Critical Points of Harmonic Functions on Domains In/?3

It is shown that the critical point relations of Morse theory, together with the maximum principle, comprise a complete set of critical point relations for harmonic functions of three variables. The proof proceeds by first constructing a simplified example and then developing techniques to modify this example to realize all admissible possibilities. Techniques used differ substantially from those used by Morse in his solution of the analogous two-variable problem. Introduction. Critical point theory is a subject which has been of pure and applied interest for many years. Morse resolved the problem associated with the study of critical points of nondegenerate functions of several variables in the late 1920's. A. Sard, a student of Morse, is credited with the discovery of a method to perturb a function with a degenerate critical point and to split or bifurcate it into several nondegenerate critical points. R. Thorn has done famous recent work on applications of bifurcation theory to many difficult problems in biology as well as other areas not usually associated with differential topology. This paper deals with an aspect of the following classical problem. A certain collection of functions is fixed and we seek a full description of the critical points theory of the functions considered as a class. If the functions are merely C°°, then the best available description of their critical point theory is contained in the Morse inequalities [3]. These inequalities relate the numbers and the kinds of critical points of a function with the Betti numbers of the domain. In his book, Topological methods in the study of functions of a complex variable [1], Morse, did a complete treatment of the critical points theory of functions of two variables which resemble harmonic functions topologically. In doing this, he proved all the standard theorems relating zeros, poles, and branch points of his class of pseudoharmonic and pseudomeromorphic functions, the latter of which he called inner transformations. It would seem that the critical point behavior of harmonic functions of two variables should then be determined solely by topological considerations. In this vein, Morse proved a completeness result which can be summarized as follows. Given integers which satisfy the Morse inequalities for a harmonic function on a topological disc (including the condition that no interior point can be an extremum), there exists a region ö in Ä2, ñ homeomorphic to a two-cell, and a function U such that the given integers describe the critical points of U on ß and its boundary. Roughly speaking, it means that the theory of critical points of Received by the editors October 3, 1978 and, in revised form, June 28, 1979 and December 26, 1979. AMS (MOS) subject classifications (1970). Primary 58E05. © 1980 American Mathematical Society 0002-9947/80/0000-0407/$06.50 137 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use