On spectral geometry of minimal surfaces in $\mathbf {C}\mathrm {P}^ n$
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- by Yi Bing Shen PDF
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Abstract:
By employing the standard isometric imbedding of $C{P^n}$ into the Euclidean space, a classification theorem for full, minimal, $2$-type surfaces in $C{P^n}$ that are not $\pm$ holomorphic is given. All such compact minimal surfaces are either totally real minimal surfaces in $C{P^2}$ or totally real superminimal surfaces in $C{P^3}$ and $C{P^4}$. In the latter case, they are locally unique. Moreover, some eigenvalue inequalities for compact minimal surfaces of $C{P^n}$ with constant Kaehler angle are shown.References
- John Bolton, Gary R. Jensen, Marco Rigoli, and Lyndon M. Woodward, On conformal minimal immersions of $S^2$ into $\textbf {C}\textrm {P}^n$, Math. Ann. 279 (1988), no. 4, 599–620. MR 926423, DOI 10.1007/BF01458531
- Bang-Yen Chen, Total mean curvature and submanifolds of finite type, Series in Pure Mathematics, vol. 1, World Scientific Publishing Co., Singapore, 1984. MR 749575, DOI 10.1142/0065
- 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
- 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
- Katsuei Kenmotsu, On minimal immersions of $\textbf {R}^2$ into $\textbf {P}^n(\textbf {C})$, J. Math. Soc. Japan 37 (1985), no. 4, 665–682. MR 806307, DOI 10.2969/jmsj/03740665
- A. Martínez and A. Ros, On real hypersurfaces of finite type of $\textbf {C}\textrm {P}^{m}$, Kodai Math. J. 7 (1984), no. 3, 304–316. MR 760040, DOI 10.2996/kmj/1138036953
- Takashi Ogata, Curvature pinching theorem for minimal surfaces with constant Kaehler angle in complex projective spaces, Tohoku Math. J. (2) 43 (1991), no. 3, 361–374. MR 1117210, DOI 10.2748/tmj/1178227460
- Yoshihiro Ohnita, Minimal surfaces with constant curvature and Kähler angle in complex space forms, Tsukuba J. Math. 13 (1989), no. 1, 191–207. MR 1003602, DOI 10.21099/tkbjm/1496161017
- Antonio Ros, Spectral geometry of CR-minimal submanifolds in the complex projective space, Kodai Math. J. 6 (1983), no. 1, 88–99. MR 698330
- Antonio Ros, On spectral geometry of Kaehler submanifolds, J. Math. Soc. Japan 36 (1984), no. 3, 433–448. MR 746704, DOI 10.2969/jmsj/03630433
- Yi Bing Shen, Spectral geometry of totally real minimal submanifolds, Chinese Ann. Math. Ser. A 12 (1991), no. 6, 745–753 (Chinese). MR 1154155
- Tsunero Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385. MR 198393, DOI 10.2969/jmsj/01840380
- Jon G. Wolfson, On minimal surfaces in a Kähler manifold of constant holomorphic sectional curvature, Trans. Amer. Math. Soc. 290 (1985), no. 2, 627–646. MR 792816, DOI 10.1090/S0002-9947-1985-0792816-3
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 3873-3889
- MSC: Primary 53C42; Secondary 58G25
- DOI: https://doi.org/10.1090/S0002-9947-1995-1308022-6
- MathSciNet review: 1308022