Whitney stability of solvability

Abstract

The set of symbols which are both of real principal type and pseudoconvex is shown to be open in the Whitney topology on the space of symbols of order k. This yields sufficient conditions for the stability of solvability of pseudodiffere ntial equations. 1. Introduction. In a previous paper [3] we studied the stability of solvability for pseudodifferential equations of real principal type in the FS> topology of Michor [9] and in an analogous topology we defined, the FSf topology. In particular, we considered the stability of real principal type and of pseudoconvexi ty in the space of principal symbols of order k. These two conditions together imply solvability and hence their stability implies stability of solvability where the conditions hold. We say that a condition is stable for a given topology if the set where the condition holds is an open set in that topology. A condition is stable at a point if it is satisfied on an open neighborhood of that point. Of course, stability in one topology implies stability in all finer topologies. In the present paper we consider the stability of real principal type and of pseudoconvexity in the Cr-coarse and Cr-fine topologies on the space of principal symbols of order k. In §2 we first review the C'-coarse and C-fine topologies. We then give examples showing that neither real principal type nor pseudoconvexity is stable in the Cr-coarse, r > 0, or C°-fine topologies, and that real principal type is not stable in any Cr-fine topology. In §3 we first outline and then give the complete proof of Whitney or Cr-fine, r > 1, stability of real principal type and pseudoconvexity jointly. This establishes the Cr-fine stability of solvability of a pseudodifferential equation with a corresponding principal symbol which is both of real prinicpal type and pseudoconvex. In §4 we consider pseudoriemannian manifolds (X, β) of dim > 3. If β is given contravariantly, then β is naturally a principal symbol of order 2. We begin §4 with some new results on sectional curvature. In general, everywhere negative timelike sectional curvature is not a C r-fine stable condition for any r > 0. However, if the Riemann-Christoffel curvature R satisfies a nonvanishing requirement on all timelike and null planes, then