I was inspired by the recent result of Ekholm, White, and Wienholtz that a stationary minimal surface in Rn, whose boundary curve has total curvature at most 4 pi, must be embedded. The paper was published in Annals of Mathematics 155 (2002), 109-234. See the Postscript version. Their result implies the well-known Fáry-Milnor theorem on the total curvature of knots in R3. Jaigyoung Choe and I took up the challenge to extend their result, with an appropriately modified hypothesis, to space forms and to manifolds of variable curvature. We showed that in a space form of constant sectional curvature K0, if the total curvature of a curve, plus the supremum of K0 times the area of cones over the curve with vertex in the convex hull, is at most 4 pi, then any minimal surface with that curve as boundary must be embedded. These hypotheses are sharp. Moreover, If K0 is nonpositive, then the same result holds in a manifold whose sectional curvatures are bounded above by K0 [44]. This implies the unknotting theorem of Alexander-Bishop and of Schmitz in Hadamard manifolds, as well as some new unknotting results. See the Postscript version or the PDF version. I reported on this work, and on possible analogous density estimates for mean-curvature flow, at the Workshop on Geometric Evolution Equations held at NCTS in Hsinchu, Taiwan in July, 2002. My paper [47] for the Proceedings of that workshop is available in the Postscript version or the PDF version.

Further open problems suggested by this work include the following. For problems (1) and (2), in a manifold whose sectional curvatures are bounded above by K0, let a smooth closed curve be given satisfying the above modified hypothesis for total curvature. (1) Is there only one minimal surface of the type of the disk and bounded by the curve? (In R3, this is Nitsche's 1973 theorem. See Invent. Math. 8 pp. 313-333.) (2) Is there any minimal surface of genus one, or higher, bounded by the curve? (an open problem even in R3.) (3) Is there an upper bound for the density of a k-dimensional stationary integral current in Rn in terms of some version of the total curvature of its boundary?

When 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.

The proof of isotopy to the standard theta-graph if the graph has net total curvature less than 4 pi is given in the second paper with Yamada (see [62]). This paper also establishes the same results for graphs which are only continuous, with a new definition of net total curvature analogous to the definition of total curvature of knots given by Milnor [1950]. In general, Milnor's description of total curvature as an integral over the unit sphere of the multiplicity of a knot is extended to describe the net total curvature of a graph. Taniyama's earlier version of total curvature TC(C) of a graph C gives different values; he shows [1998] that if a theta graph C has TC(C) ‹ 5 pi, then C is isotopic to a standard embedding. For a theta graph, TC(C) is at least equal to the net total curvature plus pi.

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 K0 be an upper bound for the sectional curvatures of M. When K0 is positive, we also assume that M has diameter at most pi/square root of K0. We also assume that S is strongly stationary with respect to C. If C has cone total curvature TC(C), then at any point p of S, 2 pi times the density of S is < TC(C) + K0 times the area of the geodesic cone over C with vertex p. See the PDF version.

[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.