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Approximation by smooth embedded hypersurfaces with positive mean curvature

Published online by Cambridge University Press:  17 April 2009

Fang Hua Lin
Fang Hua Lin
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, New York 10012
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Abstract

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Here we initiate the study of the following problem. Let Ω be a compact domain in a Riemannian manifold such that ∂Ω is of minimum area for the contained volume. Can ∂Ω be approximated by smooth hypersurfaces of positive mean curvature? It reduces to the question of whether or not a stable (or minimizing) hypercone in a Euclidian space can be approximated by smooth hypersurfaces of positive mean curvature. The positive solution to the problem may be useful for studying the curvature and topology of Ω.

We show in this paper that such approximation is possible provided that the given minimal cone satisfies some additional hypothesis.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 36 , Issue 2 , October 1987 , pp. 197 - 208
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Geometric measure theory and the calculus of variations. (Proc. Sympos. Pure Math. 44, Allard, W.K. and Almgren, F.J. Jr., Eds., American Mathematical Society, Providence, Rhode Island, 1986).CrossRefGoogle Scholar
[2]Almgren, F.J., Solomon, B., “How to connect minimal surfaces by bridges”, Abstracts Ams. Math. Soc. (1980), 775–B7.Google Scholar
[3]Caffarelli, L., Hardt, R., Simon, L., “Minimal surfaces with isolated singularities”, Manuscripta Math. 48 (1984), 1–18.CrossRefGoogle Scholar
[4]Gouzalez, E., Massari, U., Tamanini, I., “On the regularity of bodies of sets minimizing perimeter with a volume constraint”, Indiana Univ. Math. J. 32 (1983), 25–37.CrossRefGoogle Scholar
[5]Hardt, R., Simon, L., “Area minimizing hypersurfaces with isolated singularities”, J. Reine Angew. Math. 362 (1985), 102–129.Google Scholar
[6]Lawson, B., Michelsohn, M.L., “Embedding and surrounding with positive mean curvature”, Inv Math. 77 (1984), 399–419.CrossRefGoogle Scholar
[7]Lin, F.M., “Minimality and stability of minimal hypersurfaces in ℝn+1”, Bull. Austral. Math. Soc. (Paper # 6546).Google Scholar
[8]Lin, F.H., Regularity for a class of parametric obstacle problem, Ph.D. Thesis (University of Minnesota, Minneapolis, 1985).Google Scholar
[9]Meeks, W., Yau, S.T., “The existence of embedded minimal surfaces and the porblem of uniqueness”, Math. Z. 179 (1982), 151–168.CrossRefGoogle Scholar
[10]Simon, L., “Isolated singularities of extrema of geometric variational problems”, C.M.A. Research Report, (Australian National University, 1984).Google Scholar
[11]Simon, L., Lectures on Geometric Measure Theory, (Centre for Mathematical Analysis, Australian National University, 1983).Google Scholar
[12]Tomi, F., “A perturbation theorem for surfaces of constant mean curvature”, Math. Z. 141 (1975), 253–263.CrossRefGoogle Scholar