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.