Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Singularities and Topological Change for Deforming Domains in Manifolds

Wu-Hsiung Huang

Abstract

Given a $C^{0}$-deformation of domains $D(t)$ on a manifold $M^{n}$, which allows the topological types of the domains $D(t)$ to change with $t$, in what cases are the entities in analysis continuous in $t$, so that analysis techniques still work along $t$? This type of problem was addressed in our previous work [Hw] concerning domains on hypersurfaces of constant mean curvature (CMC) in $\mathbb{R}^{n+1}$. In this paper, we consider a more popular circumstance, where the deforming domains are situated in any smooth manifold $M^{n}$ equipped with an arbitrary self-adjoint strongly elliptic operator $L$ (replacing the stability operator for CMC hypersurfaces in $\mathbb R^{n+1}$ [Hw]). We define the concept of quasi-Lipschitz domains by gluing together some boundary points of a Lipschitz domain in a specific manner, allowing the topology of the deforming domain $D(t)$ to change. It is established that any ``appropriate" monotone $C^{0}$-deformation on $M^{n}$ (see Definition 1.1) exhibits Sobolev continuity and eigenvalue continuity of $L$ along $t$. As a consequence, a ``global" Morse index theorem is obtained. Furthermore, given an \emph{arbitrary} Lipschitz domain $D$ in $M^{n}$, we can find a $C^0$-deformation from a small $n$-ball to the domain $D$, such that the topology of $D(t)$ may change, yet the required continuity theorems still hold, and hence the global Morse index theorem still follows.

Singularities and Topological Change for Deforming Domains in Manifolds

Abstract

Given a -deformation of domains on a manifold , which allows the topological types of the domains to change with , in what cases are the entities in analysis continuous in , so that analysis techniques still work along ? This type of problem was addressed in our previous work [Hw] concerning domains on hypersurfaces of constant mean curvature (CMC) in . In this paper, we consider a more popular circumstance, where the deforming domains are situated in any smooth manifold equipped with an arbitrary self-adjoint strongly elliptic operator (replacing the stability operator for CMC hypersurfaces in [Hw]). We define the concept of quasi-Lipschitz domains by gluing together some boundary points of a Lipschitz domain in a specific manner, allowing the topology of the deforming domain to change. It is established that any ``appropriate" monotone -deformation on (see Definition 1.1) exhibits Sobolev continuity and eigenvalue continuity of along . As a consequence, a ``global" Morse index theorem is obtained. Furthermore, given an \emph{arbitrary} Lipschitz domain in , we can find a -deformation from a small -ball to the domain , such that the topology of may change, yet the required continuity theorems still hold, and hence the global Morse index theorem still follows.

Paper Structure

This paper contains 6 sections, 9 theorems, 66 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Quasi-Lipschitz domains
  3. Trace theorem
  4. The singularities
  5. Topological change
  6. Concluding remarks

Key Result

Theorem A

(Continuity and Existence) Let $L$ be a self-adjoint strongly elliptic operator of second order defined on $M^{n}$, which has uniqueness of the Cauchy problem. For any monotone $C^{0}$-deformation $\mathcal{D}$ on $M^{n}$ (given in Definition D1.1), we have the Sobolev continuity in $t$, i.e., where $H_{t} := H(D(t))$ for all $t \in [0,b]$. And the eigenvalues $\lambda_{k}(t):=\lambda_{k}(D(t))$

Figures (16)

  • Figure 2.1: Lipschitz triple in $\mathbb R^n$
  • Figure 2.2: gluing map ($n=3$)
  • Figure 2.3: quasi-Lipschitz domain
  • Figure 2.4: examples of quasi-Lipschitz domains
  • Figure 2.5: Can't squeez preglued set $\alpha^1$ thru interior
  • ...and 11 more figures

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.1
  • Definition 1.3
  • Definition 1.4
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • ...and 28 more