Minimal Surfaces in \(\mathbb{S}^{2} \times\mathbb{S}^{2}\)

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• 作者：Francisco Torralbo (1)
  Francisco Urbano (1)
  
  1. Departamento de Geometr铆a y Topolog铆a ; Universidad de Granada ; 18071 ; Granada ; Spain
  
• 关键词：Surfaces ; Minimal ; Complex surfaces ; 53C42 ; 53C40
• 刊名：Journal of Geometric Analysis
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：25
• 期：2
• 页码：1132-1156
• 全文大小：375 KB
• 参考文献：1. Asperti, A.C., Ferus, D., Rodr铆guez, L. (1982) Surfaces with nonzero normal curvature. Rend. Sci. Fis. Mat. Lincei 73: pp. 109-115
  2. Bryant, R. (1982) Conformal and minimal immersions of compact surfaces into the 4-sphere. J. Differ. Geom. 17: pp. 455-473
  3. Castro, I., Urbano, F. (2007) Minimal Lagrangian surfaces in $\mathbb {S}^{2}\times \mathbb {S}^{2}$. Commun. Anal. Geom. 15: pp. 217-248 external" href="http://dx.doi.org/10.4310/CAG.2007.v15.n2.a1" target="_blank" title="It opens in new window">CrossRef
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  6. Eschenburg, J.-T., Guadalupe, I.V., Tribuzy, R. (1985) The fundamental equations of minimal surfaces in $\mathbb{C}P^{2}$. Math. Ann. 270: pp. 571-598 external" href="http://dx.doi.org/10.1007/BF01455305" target="_blank" title="It opens in new window">CrossRef
  7. Hauswirth, L., Kilian, M., Schmidt, M.U.: Finite type minimal annuli in e-equation id-i-eq374"> equation-source format-t-e-x">\(\mathbb{S}^{2} \times\mathbb{R}\) . ef="http://arxiv.org/abs/arXiv:1210.5606v1" class="a-plus-plus">arXiv:1210.5606v1 [math.DG]
  8. Hitchin, N.J. (1990) Harmonic maps from a 2-torus to the 3-sphere. J. Differ. Geom. 31: pp. 627-710
  9. Lawson, H.B. (1970) Complete minimal surfaces in $\mathbb {S}^{3}$. Ann. Math. 92: pp. 335-374 external" href="http://dx.doi.org/10.2307/1970625" target="_blank" title="It opens in new window">CrossRef
  10. Li, P., Yau, S.-T. (1982) A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math. 69: pp. 269-291 external" href="http://dx.doi.org/10.1007/BF01399507" target="_blank" title="It opens in new window">CrossRef
  11. Micallef, M.J., Wolfson, J.G. (1993) The second variation of area of minimal surfaces in four-manifolds. Math. Ann. 295: pp. 245-267 external" href="http://dx.doi.org/10.1007/BF01444887" target="_blank" title="It opens in new window">CrossRef
  12. Montiel, S., Urbano, F. (1997) Second variation of superminimal surfaces into self-dual Einstein four-manifolds. Trans. Am. Math. Soc. 349: pp. 2253-2269 external" href="http://dx.doi.org/10.1090/S0002-9947-97-01933-8" target="_blank" title="It opens in new window">CrossRef
  13. Ruh, E.A., Vilms, J. (1970) The tension field of the Gauss map. Trans. Am. Math. Soc. 149: pp. 569-573 external" href="http://dx.doi.org/10.1090/S0002-9947-1970-0259768-5" target="_blank" title="It opens in new window">CrossRef
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  15. Torralbo, F., Urbano, F. (2012) Surfaces with parallel mean curvature vector in $\mathbb {S}^{2} \times \mathbb {S}^{2}$ and $\mathbb {H}^{2} \times \mathbb {H}^{2}$. Trans. Am. Math. Soc. 364: pp. 785-813 external" href="http://dx.doi.org/10.1090/S0002-9947-2011-05346-8" target="_blank" title="It opens in new window">CrossRef
  16. Torralbo, F., Urbano, F.: Compact stable minimal submanifolds. To appear in Proc. Am. Math. Soc.
  17. Wolfson, J.G. (1989) Minimal surfaces in K盲hler surfaces and Ricci curvature. J. Differ. Geom. 29: pp. 281-294
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Differential Geometry
  Convex and Discrete Geometry
  Fourier Analysis
  Abstract Harmonic Analysis
  Dynamical Systems and Ergodic Theory
  Global Analysis and Analysis on Manifolds
  
• 出版者：Springer New York
• ISSN：1559-002X

文摘

A general study of minimal surfaces of the Riemannian product of two spheres \(\mathbb {S}^{2}\times \mathbb {S}^{2}\) is tackled. We establish a local correspondence between (non-complex) minimal surfaces of \(\mathbb {S}^{2} \times \mathbb {S}^{2}\) and a certain pair of minimal surfaces of the sphere \(\mathbb {S}^{3}\) . This correspondence also allows us to link minimal surfaces in \(\mathbb{S}^{3}\) and in the Riemannian product \(\mathbb {S}^{2} \times \mathbb {R}\) . Some rigidity results for compact minimal surfaces are also obtained.