# REGULARITY OF SOAP FILM-LIKE SURFACES SPANNING GRAPHS IN A RIEMANNIAN MANIFOLD

• Gulliver, Robert (ROBERT GULLIVER SCHOOL OF MATHEMATICS UNIVERSITY OF MINNESOTA) ;
• Park, Sung-Ho (GRADUATE SCHOOL OF EDUCATION HANKUK UNIVERSITY OF FOREIGN STUDIES) ;
• Pyo, Jun-Cheol (DEPARTMENT OF MATHEMATICS SEOUL NATIONAL UNIVERSITY) ;
• Seo, Keom-Kyo (DEPARTMENT OF MATHEMATICS SOOKMYUNG WOMEN'S UNIVERSITY)
• Published : 2010.09.01
• 76 4

#### Abstract

Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-{\kappa}^2$. Using the cone total curvature TC($\Gamma$) of a graph $\Gamma$ which was introduced by Gulliver and Yamada [8], we prove that the density at any point of a soap film-like surface $\Sigma$ spanning a graph $\Gamma\;\subset\;M$ is less than or equal to $\frac{1}{2\pi}\{TC(\Gamma)-{\kappa}^2Area(p{\times}\Gamma)\}$. 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 $TC(\Gamma)$ < $3.649{\pi}\;+\;{\kappa}^2\inf\limits_{p{\in}F}Area(p{\times}{\Gamma})$, then the only possible singularities of a piecewise smooth (M, 0, $\delta$)-minimizing set $\Sigma$ are the Y-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $\pi$/b, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.

#### Keywords

soap film-like surface;graph;density

#### Acknowledgement

Supported by : NRF

#### References

1. F. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165, viii+199 pp.
2. F. Almgren and J. Taylor, The geometry of soap films and soap bubbles, Scientific American 235 (1976), 82-93. https://doi.org/10.1038/scientificamerican0776-82
3. J. Choe, The isoperimetric inequality for minimal surfaces in a Riemannian manifold, J. Reine Angew. Math. 506 (1999), 205-214.
4. J. Choe and R. Gulliver, Isoperimetric inequalities on minimal submanifolds of space forms, Manuscripta Math. 77 (1992), no. 2-3, 169-189. https://doi.org/10.1007/BF02567052
5. J. Choe and R. Gulliver, Embedded minimal surfaces and total curvature of curves in a manifold, Math. Res. Lett. 10 (2003), no. 2-3, 343-362. https://doi.org/10.4310/MRL.2003.v10.n3.a5
6. T. Ekholm, B. White, and D. Wienholtz, Embeddedness of minimal surfaces with total boundary curvature at most $4{\pi}$, Ann. of Math. (2) 155 (2002), no. 1, 209-234. https://doi.org/10.2307/3062155
7. H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.
8. R. Gulliver and S. Yamada, Area density and regularity for soap film-like surfaces spanning graphs, Math. Z. 253 (2006), no. 2, 315-331. https://doi.org/10.1007/s00209-005-0903-9
9. D. Kinderlehrer, L. Nirenberg, and J. Spruck, Regularity in elliptic free boundary problems, J. Analyse Math. 34 (1978), 86-119 https://doi.org/10.1007/BF02790009
10. O. Ore, Graphs and Their Uses, Random House, New York, 1963.
11. L. Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, 1983. vii+272 pp.
12. J. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489-539. https://doi.org/10.2307/1970949