I was inspired by the recent result of Ekholm, White,
and Wienholtz that a stationary minimal surface in |

Sumio Yamada heard me talk about the work with Choe, he asked the
very interesting question whether it only works for the disjoint union of
simple closed curves, or whether it can be generalized to graphs, or networks.
In joint work, we showed that the methods, with suitable modifications, can be
extended to the case of a curved polyhedral surface whose variational boundary
is a graph. This also required finding the right definition of total curvature
for a graph. The paper with Yamada (see [57])
shows in particular that a graph which is not curved too much, having total
curvature less than 3.6 π, cannot be the boundary of an area-minimizing
surface with T singularities.Sumio and I have writen a more general paper [59], based on a different notion of total curvature. Anticipating this second paper, I wrote up a partial summary of the results in the summary of total curvature of graphs (see [58]). In particular, different notions of total curvature are compared, each useful in a different context. One of these is Taniyama's TC(C) for a graph C. |

In joint work [51] with Sung-ho Park, Juncheol Pyo, and Keomkyo Seo, the techniques applied in my papers [44] with Jaigyoung Choe and [48] with Sumio Yamada are brought to bear on the more general problem of a soap film-like surface S spanning a graph C in a simply connected 3-dimensional Riemannian manifold M. Let K

[44]. Embedded Minimal surfaces and Total Curvature of Curves in a
Manifold (with Jaigyoung Choe). * Math. Research
Letters * **10**, 343--362 (2003).
Postscript version or
PDF version.

[47]. "Density Estimates for Minimal Surfaces and Surfaces Flowing by
Mean Curvature." In Proceedings of Workshop on Geometric
Evolution Equations, NCTS, Taiwan (July 2002). Contemporary
Mathematics v. 367, 129-140 (2005).
Postscript version or
PDF version.

[48]. Area Density and Regularity for Soap Film-Like Surfaces Spanning
Graphs (with Sumio Yamada). *Math. Zeitschrift* **253**,
315--331 (2006). PDF version.

[49]. Total Curvature of Graphs in Space.
*Quarterly Journal Pure and Appl. Math.* **3**, 773--783 (2007).
PDF version.

[62]. Total Curvature and isotopy of graphs in $R^3$ (with Sumio
Yamada), ArXiv:0806.0406. PDF version.

[51]. Regularity of soap film-like surfaces spanning graphs in
a Riemannian manifold (with Sung-ho Park, Juncheol Pyo, and Keomkyo Seo)
to appear in the Journal of the Korean Math Society.