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The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds. (English) Zbl 0867.53007
Let \(M\) be an orientable surface immersed in a flat three-dimensional manifold \(N\). Minimal immersions and nonzero constant mean curvature immersions are critical points for the area variation of \(M\), but while the former are critical points for all variations with compact support, the latter are critical points only for compactly supported variations that preserve the volume enclosed by the surface \(M\). Variational problems of this kind are called isoperimetric problems. For the analysis of the stability of these immersions, we need to consider the second variation formula for the area, namely the quadratic formula called index form. If \(M\) is an orientable surface immersed with constant mean curvature in an orientable flat three manifold, then \(M\) is stable if for any smooth function \(u\) with compact support such that \(\int_M u.dA=0\) we have \(I(u)\geq 0\), where \(I\) is expressed by means of the Laplacian on \(M\) and by the squared norm of the second fundamental form and \(u\) is the normal component of the variation vector.
After reporting some of the major results existing in the literature in the past fifteen years on the relationship between the values of the index, the genus \(g\), the total curvature and the geometry of the immersions, the authors study the spaces of index one minimal surfaces and the space of constant mean curvature surfaces with \(g\geq 2\) in nonfixed flat 3-manifolds. Using appropriate topological, group-theoretical, analytical and differential methods, the authors prove important compactness theorems, namely:
Theorem. From every sequence of compact orientable index one minimal surfaces embedded in flat 3-tori one can extract, up to scaling, a convergent subsequence from both the tori and the surfaces to an index one minimal surface embedded in the limit three-torus. The above surfaces converge in the \(C^k\) topology, \(k\geq 2\), and their topology is preserved in the limit.
For the space of constant mean curvature surfaces the authors prove the following Theorem. Let \(M_n\subset N_n\) be a sequence of compact, orientable, stable surfaces embedded in complete, orientable, flat 3-manifolds. Assume that \(\text{genus}(M_n)\geq 2\) and that the induced morphisms \(\pi_1(M_n)\to \pi_1(N_n)\) between the fundamental groups are surjective. Then one can extract, up to dilatations, convergent subsequences of both the surfaces and the ambient manifolds to a compact stable surface weakly embedded in the limit manifold. Moreover, the compactness is strong in the sense that these limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds, but it is proved that there is no compactness in a strong sense for the space of genus one stable embedded surfaces.
A corollary of these important theorems of compactness is a nonexistence result for a certain kind of stable surfaces with \(g\geq 2\) embedded in most flat 3-manifolds, which is very interesting for isoperimetric problems as it implies restrictions on the type of solutions of these problems.

MSC:
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q20 Variational problems in a geometric measure-theoretic setting
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[1] A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. I, Amer. Math. Soc. Transl. (2) 21 (1962), 341 – 354. A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. II, Amer. Math. Soc. Transl. (2) 21 (1962), 354 – 388. A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. III, Amer. Math. Soc. Transl. (2) 21 (1962), 389 – 403. A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. IV, Amer. Math. Soc. Transl. (2) 21 (1962), 403 – 411. A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. V, Amer. Math. Soc. Transl. (2) 21 (1962), 412 – 416. · Zbl 0122.39601
[2] Hiroshi Mori, On surfaces of right helicoid type in \?³, Bol. Soc. Brasil. Mat. 13 (1982), no. 2, 57 – 62. , https://doi.org/10.1007/BF02584676 Hiroshi Mori, Stable complete constant mean curvature surfaces in \?³ and \?³, Trans. Amer. Math. Soc. 278 (1983), no. 2, 671 – 687. , https://doi.org/10.1090/S0002-9947-1983-0701517-7 João Lucas Barbosa and Manfredo do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984), no. 3, 339 – 353. · Zbl 0513.53002 · doi:10.1007/BF01215045 · doi.org
