorientable minimal submanifolds

Author: Marty Ross
Journal: Trans. Amer. Math. Soc. 349 (1997), 3093-3104
MSC (1991): Primary 53C45; Secondary 58E12
MathSciNet review: 1422909
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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.

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  • [A] F.J. Almgren, Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. of Math. 85 (1966), 277-292. MR 34:702
  • [BCE] J.L. Barbosa, M.P. do Carmo and J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds, Math. Zeit. 197 (1988), 123-138. MR 88m:53109
  • [BW] G. E. Bredon and J. W. Wood, Non-orientable surfaces in orientable 3-manifolds, Invent. Math. 7 (1969), 83-110. MR 39:7616
  • [Ca1] M.P. doCarmo, Stability of minimal submanifolds, Global Differential Geometry and Global Analysis, Lect. Notes in Math. 838, Springer-Verlag, Berlin, 1981, pp. 129-139. MR 82k:53073
  • [Ca2] -, Riemannian Geometry, Birkhäuser, Boston, 1992. MR 92i:53001
  • [Ch] J. Choe, Index, vision number and stability of complete minimal surfaces, Arch. Rational Mech. Anal. 109 (1990), 195-212. MR 91b:53007
  • [EI] A. El Soufi and S. Ilias, Majoration de la seconde valeur propre d'un operateur de Schrödinger sur une variete compacte et applications, J. Funct. Anal. 103 (1992), 294-316. MR 93g:58150
  • [F] D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three manifolds, Invent. Math. 82 (1985), 121-132. MR 87b:53090
  • [FR] S. Fornari and J. Ripoll, Stability of compact hypersurfaces with constant mean curvature, Indiana Univ. Math. Jour. 43 (1994), 367-381. MR 95g:53075
  • [FS] D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199-211. MR 81i:53044
  • [GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978. MR 80b:14001
  • [GL] R. Gulliver and H. B. Lawson, The structure of stable minimal hypersurfaces near a singularity, Proc. Symp. Pure Math., A.M.S. 44 (1986), 213-239. MR 87g:53091
  • [H] J. Hempel, 3-Manifolds, Annals of Math. Studies 86, Princeton Univ. Press, New Jersey, 1976. MR 54:3702
  • [K1] R. Kusner, Conformal geometry and complete minimal surfaces, Bull. Amer. Math. Soc. 17 (1987), 296-300. MR 88j:53008
  • [K2] -, Comparison surfaces for the Willmore problem, Pac. J. Math. 138 (1989), 317-345. MR 90e:53013
  • [La] H.B. Lawson, Complete minimal surfaces in $\mathbb S^3$, Ann. of Math. 92 (1970), 335-374. MR 42:5170
  • [LaS] H.B. Lawson and J. Simons, On stable currents and their applications to global problems in real and complex geometry, Ann. of Math. 98 (1973), 427-450. MR 48:2881
  • [LiS] I.C. Lima and A.M. da Silveira, Stability of complete nonorientable minimal surfaces in $\mathbf {R}^{3}$, preprint.
  • [LR] F.J. Lopez and A. Ros, Complete minimal surfaces with index one and stable constant mean curvature surfaces, Comm. Math. Helv. 64 (1989), 34-43. MR 90b:53006
  • [LY] P. Li and S.-T. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invet. Math. 69 (1982), 269-291. MR 84f:53049
  • [Ma] G. Martens, Minimale Blätterzahl bei Überlagerungen Kleinscher Flächen über der projectiven Ebene, Arch. Math. 30 (1978), 481-486. MR 58:16697
  • [MSY] W. Meeks III, L. Simon and S.-T. Yau, Embedded minimal surfaces, exotic spheres and manifolds with positive Ricci curvature, Ann. of Math. 116 (1982), 621-659. MR 84f:53053
  • [Mi] M.J. Micallef, Stable minimal surfaces in Euclidean space, J. Diff. Geom. 19 (1984), 64-76. MR 85e:53009
  • [Mu] J.R. Munkres, Topology, Prentice-Hall, Eaglewood Cliffs, New Jersey, 1975. MR 57:4063
  • [P] B. Palmer, Stability of minimal hypersurfaces, Comm. Math. Helv. 66 (1991), 185-188. MR 92m:58023
  • [R1] M. Ross, Complete nonorientable minimal surfaces in $\mathbf {R}^{3}$, Comm. Math. Helv. 67 (1992), 64-76. MR 92k:53022
  • [R2] M. Ross, Schwarz' $P$ and $D$ surfaces are stable, Diff. Geom. Appl. 2 (1992), 179-195. MR 94j:53010
  • [RR] M. Ritore and A. Ros, Stable constant mean curvature tori and the isoperimetric problem in three space forms, Comm. Math. Helv. 67 (1992), 293-305. MR 93a:53055
  • [RS] M. Ross and C. Schoen, Stable quotients of periodic minimal surfaces, Comm. Anal. Geom. 2 (1994), 451-459. MR 95j:53016
  • [Sh] K. Shiohama, Total curvatures and minimal areas of complete surfaces, Proc. A.M.S. 94 (1985), 310-316. MR 86h:53047
  • [Si1] L. Simon, First and second variation in geometry and topology, University of Melbourne Research Report, 1979.
  • [Si2] -, Lectures on Geometric Measure Theory, Proc. Centre Math. Anal., Aust. Nat. Univ., Canberra, Australia, 1984. MR 87a:49001
  • [Sm] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968), 62-105. MR 38:1617
  • [Sy] J.L. Synge, On the connectivity of spaces of positive curvature, Quart. J. Math. (Oxford Series) 7 (1936), 316-320.
  • [SY] R. Schoen and S.-T. Yau, Existence of incompressible minimal surfaces and the topolgy of three-dimensional manifolds with non-negative scalar curvature, Ann. of Math. 110 (1979), 127-142. MR 81k:53029
  • [Ya] S.-T. Yau, Nonlinear analysis in geometry, L'Enseignement Math. 33 (1987), 109-158. MR 88g:58003;MR 88e:53001
  • [Yp] S. Yaprak, Bernstein type theorems for minimal surfaces, Geometry and Toplogy of Submanifolds I V, World Scientific, River Edge, New Jersey, 1992, pp. 30-42. MR 93j:53011

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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

Keywords: Nonorientable minimal surface, stable, Bernstein Theorem, second variation
Received by editor(s): July 21, 1994
Article copyright: © Copyright 1997 American Mathematical Society

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