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Transactions of the American Mathematical Society

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

 
 

 

A construction of
homologically area minimizing hypersurfaces
with higher dimensional singular sets


Author: Nathan Smale
Journal: Trans. Amer. Math. Soc. 352 (2000), 2319-2330
MSC (2000): Primary 53A10; Secondary 49Q05
DOI: https://doi.org/10.1090/S0002-9947-99-02576-3
Published electronically: November 17, 1999
MathSciNet review: 1695037
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Abstract: We show that a large variety of singular sets can occur for homologically area minimizing codimension one surfaces in a Riemannian manifold. In particular, as a result of Theorem A, if $N$ is smooth, compact $n+1$ dimensional manifold, $n\geq 7$, and if $S$ is an embedded, orientable submanifold of dimension $n$, then we construct metrics on $N$ such that the homologically area minimizing hypersurface $M$, homologous to $S$, has a singular set equal to a prescribed number of spheres and tori of codimension less than $n-7$. Near each component $\Sigma $ of the singular set, $M$ is isometric to a product $C\times \Sigma $, where $C$ is any prescribed, strictly stable, strictly minimizing cone. In Theorem B, other singular examples are constructed.


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Additional Information

Nathan Smale
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

DOI: https://doi.org/10.1090/S0002-9947-99-02576-3
Received by editor(s): January 30, 1998
Published electronically: November 17, 1999
Article copyright: © Copyright 2000 American Mathematical Society