JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.Introduction. With any smooth mapping of one Riemannian manifold into another it is possible to associate a variety of invariantly defined functionals. Each such functional of course determines a class of extremal mappings, in the sense of the calculus of variations, and those extremals, in the very special cases thus far considered, play an important role in a number of familiar differential-geometric theories.The present paper is devoted to a rather general study of a functional E of geometrical and physical interest, analogous to energy. Our central problem is that of deforming a given mapping into an extremal of E. Following an infinite-dimensional analogue of the Morse critical point theory, we construct gradient lines of E (in a suitable function space) ; and E is a decreasing function along those lines. With suitable metric and curvature assumptions on the target manifold (assumptions which cannot be entirely circumvented, in view of the examples of ?? 4E and lOD), we prove that the gradient lines do in fact lead to extremals (see Theorem 11A). If f: M -> M' is a smooth mapping of manifolds whose metrics are gjjdxtdxj resp. g,6'dyadyfl, then the energy E (f) is defined by the integral E (f ) =21g go" ox! ,xi where the fi" are local coordinates of the point f (x), * 1 being the volume element of M (assumed compact). Thus E (f) can be considered as a generalization of the classical integral of Dirichlet. The Euler-Lagrange equations for E are a system of non-linear partial differential equations of elliptic type: Afc + rpy'a af3 i9f 01 ; A is the Laplace-Beltrami operator on M and the r,,y' are the Christoffel symbols on M'. Although this system is suggestive of the simple equation Au + + (u) * grad2 ut = 0for one unknown, there is in general very little connection between the two because of the phenomenon of curvature.It has been necessary to go into the question of existence of solutions in rather great detail, owing to the want of general results for non-linear systems. Direct methods of the calculus of variations seem to lead to severe difficulties, and that is one reason why we have preferred to approach the problem through the gradient-line technique, which amounts to replacing the equations above by a system of parabolic equations whose relation to the elliptic system is analogous to that of Fourier's equation to Laplace's equation. This approach is of independent interest, in any case. Our methods are strongly potential-theoretic in nature. The local equations are first replaced by global equations of essentially the same form, embed...
