Singularities and Topological Change for Deforming Domains in Manifolds
The main purpose of this paper is to investigate when topology jumps while analysis remains continuous. 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 are the analysis notions associated with $D(t)$ continuous in $t$, so that the machinery of analysis is still working along the deformation? This type of problem arose 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. The continuity theorems and the existence of the required deformations are proved. We establish that any monotone $C^{0}$-deformation of quasi-Lipschitz domains in $M^{n}$ satisfies Sobolev continuity and eigenvalue continuity for the operator $L$ along the deformation parameter $t$. As a consequence, a \emph{global} Morse index theorem is obtained. Furthermore, given any quasi-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 remain valid, and the Morse index theorem holds for the deformation.