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
[6] (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 [6] 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 [17].
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 [17] are in [13] and [15].
In particular, [13] 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 [15] 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
surface-with-boundary.
- 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 [9].
- 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 [34]). 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)].
[6]. Regularity of Minimizing Surfaces of Prescribed Mean Curvature,
Annals of Mathematics 97, 275-305 (1973).
[13]. Branched Immersions of Surfaces and Reduction of
Topological Type, I: Math. Z. 145, 267-288 (1975).
[15]. Finiteness of the Ramified Set for Branched Immersions of
Surfaces, Pacific J. Math. 64, 153-166 (1976).
[17]. Branched Immersions of Surfaces and Reduction of Topological
Type, II: Math. Annalen 230, 25-48 (1977).
[9]. On Boundary Branch Points of Minimizing Surfaces, Archive
Rational Mech. Anal. 52, 20-25 (1973)
(with Frank David Lesley).
[34]. 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.