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Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant −κ2. Using the cone total curvature TC(Γ) of a graph ¡ which was introduced by Gulliver and Yamada [8], we prove that the density at any point of a soap film-like surface Σ spanning a graph ¡ Γ M is less than or equal to [수식]. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when n = 3,this density estimate implies that if [수식] then the only possible singularities of a piecewise smooth (M, 0, δ)-minimizing set Σ are the Y -singularity cone. In a manifold with sectional curvature bounded above by b2 and diameter bounded by π/b, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.
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참고문헌 (12)
참고문헌 신청
D. Kinderlehrer / 1978 / Regularity in elliptic free boundary problems / J. Analyse Math. 34 : 86 ~ 119
F. Almgren / 1976 / Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints / Mem. Amer. Math. Soc. 4 (165) : 8 ~ 199
F. Almgren / 1976 / The geometry of soap films and soap bubbles / Scientific American 235 : 82 ~ 93
H. Federer / 1969 / Geometric Measure Theory / Springer-Verlag
J. Choe / 1992 / Isoperimetric inequalities on minimal submanifolds of space forms / Manuscripta Math. 77 (2) : 169 ~ 189
F. Almgren / 1976 / Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints / Mem. Amer. Math. Soc. 4 (165) : 8 ~ 199
F. Almgren / 1976 / The geometry of soap films and soap bubbles / Scientific American 235 : 82 ~ 93
H. Federer / 1969 / Geometric Measure Theory / Springer-Verlag
J. Choe / 1992 / Isoperimetric inequalities on minimal submanifolds of space forms / Manuscripta Math. 77 (2) : 169 ~ 189
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