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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A variational problem for submanifolds of Euclidean space


Author: Joseph A. Erbacher
Journal: Proc. Amer. Math. Soc. 36 (1972), 467-470
MSC: Primary 53C30; Secondary 53C70
DOI: https://doi.org/10.1090/S0002-9939-1972-0309020-1
MathSciNet review: 0309020
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Abstract: Let $ {M^n}$ be a compact differentiable manifold and $ {R^{n + k}}$ Euclidean space. A necessary and sufficient condition is given for an immersion $ \psi :{M^n} \to {R^{n + k}}$ to be a stationary immersion for $ J = \smallint M_\psi ^n\langle x - {x_c},x - {x_c}\rangle $ dv subject to the side condition $ V = \smallint M_\psi ^n$ dv= a fixed constant, where $ {x_c}$ is the center of mass. In particular, minimal submanifolds of spheres satisfy this condition.


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

  • [1] E. Hopf, Lectures on differential geometry in the large, Notes, Stanford University, Stanford, Calif.
  • [2] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0309020-1
Keywords: Compact differentiable manifold, Riemannian, Euclidean space, center of mass, stationary immersion, mean curvature normal, minimal submanifold
Article copyright: © Copyright 1972 American Mathematical Society

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