Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Minimal hypersurfaces of $ {\bf R}\sp {2m}$ invariant by $ {\rm SO}(m)\times {\rm SO}(m)$

Author: Hilário Alencar
Journal: Trans. Amer. Math. Soc. 337 (1993), 129-141
MSC: Primary 53C42
MathSciNet review: 1091229
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G = {\text{SO}}(m) \times {\text{SO}}(m)$ act in the standard way on $ {{\mathbf{R}}^m} \times {{\mathbf{R}}^m}$. We describe all complete minimal hypersurfaces of $ {{\mathbf{R}}^m}\backslash \{ 0\} $ which are invariant under $ G$ for $ m = 2$, $ 3$ . We also show that the unique minimal hypersurface of $ {{\mathbf{R}}^{2m}}$ which is invariant under $ G$ and passes through the origin of $ {{\mathbf{R}}^{2m}}$ is the minimal quadratic cone.

References [Enhancements On Off] (What's this?)

  • [1] H. Alencar, Hipersuperfícies mínimas de $ {{\mathbf{R}}^{2m}}$ invariantes por $ {\text{SO}}(m) \times {\text{SO}}(m)$, Tese de Doutorado, IMPA, 1988.
  • [2] A. Back, M. do Carmo, and W. Y. Hsiang, On some fundamental equations in equivariant Riemannian geometry, Preprint.
  • [3] J. L. Barbosa and M. do Carmo, Helicoids, catenoids, and minimal hypersurfaces of 𝑅ⁿ invariant by an ℓ-parameter group of motions, An. Acad. Brasil. Ciênc. 53 (1981), no. 3, 403–408. MR 663233
  • [4] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. MR 0250205
  • [5] J. de M. Gomes, Sobre hipersuperfícies com curvatura média constante no espaço hiperbólico, Tese de Doutorado, IMPA, 1984.
  • [6] Jack K. Hale, Ordinary differential equations, 2nd ed., Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. MR 587488
  • [7] Wu-teh Hsiang and Wu-yi Hsiang, On the existence of codimension-one minimal spheres in compact symmetric spaces of rank 2. II, J. Differential Geom. 17 (1982), no. 4, 583–594 (1983). MR 683166
  • [8] Wu-yi Hsiang, Zhen Huan Teng, and Wen Ci Yu, New examples of constant mean curvature immersions of (2𝑘-1)-spheres into Euclidean 2𝑘-space, Ann. of Math. (2) 117 (1983), no. 3, 609–625. MR 701257, 10.2307/2007036
  • [9] Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C42

Retrieve articles in all journals with MSC: 53C42

Additional Information

Keywords: Minimal hypersurface, orbit space, invariant hypersurface, minimal quadratic cone
Article copyright: © Copyright 1993 American Mathematical Society