My first mathematical research largely resolved the problem of branch
points of parametric two-dimensional minimal surfaces and surfaces
of prescribed mean curvature (prescribed mean curvature vector,
for codimension > 1).
- My thesis, written at Stanford under Bob Osserman, showed that a conformally
parameterized disk of prescribed mean curvature which
minimizes the corresponding functional (area, or area + volume
integral, resp.) is free of interior branch points in codimension
one, and has only "true" branch points in arbitrary codimension
 (a branch point is "false" if it is a branched covering of an
embedded surface, otherwise it is a "true" branch point).
- In order to eliminate false branch points,
the paper  assumed the surface had the topological type of
the disk; in later papers, I extended the result to oriented
surfaces of arbitrary genus with any finite number of boundary
components, assuming the Douglas hypothesis holds .
The Douglas hypothesis requires that the minimum of the
functional among surfaces of the same topological
type be strictly smaller than among surfaces of lower topological
type, which means: having smaller total genus and more connected
components, one of these holding strictly. Jesse Douglas had used this
hypothesis to show the existence of a minimal surface of given
topological type with a given family of Jordan curves as boundary
(J. Math. Phys. 15, 105-123 (1936)).
- Theorems leading up to the result of  are in  and .
In particular,  proves the fundamental theorem of branched
immersions. To describe the fundamental theorem, call a mapping
"ramified" if it describes the same germ of surface at two
distinct points. A point in the domain of the mapping is
"ramified" if the mapping is ramified in every neighborhood. In
particular, a false branch point must be a ramified point. The
fundamental theorem of branched immersions states that a branched
immersion with the unique continuation property (such as a
surface of prescribed mean curvature vector) from an oriented
compact surface-with-boundary, which is injective on the boundary,
factors through an unramified branched immersion defined on
another compact surface-with-boundary.
- The paper  shows
that a branched immersion from the interior of a compact oriented
surface-with-boundary, whose boundary mapping is injective, is
topologically equivalent to a branched immersion on the closed
- In joint work with Frank David Lesley, I showed that these results
on interior branch points of surfaces of prescribed mean curvature
vector are also valid for boundary branch points along a real-analytic
segment of the boundary curve .
- Three questions were not
succesfully addressed in those papers, even in codimension one,
and remain open problems: (1) false branch points
in the free-boundary problem (first results by Alt and Tomi, and
by Ye); (2) false branch points on non-orientable surfaces; and
(3) true boundary branch points for smooth (but not analytic)
Plateau boundary conditions (see my perplexing example in the do
Carmo Festschrift ). Another fascinating problem which was
open for a long time (I heard about it from Blaine Lawson in 1972)
is the behavior near a branch point of an area-minimizing surface
in higher codimension. Mario Micallef and Brian White solved this
problem, showing that the surface closely resembles a holomorphic
curve for some orthogonal complex structure on an even-dimensional
submanifold [Ann. of Math. 141, 35-85 (1995)]. White then went
on to show that even in higher codimension, there can be no "true"
branch points along a real-analytic segment of the boundary curve
[Acta Math. 179, 295-305 (1997)].
. Regularity of Minimizing Surfaces of Prescribed Mean Curvature,
Annals of Mathematics 97, 275-305 (1973).
. Branched Immersions of Surfaces and Reduction of
Topological Type, I: Math. Z. 145, 267-288 (1975).
. Finiteness of the Ramified Set for Branched Immersions of
Surfaces, Pacific J. Math. 64, 153-166 (1976).
. Branched Immersions of Surfaces and Reduction of Topological
Type, II: Math. Annalen 230, 25-48 (1977).
. On Boundary Branch Points of Minimizing Surfaces, Archive
Rational Mech. Anal. 52, 20-25 (1973)
(with Frank David Lesley).
. A Minimal Surface with an Atypical Boundary Branch Point, pp. 211-228
of Differential Geometry: a Symposium in honor of Manfredo P. do
Carmo, B. Lawson and K. Tenenblat, eds., Longman, Harlow 1991.