A generalization of minimal cones
Author:
Norio Ejiri
Journal:
Trans. Amer. Math. Soc. 276 (1983), 347360
MSC:
Primary 53C42
MathSciNet review:
684514
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a positive real line, an dimensional unit sphere. We denote by the polar coordinate of an dimensional Euclidean space . It is well known that if is a minimal submanifold in , then is minimal in . is called a minimal cone. We generalize this fact and give many minimal submanifolds in real and complex space forms.
 [1]
R.
L. Bishop and B.
O’Neill, Manifolds of negative
curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 0251664
(40 #4891), http://dx.doi.org/10.1090/S00029947196902516644
 [2]
David
E. Blair, On a generalization of the catenoid, Canad. J. Math.
27 (1975), 231–236. MR 0380637
(52 #1534)
 [3]
Bangyen
Chen and Koichi
Ogiue, On totally real submanifolds,
Trans. Amer. Math. Soc. 193 (1974), 257–266. MR 0346708
(49 #11433), http://dx.doi.org/10.1090/S00029947197403467087
 [4]
Norio
Ejiri, Totally real minimal submanifolds in a
complex projective space, Proc. Amer. Math.
Soc. 86 (1982), no. 3, 496–497. MR 671223
(84g:53065), http://dx.doi.org/10.1090/S00029939198206712230
 [5]
Joseph
Erbacher, Isometric immersions of constant mean curvature and
triviality of the normal connection, Nagoya Math. J.
45 (1972), 139–165. MR 0380679
(52 #1576)
 [6]
Chorngshi
Houh, Some totally real minimal surfaces in
𝐶𝑃², Proc. Amer. Math.
Soc. 40 (1973),
240–244. MR 0317189
(47 #5737), http://dx.doi.org/10.1090/S00029939197303171899
 [7]
Wuyi
Hsiang and H.
Blaine Lawson Jr., Minimal submanifolds of low cohomogeneity,
J. Differential Geometry 5 (1971), 1–38. MR 0298593
(45 #7645)
 [8]
H.
Blaine Lawson Jr., Complete minimal surfaces in
𝑆³, Ann. of Math. (2) 92 (1970),
335–374. MR 0270280
(42 #5170)
 [9]
Hiroo
Naitoh, Totally real parallel submanifolds in
𝑃ⁿ(𝑐), Tokyo J. Math. 4
(1981), no. 2, 279–306. MR 646040
(83h:53072), http://dx.doi.org/10.3836/tjm/1270215155
 [10]
Katsumi
Nomizu and Brian
Smyth, A formula of Simons’ type and hypersurfaces with
constant mean curvature, J. Differential Geometry 3
(1969), 367–377. MR 0266109
(42 #1018)
 [11]
Koichi
Ogiue, On fiberings of almost contact manifolds, Kōdai
Math. Sem. Rep. 17 (1965), 53–62. MR 0178428
(31 #2685)
 [12]
Tominosuke
Ôtsuki, Minimal hypersurfaces in a Riemannian manifold of
constant curvature., Amer. J. Math. 92 (1970),
145–173. MR 0264565
(41 #9157)
 [13]
James
Simons, Minimal varieties in riemannian manifolds, Ann. of
Math. (2) 88 (1968), 62–105. MR 0233295
(38 #1617)
 [14]
Shûkichi
Tanno, Sasakian manifolds with constant 𝜑holomorphic
sectional curvature, Tôhoku Math. J. (2) 21
(1969), 501–507. MR 0251667
(40 #4894)
 [15]
Kentaro
Yano and Shigeru
Ishihara, Submanifolds with parallel mean curvature vector, J.
Differential Geometry 6 (1971/72), 95–118. MR 0298598
(45 #7650)
 [16]
Kentaro
Yano and Masahiro
Kon, Antiinvariant submanifolds, Marcel Dekker, Inc., New
YorkBasel, 1976. Lecture Notes in Pure and Applied Mathematics, No. 21. MR 0425849
(54 #13799)
 [17]
S. T. Yau, Submanifolds with constant mean curvature. I, Amer. J. Math. 97 (1975), 76100.
 [1]
 R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 149. MR 0251664 (40:4891)
 [2]
 D. Blair, A generalization of the catenoid, Canad. J. Math. 27 (1975), 231236. MR 0380637 (52:1534)
 [3]
 B. Y. Chen and K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1973), 257266. MR 0346708 (49:11433)
 [4]
 N. Ejiri, Totally real minimal submanifolds in a complex projective space, preprint. MR 671223 (84g:53065)
 [5]
 J. A. Erbacher, Isometric immersions with constant mean curvature and triviality of the normal bundle, Nagoya Math. J. 45 (1972), 139165. MR 0380679 (52:1576)
 [6]
 C. S. Houh, Some totally real minimal surface in , Proc. Amer. Math. Soc. 40 (1973), 240244. MR 0317189 (47:5737)
 [7]
 W. Y. Hsiang and H. B. Lawson, Jr., Minimal submanifolds of lowcohomogeneity, J. Differential Geom. 5 (1971), 138. MR 0298593 (45:7645)
 [8]
 H. B. Lawson, Jr., Complete minimal surfaces in , Ann. of Math. (2) 90 (1970), 335374. MR 0270280 (42:5170)
 [9]
 H. Naitoh, Totally real parallel submanifolds in , Tokyo J. Math. 4 (1981), 279306. MR 646040 (83h:53072)
 [10]
 K. Nomizu and B. Smyth, A formula of Simon's type and hypersurfaces with constant mean curvature, J. Differential Geom. 3 (1969), 367377. MR 0266109 (42:1018)
 [11]
 K. Ogiue, On fiberings of almost contact manifolds, Kōdai Math. Sem. Rep. 17 (1965), 5362. MR 0178428 (31:2685)
 [12]
 T. Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math. 92 (1970), 145173. MR 0264565 (41:9157)
 [13]
 J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62105. MR 0233295 (38:1617)
 [14]
 S. Tanno, Sasakian manifolds with constant holomorphic sectional curvature, Tôhoku Math. J. 21 (1969), 501507. MR 0251667 (40:4894)
 [15]
 K. Yano and S. Ishihara, Submanifolds with parallel mean curvature vector, J. Differential Geom. 6 (1971), 95118. MR 0298598 (45:7650)
 [16]
 K. Yano and M. Kon, Antiinvariant submanifolds, Marcel Dekker, New York, 1976. MR 0425849 (54:13799)
 [17]
 S. T. Yau, Submanifolds with constant mean curvature. I, Amer. J. Math. 97 (1975), 76100.
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
53C42
Retrieve articles in all journals
with MSC:
53C42
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719830684514X
PII:
S 00029947(1983)0684514X
Keywords:
Minimal cone,
cohomogeneity,
catenoid,
warped product,
Sasakian structure,
holomorphic sectional curvature,
totally real submanifold
Article copyright:
© Copyright 1983
American Mathematical Society
