Second variation of superminimal surfaces into self-dual Einstein four-manifolds

Montiel, Sebastián; Urbano, Francisco

doi:10.1090/S0002-9947-97-01933-8

Second variation of superminimal surfaces into self-dual Einstein four-manifolds
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by Sebastián Montiel and Francisco Urbano PDF

Trans. Amer. Math. Soc. 349 (1997), 2253-2269 Request permission

Abstract:

The index of a compact orientable superminimal surface of a self-dual Einstein four-manifold $M$ with positive scalar curvature is computed in terms of its genus and area. Also a lower bound of its nullity is obtained. Applications to the cases $M=\mathbb {S}^4$ and $M=\mathbb {C}\mathbb {P}^2$ are given, characterizing the standard Veronese immersions and their twistor deformations as those with lowest index.

References

M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143
João Lucas Marquês Barbosa, On minimal immersions of $S^{2}$ into $S^{2m}$, Trans. Amer. Math. Soc. 210 (1975), 75–106. MR 375166, DOI 10.1090/S0002-9947-1975-0375166-2
Lionel Bérard-Bergery, Marcel Berger, and Christian Houzel (eds.), Géométrie riemannienne en dimension 4, Textes Mathématiques [Mathematical Texts], vol. 3, CEDIC, Paris, 1981 (French). Papers from the Arthur Besse seminar held at the Université de Paris VII, Paris, 1978/1979. MR 769127
Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
Robert L. Bryant, Conformal and minimal immersions of compact surfaces into the $4$-sphere, J. Differential Geometry 17 (1982), no. 3, 455–473. MR 679067
M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry 3 (1969), 379–392. MR 266084, DOI 10.4310/jdg/1214429060
Eugenio Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111–125. MR 233294
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Shiing Shen Chern and Jon Gordon Wolfson, Minimal surfaces by moving frames, Amer. J. Math. 105 (1983), no. 1, 59–83. MR 692106, DOI 10.2307/2374381
Norio Ejiri, The index of minimal immersions of $S^{2}$ into $S^{2n}$, Math. Z. 184 (1983), no. 1, 127–132. MR 711733, DOI 10.1007/BF01162011
J. Eells and S. Salamon, Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 4, 589–640 (1986). MR 848842
Jost-Hinrich Eschenburg, Irwen Válle Guadalupe, and Renato de Azevedo Tribuzy, The fundamental equations of minimal surfaces in $\textbf {C}\textrm {P}^2$, Math. Ann. 270 (1985), no. 4, 571–598. MR 776173, DOI 10.1007/BF01455305
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
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
Thomas Friedrich, On surfaces in four-spaces, Ann. Global Anal. Geom. 2 (1984), no. 3, 257–287. MR 777909, DOI 10.1007/BF01876417
Th. Friedrich and R. Grunewald, On Einstein metrics on the twistor space of a four-dimensional Riemannian manifold, Math. Nachr. 123 (1985), 55–60. MR 809333, DOI 10.1002/mana.19851230106
Théorie des variétés minimales et applications, Société Mathématique de France, Paris, 1988 (French). Minimal submanifolds; Papers from the seminar held at the École Polytechnique, Palaiseau, October 1983–June 1984; Astérisque No. 154-155 (1987) (1987). MR 955056
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
N. J. Hitchin, Kählerian twistor spaces, Proc. London Math. Soc. (3) 43 (1981), no. 1, 133–150. MR 623721, DOI 10.1112/plms/s3-43.1.133
Motoko Kotani, Connectedness of the space of minimal $2$-spheres in $S^{2m}(1)$, Proc. Amer. Math. Soc. 120 (1994), no. 3, 803–810. MR 1169040, DOI 10.1090/S0002-9939-1994-1169040-9
J.-L. Koszul and B. Malgrange, Sur certaines structures fibrées complexes, Arch. Math. (Basel) 9 (1958), 102–109 (French). MR 131882, DOI 10.1007/BF02287068
Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
Bonaventure Loo, The space of harmonic maps of $S^2$ into $S^4$, Trans. Amer. Math. Soc. 313 (1989), no. 1, 81–102. MR 962283, DOI 10.1090/S0002-9947-1989-0962283-9
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. MR 924677, DOI 10.2307/1971420
Mario J. Micallef and Jon G. Wolfson, The second variation of area of minimal surfaces in four-manifolds, Math. Ann. 295 (1993), no. 2, 245–267. MR 1202392, DOI 10.1007/BF01444887
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. MR 1178529, DOI 10.1007/BFb0083639
Yong-Geun Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101 (1990), no. 2, 501–519. MR 1062973, DOI 10.1007/BF01231513
R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI 10.2307/1971247
Takao Sasai, The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations, Tohoku Math. J. (2) 36 (1984), no. 1, 17–24. MR 733615, DOI 10.2748/tmj/1178228899
J. Simons, Minimal varieties in Riemaniann manifolds, Ann. of Math., 88(1968), 62–105.
Yum Tong Siu and Shing Tung Yau, Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), no. 2, 189–204. MR 577360, DOI 10.1007/BF01390043
Francisco Urbano, Minimal surfaces with low index in the three-dimensional sphere, Proc. Amer. Math. Soc. 108 (1990), no. 4, 989–992. MR 1007516, DOI 10.1090/S0002-9939-1990-1007516-1