Singularities and Topological Change for Deforming Domains in Manifolds

Given a $C^0$-deformation of domains $D(t)$ in a manifold $M^n$, allowing the topological type of the domains $D(t)$ to vary with $t$, in what situations do the analysis objects associated with $D(t)$ remain continuous in $t$, so that analysis techniques continue to work along the deformation? This type of problem was studied in our previous work [Hw] for domains on constant mean curvature (CMC) hypersurfaces in ${\mathbb{R}}^{n+1}$. In the present paper, we consider a general setting in which the deforming domains are situated in an arbitrary smooth manifold $M^n$ equipped with a self-adjoint strongly elliptic operator $L$, replacing the stability operator for CMC hypersurfaces in ${\mathbb{R}}^{n+1}$ considered in [Hw]. We introduce the notion of quasi-Lipschitz domains by gluing certain boundary points of a Lipschitz domain in a specific manner, thereby allowing the topology of the deforming domain $D(t)$ to change. It is established that any “appropriate” monotone $C^0$-deformation on Mn (see Definition 1.1) satisfies Sobolev continuity and eigenvalue continuity for the operator $L$ along the deformation parameter $t$. As a consequence, a “global” Morse index theorem is obtained. Furthermore, given an arbitrary Lipschitz domain $D$ in $M^n$, we construct a $C^0$-deformation from a small n-ball to the domain $D$, along which the topology of $D(t)$ may change, while the required continuity properties continue to hold. Consequently, the Morse index theorem remains valid as well.