Minimal hypersurfaces of $\textbf {R}^ {2m}$ invariant by $\textrm {SO}(m)\times \textrm {SO}(m)$
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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
-
H. Alencar, Hipersuperfícies mínimas de ${{\mathbf {R}}^{2m}}$ invariantes por ${\text {SO}}(m) \times {\text {SO}}(m)$, Tese de Doutorado, IMPA, 1988.
A. Back, M. do Carmo, and W. Y. Hsiang, On some fundamental equations in equivariant Riemannian geometry, Preprint.
- J. L. Barbosa and M. do Carmo, Helicoids, catenoids, and minimal hypersurfaces of $\textbf {R}^{n}$ invariant by an $\ell$-parameter group of motions, An. Acad. Brasil. Ciênc. 53 (1981), no. 3, 403–408. MR 663233
- E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. MR 250205, DOI 10.1007/BF01404309 J. de M. Gomes, Sobre hipersuperfícies com curvatura média constante no espaço hiperbólico, Tese de Doutorado, IMPA, 1984.
- Jack K. Hale, Ordinary differential equations, 2nd ed., Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. MR 587488
- 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 Geometry 17 (1982), no. 4, 583–594 (1983). MR 683166
- 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, DOI 10.2307/2007036
- 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
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 337 (1993), 129-141
- MSC: Primary 53C42
- DOI: https://doi.org/10.1090/S0002-9947-1993-1091229-1
- MathSciNet review: 1091229