The Morse Lemma in Infinite Dimensions via Singularity Theory

An infinite dimensional Morse lemma is proved using the deformation lemma from singularity theory. It is shown that the versions of the Morse lemmas due to Palais and Tromba are special cases. An infinite dimensional splitting lemma is proved. The relationship of the work here to other approaches in the literature in discussed. Introduction. This paper shows that when the singularity theory proof of the Morse lemma is extended to infinite dimensions, it gives a result better than the best available. The best available Morse lemma is that of Tromba [1976], [1981] which improves upon the usual Morse-Palais lemma (cf. Palais [1963], [1969]) for the following crucial reason: The Morse-Palais lemma assumes that the second derivative of the function at its critical point is strongly nondegenerate in the sense of defining an isomorphism between the space and its dual. Such a hypothesis is not satisfied in standard elliptic variational problems; however, the hypotheses of Tromba’s Morse lemma are normally verified in such problems. Specific examples are presented in Buchner, Marsden and Schecter [1983]; for others see (a) Tromba 1976], 1981 for geodesics and minimal surfaces; (b) Choquet-Bruhat and Marsden [1976] and Arms, Marsden and Moncrief [1982] for general relativity; (c) Ball, Knops and Marsden [1978] and Marsden and Hughes [1983] for elasticity. In conjunction with the Morse lemma are questions of 1. normal forms for more degenerate singularities and 2. a splitting lemma and reduction to finite dimensional catastrophe theory. Such questions have been studied by Magnus [1976], [1978], [1979], Arkeryd [1979] and Chillingworth [1980], but under hypotheses similar to those of the Morse-Palais lemma. In view of the difficulties with these hypotheses, it is important to also carry this program out under more applicable hypotheses. Such a setting is provided here. A related setting for a normal form theory in infinite dimensions is presented in Beeson and Tromba [1981]. Their situation is further complicated by the presence of a group action. A closely related setting is given in Dangelmayr [1979] and Magnus [1980]. The plan of the paper is as follows: 1. Theorem A in 2 gives conditions under which two given functions are related by a diffeomorphism in a neighborhood of a singular point. 2. Theorem B in [}3 is the Morse-Tromba lemma and is shown to be a straightforward consequence of Theorem A. 3. Section 4 discusses the splitting lemma and the associated reduction to finite dimensional catastrophe theory. Finally, we note that the ideas in Theorem A below are useful in the study of vector fields. In particular, the methods can be used to deal with some C-flat ambiguities in normal forms of vector fields at a singular point. These topics will be the subject of other publications. *Received by the editors June 28, 1982, and in revised form October 29, 1982. *Department of Mathematics, University of Houston, Houston, Texas 77004. *Department of Mathematics, University of California, Berkeley, California 94720. 103 D ow nl oa de d 12 /0 2/ 12 to 1 40 .2 54 .8 7. 10 3. R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp :// w w w .s ia m .o rg /jo ur na ls /o js a. ph p 1038 MARTIN GOLUBITSKY AND JERROLD MARSDEN 1. The singularity theory method. To put the methods in perspective, we shall recall some of the ideas of singularity theory with a view towards the Morse lemma. The basic methods of singularity theory under the notion of k-determining are contained in Mather 1970], Siersma [1974] and Wasserman [1974], though they are not stated there in precisely the form we use here. One of the goals of singularity theory is to bring functions into normal form in a neighborhood of a singular point. The procedure for doing so involves two steps: 1. The analytical step. This step gives criteria for when two functions are related by a diffeomorphism. This is done using what is called the deformation method and involves the integration of ordinary differential equations. 2. The algebraic step. The verification of the hypotheses needed to guarantee that a function is related to a specific normal form by a diffeomorphism usually reduces to a problem in linear algebra. Let us formalize these steps somewhat, with a view toward the Morse lemma in R n. Let g and h be smooth real valued functions defined on a neighborhood of the origin in R with g(0)= h(0)=0. We say that g and h are right equivalent if there is a C diffeomorphism defined on a neighborhood of 0 in n with (0)--0 such that g(x)h(C(x)). If D(0)=I-identity, we say that g and h are strongly right equivalent. The Morse lemma in. n states that if g is a C function satisfying g(0)--0 and Dg(O) 0 and if DEg(0) is a nondegenerate symmetric bilinear form of index k then g is strongly right equivalent to