(1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be deformed to a harmonic map with minimum energy in its homotopy class in the following cases: (a) Riem ' f t ^0; (b) dim M = 2 and n2(N) = 0. (3) Any map 0O: S m -> S can be deformed to a harmonic map provided m ^ 7. More generally, suitably restricted harmonic polynomial maps can be joined to provide harmonic maps between spheres. (4) The homotopy class of maps of degree 1 from the 2-torus T to the 2-sphere S has no harmonic representative, whatever Riemannian metrics are put on T and S. (5) If in (2) M has a smooth boundary, then various Dirichlet problems have solutions in case (a) and (b); and also when the boundary data is sufficiently small.

Another Report on Harmonic Maps

The problem of Plateau or the proof of the existence of a surface of least area bounded by a given Jordan contour in Euclidean space is one of the classical problems of the Calculus of Variations. This problem has been studied at least since the time of Riemann but it was not until 1930 that satisfactory proofs for a general contour were given independently by Douglas [25]' and Rado [15]. However, before that time, considerable information was gathered and the problem was solved in many interesting special cases. For the literature of this period, we refer to the excellent report of Rad6 [38] on the problem of Plateau. Shortly after the appearance of the solutions of Douglas and Rado, McShane presented an interesting solution which was perhaps more along the lines of the straightforward "direct methods of the Calculus of Variations" (see [30]). In the past seventeen years the problem has been generalized by replacing r by several contours r1, * * *, rk and by prescribing the topological structure of the desired surface ([5], [19], [27], [28], 129], [33], 135], [41], [43]), or by replacing the condition that the surface be bounded by a given contour by the condition that only parts of the boundary be prescribed, the remainder of the boundary being merely restricted to lie along certain manifolds (see [18], [22], [23], [24]). Finally minimal surfaces which do not furnish even relative minima have been discussed (see [20], [21], [31], [32], [33], [34], [35], [39], [40], [42], [43]). The present paper generalizes the problem by replacing the underlying Euclidean space by a Riemannian manifold of considerable generality. The writer knows of only one paper in which the Plateau problem has been solved in any space other than Euclidean space, namely the recent paper by A. Lonseth [7] where the space considered is "hyperbolic space". However, S. Bochner [2] has studied the "harmonic" functions in a general Riemannian metric (of sufficient differentiability) but has not demonstrated their minimizing property in general. This work is, of course of great interest as it is still true in a Riemannian space (of sufficient differentiability) that if a minimal surface is mapped conformally on a plane region, then the representing vector is "harmonic" in Bochner's sense and also that a surface of least area (if of sufficient differentiability) is a minimal surface (in the sense of Differential Geometry). The present writer does not attempt to generalize all of the preceding results on the Plateau problem but restricts himself to the case of a surface of the type of a k-fold connected plane region bounded by k given contours. The space is a "homogeneously regular" (see Def. 3.1) Riemannian manifold 9) of class C'. Every compact, regular Riemannian manifold of class C' is seen to be homo-

The Problem of Plateau on a Riemannian Manifold

It was in the early nineteen thirties that Douglas and Radd solved the Plateau problem for a rectifiable Jordan curve in Euclidean space. The minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk. In 1948 Morrey generalized the theorem of Douglas and Radd to minimal surfaces in a homogeneously regular Riemannian manifold. In both cases the solutions are defective as they may possess branch points. It was not until 1968 that Osserman [-24] was able to prove that interior branch points do not exist on the Douglas-Radd-Morrey solution. (Osserman's theorem in this full generality was proved by Alt [-2, 3] and Gulliver [8].) If the Jordan curve is real analytic, Gulliver and Lesley [9] showed that the solution surface is free of branch points even on its boundary. Hence, at least in this case, we know that the Douglas-Radd-Morrey solution is an immersion and represents a classical surface. However, the Douglas-Radd-Morrey solution may in general have selfintersections. It was then a classical problem to find conditions on the Jordan curve to guarantee that the solution is an embedded disk. In 1932 Radd was the first to produce a general condition of this type. He proved that if the curve can be projected onto a convex curve in the plane, then it bounds a unique embedded minimal disk which is a graph over this plane. It was not until the seventies that there was further progress on the problem of embeddedness. Osserman had conjectured that if a Jordan curve is extreme, that is, lies on the boundary of its convex hull, then the Douglas-Rad6 solution is embedded. Gulliver and Spruck [10] proved this conjecture under the additional assumption that the absolute total curvature of the Jordan curve is less than 4~z. Right after this work, Tomi and Tromba [32] showed that an extreme curve must bound an embedded minimal disk which need not be stable. In 1976 Almgren and Simon [1] established the existence for such a curve of an embedded stable minimal disk (which need not be a Douglas-Radd solution). Finally in the same year, the authors [18] proved that any Douglas-Rad6 solution must be embedded.

The existence of embedded minimal surfaces and the problem of uniqueness

w 1. In t roduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 w 2. The minimal surface system . . . . . . . . . . . . . . . . . . . . . . . . . 2 w 3. A brief summary of known results in eodimension one . . . . . . . . . . . . . . 4 w 4. The existence of solutions of dimension two . . . . . . . . . . . . . . . . . . . 5 w 5. The non-uniqueness and non-stabi l i ty of solutions of dimension two . . . . . . . . 6 w 6. The non-existence of solutions in dimensions >/4 . . . . . . . . . . . . . . . . . 11 w 7. The existence of non-parametric minimal cones . . . . . . . . . . . . . . . . . 13

https://projecteuclid.org/download/pdf_1/euclid.acta/1485889963

Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system

A new uniqueness theorem for minimal surfaces

Corrections to Our Paper on the Existence of Minimal Surfaces of General Critical Types

Unstable minimal surfaces of higher topological structure

Minimal Surfaces Not of Minimum Type by a New Mode of Approximation

The Continuity of the Area of Harmonic Surfaces as a Function of the Boundary Representations

test for exhaustiveness calls for reasoning so circuitous that it is omitted here. In any case, such a conclusion requires independent checking. These diagrams are of interest, as exhibiting schematic symmetries, or rotation periods 2 or 3 or 7. I think it correct to assert further (cf. §3) that the completion of each pattern by inclusion of its minor polyedra is uniquely determinate; so that 11 appears to be the total number of systems unlike in respect of numbers and contiguities. Not inapposite may be an obiter dictum from the late mathematician of wide vision, Maxime Bocher. \"Moreover, although the mathematical method is the traditional one for arriving at the truth concerning geometrical facts, it is not the only one. Direct appeal to the intuition is often a short and fairly safe cut to geometric results.\"*

On the Existence of Minimal Surfaces of General Critical Types.

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