[3] 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. · Zbl 0653.53045 · doi:10.1007/BF01161634 · doi.org
[4] M. do Carmo and C. K. Peng, Stable complete minimal surfaces in \?³ are planes, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 903 – 906. · Zbl 0442.53013
[5] 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). · Zbl 0766.58055 · doi:10.1016/0022-1236(92)90123-Z · doi.org
[6] D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math. 82 (1985), no. 1, 121 – 132. · Zbl 0573.53038 · doi:10.1007/BF01394782 · doi.org
[7] 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. · Zbl 0439.53060 · doi:10.1002/cpa.3160330206 · doi.org
[8] Katia Rosenvald Frensel, Stable complete surfaces with constant mean curvature, An. Acad. Brasil. Ciênc. 60 (1988), no. 2, 115 – 117 (1989). · Zbl 0678.53056
[9] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. · Zbl 0408.14001
[10] Joel Hass, Jon T. Pitts, and J. H. Rubinstein, Existence of unstable minimal surfaces in manifolds with homology and applications to triply periodic minimal surfaces, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 147 – 162. · Zbl 0798.53009
[11] Ernst Heintze, Extrinsic upper bounds for \?\(_{1}\), Math. Ann. 280 (1988), no. 3, 389 – 402. · Zbl 0628.53044 · doi:10.1007/BF01456332 · doi.org
[12] D. Hoffman, W.H. Meeks, Limits of minimal surfaces and Scherk’s second surface, Univ. Massachusetts. · Zbl 0709.53006
[13] D. Hoffman and W. H. Meeks III, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990), no. 2, 373 – 377. · Zbl 0722.53054 · doi:10.1007/BF01231506 · doi.org
[14] Hermann Karcher, The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscripta Math. 64 (1989), no. 3, 291 – 357. · Zbl 0687.53010 · doi:10.1007/BF01165824 · doi.org
[15] Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465 – 503. · Zbl 0726.53007
[16] 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. · Zbl 0679.53047 · doi:10.1007/BF02564662 · doi.org
[17] William H. Meeks III, The theory of triply periodic minimal surfaces, Indiana Univ. Math. J. 39 (1990), no. 3, 877 – 936. · Zbl 0721.53057 · doi:10.1512/iumj.1990.39.39043 · doi.org
[18] William H. Meeks III and Harold Rosenberg, The global theory of doubly periodic minimal surfaces, Invent. Math. 97 (1989), no. 2, 351 – 379. · Zbl 0676.53068 · doi:10.1007/BF01389046 · doi.org
[19] William H. Meeks III and Harold Rosenberg, The geometry of periodic minimal surfaces, Comment. Math. Helv. 68 (1993), no. 4, 538 – 578. · Zbl 0807.53049 · doi:10.1007/BF02565835 · doi.org
[20] Sebastián Montiel and Antonio Ros, Schrödinger operators associated to a holomorphic map, Global differential geometry and global analysis (Berlin, 1990) Lecture Notes in Math., vol. 1481, Springer, Berlin, 1991, pp. 147 – 174. · Zbl 0744.58007 · doi:10.1007/BFb0083639 · doi.org
[21] Jon T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes, vol. 27, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. · Zbl 0462.58003
[22] Jon T. Pitts and J. H. Rubinstein, Equivariant minimax and minimal surfaces in geometric three-manifolds, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 1, 303 – 309. · Zbl 0665.49034
[23] M. Ritoré, Complete orientable index one minimal surfaces embedded in complete orientable flat three manifolds, preprint, Univ. Granada, 1994.
[24] 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. · Zbl 0760.53037 · doi:10.1007/BF02566501 · doi.org
[25] Marty Ross, Schwarz’ \? and \? surfaces are stable, Differential Geom. Appl. 2 (1992), no. 2, 179 – 195. · Zbl 0747.53010 · doi:10.1016/0926-2245(92)90032-I · doi.org
[26] Richard M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), no. 4, 791 – 809 (1984). · Zbl 0575.53037
[27] Alexandre M. Da Silveira, Stability of complete noncompact surfaces with constant mean curvature, Math. Ann. 277 (1987), no. 4, 629 – 638. · Zbl 0627.53045 · doi:10.1007/BF01457862 · doi.org
[28] B. White, Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math. 88 (1987), no. 2, 243 – 256. · Zbl 0615.53044 · doi:10.1007/BF01388908 · doi.org
[29] J.A. Wolf, Spaces of constant curvature, 1st ed., Publish or Perish, Inc., 1984.
[30] Shing-Tung Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), no. 1-2, 109 – 158. · Zbl 0631.53002
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