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.
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
(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?
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.
Their result implies the well-known Fáry-Milnor theorem on
the total curvature of knots in R3.
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 . 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
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  for the Proceedings of that workshop is available in the
Postscript version or the
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 ).
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 .
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  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.
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 )
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 , 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 ). 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  with Sung-ho Park, Juncheol Pyo, and Keomkyo Seo, the
techniques applied in my papers  with Jaigyoung Choe and  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.
. Embedded Minimal surfaces and Total Curvature of Curves in a
Manifold (with Jaigyoung Choe). Math. Research
Letters 10, 343--362 (2003).
Postscript version or
. "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
. Area Density and Regularity for Soap Film-Like Surfaces Spanning
Graphs (with Sumio Yamada). Math. Zeitschrift 253,
315--331 (2006). PDF version.
. Total Curvature of Graphs in Space.
Quarterly Journal Pure and Appl. Math. 3, 773--783 (2007).
. Total Curvature and isotopy of graphs in $R^3$ (with Sumio
Yamada), ArXiv:0806.0406. PDF version.
. 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.