where A is the area enclosed by a curve C of length L, and where equality holds if and only if C is a circle. The purpose of this paper is to recount some of the most interesting of the many sharpened forms, generalizations, and applications of this inequality, with emphasis on recent contributions. Earlier work is summarized in the book of Hadwiger [1], Other general references, varying from very elementary to quite technical are Kazarinoff [1], Pólya [2, Chapter X], Porter [1], and the books of Blaschke listed in the bibliography. Most books on convexity also contain a discussion of the isoperimetric inequality from that perspective. One aspect of the subject is given by Burago [1]. Others may be found in a recent paper of the author [4] on Bonnesen inequalities and in the book of Santaló [4] on integral geometry and geometric probability. An important note: we shall not go into the area of so-called \"isoperimetric problems\". Those are simply variational problems with constraints, whose name derives from the fact that inequality (1) corresponds to the first example of such a problem: maximize the area of a domain under the constraint that the length of its boundary be fixed. There are also the \"isoperimetric inequalities\" of mathematical physics. They are special cases of isoperimetric problems in which typically some physical quantity, usually represented by the eigenvalues of a differential equation, is shown to be extremal for a circular or spherical domain. Extensive discussions of such problems can be found in the book of Pólya and Szegö [1] and the review article by Payne [1]. We shall discuss them here only insofar as they relate to the main subject of this paper. What we shall concentrate on here is \"the\" isoperimetric inequality (1) and other geometric versions and generalizations of it. We shall also consider

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The isoperimetric inequality

0. Introduction In this paper we develop a regularity theory for energy minimizing harmonic maps into Riemannian manifolds. Let u: M -> N be a map between Riemannian manifolds of dimension n and k. It was shown by C. B. Morrey [17] in 1948 that if n — 2, then an energy minimizing harmonic map is Holder continuous (and smooth if M and N are smooth). Since that time results have been found under special assumptions on N. Eells and Sampson [5] proved in 1963 that if N is compact and has nonpositive curvature, then every homotopy class of maps from a closed manifold M into N has a smooth harmonic representative. In the case where the image of the map is contained in a convex ball of N9 there is a complete existence and regularity theory due to Hildebrandt and Widman [15] as well as Hildebrandt, Kaul and Widman [13]. Recently Giaquinta and Giusti obtained results for the case in which the image lies in a coordinate chart [9], [10]. In this paper we show that a bounded, energy minimizing map u: M -» N is regular (in the interior) except for a closed set S of Hausdorff dimension at most n — 3. We also show S is discrete for n = 3. Moreover, we derive techniques (see Theorem IV) for lowering the dimension of S under the condition that certain smooth harmonic maps of spheres into N are trivial. This can be checked in some interesting cases, for example if N has nonpositive curvature or if the image of the map lies in a convex ball of N, we show S = 0 and any minimizing harmonic map into such a manifold is smooth. Using our methods, it is possible to reduce the dimension of S if N is a sphere or Lie group by studying harmonic spheres in N. Our methods work for functional which are the energy plus lower order terms, and thus have direct bearing on the question of the existence of global Coulomb gauges in nonabelian gauge theories. We point out that there is a strong historical precedent for partial regularity results in problems involving elliptic systems (see Almgren [1], De Giorgi [3],

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A regularity theory for harmonic maps

https://www.jstor.org/stable/pdfplus/1971247.pdf

Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature

(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 authors discuss a generalization of the usual Green function to equations with only measurable and bounded coefficients The existence and uniqueness as well as several other important properties are shown Such a Green function proves useful in connection with quasilinear elliptic systems of “diagonal type”

The Green function for uniformly elliptic equations

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An existence theorem for harmonic mappings of Riemannian manifolds

Two-Dimensional variational problems with obstructions and Plateau's problem for H-surfaces in a Riemannian manifold

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Harmonic mappings into Riemannian manifolds with non-positive sectional curvature.

The Christoffelsymbols with respect to a normal chart of a Riemannian manifold will be estimated in absolute value by using comparison theorems.

Helmut Kaul

Papers

An existence theorem for harmonic mappings of Riemannian manifolds

Two-Dimensional variational problems with obstructions and Plateau's problem for H-surfaces in a Riemannian manifold

Harmonic mappings into Riemannian manifolds with non-positive sectional curvature.

Schranken für die Christoffelsymbole

Dirichlet's boundary value problem for harmonic mappings of Riemannian manifolds