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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
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?)

    E. Hopf, Lectures on differential geometry in the large, Notes, Stanford University, Stanford, Calif.
  • 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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Keywords: Compact differentiable manifold, Riemannian, Euclidean space, center of mass, stationary immersion, mean curvature normal, minimal submanifold
Article copyright: © Copyright 1972 American Mathematical Society