On spectral geometry of minimal surfaces in

Author:
Yi Bing Shen

Journal:
Trans. Amer. Math. Soc. **347** (1995), 3873-3889

MSC:
Primary 53C42; Secondary 58G25

MathSciNet review:
1308022

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: By employing the standard isometric imbedding of into the Euclidean space, a classification theorem for full, minimal, -type surfaces in that are not holomorphic is given. All such compact minimal surfaces are either totally real minimal surfaces in or totally real superminimal surfaces in and . In the latter case, they are locally unique. Moreover, some eigenvalue inequalities for compact minimal surfaces of with constant Kaehler angle are shown.

**[1]**John Bolton, Gary R. Jensen, Marco Rigoli, and Lyndon M. Woodward,*On conformal minimal immersions of 𝑆² into 𝐶𝑃ⁿ*, Math. Ann.**279**(1988), no. 4, 599–620. MR**926423**, 10.1007/BF01458531**[2]**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****[3]**Shiing Shen Chern and Jon Gordon Wolfson,*Minimal surfaces by moving frames*, Amer. J. Math.**105**(1983), no. 1, 59–83. MR**692106**, 10.2307/2374381**[4]**Jost-Hinrich Eschenburg, Irwen Válle Guadalupe, and Renato de Azevedo Tribuzy,*The fundamental equations of minimal surfaces in 𝐶𝑃²*, Math. Ann.**270**(1985), no. 4, 571–598. MR**776173**, 10.1007/BF01455305**[5]**Katsuei Kenmotsu,*On minimal immersions of 𝑅² into 𝑃ⁿ(𝐶)*, J. Math. Soc. Japan**37**(1985), no. 4, 665–682. MR**806307**, 10.2969/jmsj/03740665**[6]**A. Martínez and A. Ros,*On real hypersurfaces of finite type of 𝐶𝑃^{𝑚}*, Kodai Math. J.**7**(1984), no. 3, 304–316. MR**760040**, 10.2996/kmj/1138036953**[7]**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**, 10.2748/tmj/1178227460**[8]**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****[9]**Antonio Ros,*Spectral geometry of CR-minimal submanifolds in the complex projective space*, Kodai Math. J.**6**(1983), no. 1, 88–99. MR**698330****[10]**Antonio Ros,*On spectral geometry of Kaehler submanifolds*, J. Math. Soc. Japan**36**(1984), no. 3, 433–448. MR**746704**, 10.2969/jmsj/03630433**[11]**Yi Bing Shen,*Spectral geometry of totally real minimal submanifolds*, Chinese Ann. Math. Ser. A**12**(1991), no. 6, 745–753 (Chinese). MR**1154155****[12]**Tsunero Takahashi,*Minimal immersions of Riemannian manifolds*, J. Math. Soc. Japan**18**(1966), 380–385. MR**0198393****[13]**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**, 10.1090/S0002-9947-1985-0792816-3

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
53C42,
58G25

Retrieve articles in all journals with MSC: 53C42, 58G25

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1995-1308022-6

Keywords:
Minimal surface,
Kaehler angle,
complex projective space,
finite type submanifolds

Article copyright:
© Copyright 1995
American Mathematical Society