Abstract
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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 197-208 |
| Number of pages | 12 |
| Journal | Bulletin of the Australian Mathematical Society |
| Volume | 36 |
| Issue number | 2 |
| DOIs | |
| State | Published - 1987 |
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- Mathematics(all)
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Approximation by smooth embedded hypersurfaces with positive mean curvature. / Lin, Fang-Hua.
In: Bulletin of the Australian Mathematical Society, Vol. 36, No. 2, 1987, p. 197-208.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Approximation by smooth embedded hypersurfaces with positive mean curvature
AU - Lin, Fang-Hua
PY - 1987
Y1 - 1987
N2 - 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.
AB - 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.
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U2 - 10.1017/S0004972700026472
DO - 10.1017/S0004972700026472
M3 - Article
VL - 36
SP - 197
EP - 208
JO - Bulletin of the Australian Mathematical Society
JF - Bulletin of the Australian Mathematical Society
SN - 0004-9727
IS - 2
ER -