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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Fine structure of the space of
spherical minimal immersions


Authors: Hillel Gauchman and Gabor Toth
Journal: Trans. Amer. Math. Soc. 348 (1996), 2441-2463
MSC (1991): Primary 53C42
MathSciNet review: 1348151
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Abstract: The space of congruence classes of full spherical minimal immersions $f:S^m\to S^n$ of a given source dimension $m$ and algebraic degree $p$ is a compact convex body $\mathcal {M}_m^p$ in a representation space $\mathcal {F}_m^p$ of the special orthogonal group $SO(m+1)$. In Ann. of Math. 93 (1971), 43--62 DoCarmo and Wallach gave a lower bound for $\mathcal {F}_m^p$ and conjectured that the estimate was sharp. Toth resolved this ``exact dimension conjecture'' positively so that all irreducible components of $\mathcal {F}_m^p$ became known. The purpose of the present paper is to characterize each irreducible component $V$ of $\mathcal {F}_m^p$ in terms of the spherical minimal immersions represented by the slice $V\cap \mathcal {M}_m^p$. Using this geometric insight, the recent examples of DeTurck and Ziller are located within $\mathcal {M}_m^p$.


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Additional Information

Hillel Gauchman
Affiliation: Department of Mathematics, Eastern Illinois University, Charleston, Illinois 61920
Email: cfhvg@ux1.cts.eiu.edu

Gabor Toth
Affiliation: Department of Mathematics, Rutgers University, Camden, New Jersey 08102
Email: gtoth@crab.rutgers.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01588-7
PII: S 0002-9947(96)01588-7
Received by editor(s): March 20, 1995
Article copyright: © Copyright 1996 American Mathematical Society