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Quasi-fibered boundary pseudodifferential operators

Chris Kottke, Frédéric Rochon

Abstract

We develop a pseudodifferential calculus for differential operators associated to quasi-fibered boundary metrics (QFB metrics), a class of metrics including the quasi-asymptotically conical metrics (QAC metrics) of Degeratu-Mazzeo and the quasi-asymptotically locally Euclidean metrics (QALE metrics) of Joyce. Introducing various principal symbols, we introduce the notion of fully elliptic QFB operators and show that those are Fredholm when acting on QFB Sobolev spaces. For QAC metrics, we also develop a pseudodifferential calculus for the conformally related class of Qb metrics. We use these calculi to construct a parametrix for the Hodge-deRham operator of certain QFB metrics, allowing us to show that it is Fredholm on suitable Sobolev spaces and that the space of $L^2$ harmonic forms is finite dimensional. Our parametrix is obtained by inverting certain model operators at infinity, inversions that we achieve in part through a fine understanding of the low energy limit of the resolvent of the Hodge-deRham operator. Our parametrix also implies that $L^2$ harmonic forms decay faster at infinity than an arbitrary $L^2$ form, the extra decay being quantified in terms of a small negative power of the distance function. This decay of $L^2$ harmonic forms is used in a companion paper to study the $L^2$ cohomology of some $QFB$ metrics.

Quasi-fibered boundary pseudodifferential operators

Abstract

We develop a pseudodifferential calculus for differential operators associated to quasi-fibered boundary metrics (QFB metrics), a class of metrics including the quasi-asymptotically conical metrics (QAC metrics) of Degeratu-Mazzeo and the quasi-asymptotically locally Euclidean metrics (QALE metrics) of Joyce. Introducing various principal symbols, we introduce the notion of fully elliptic QFB operators and show that those are Fredholm when acting on QFB Sobolev spaces. For QAC metrics, we also develop a pseudodifferential calculus for the conformally related class of Qb metrics. We use these calculi to construct a parametrix for the Hodge-deRham operator of certain QFB metrics, allowing us to show that it is Fredholm on suitable Sobolev spaces and that the space of harmonic forms is finite dimensional. Our parametrix is obtained by inverting certain model operators at infinity, inversions that we achieve in part through a fine understanding of the low energy limit of the resolvent of the Hodge-deRham operator. Our parametrix also implies that harmonic forms decay faster at infinity than an arbitrary form, the extra decay being quantified in terms of a small negative power of the distance function. This decay of harmonic forms is used in a companion paper to study the cohomology of some metrics.

Paper Structure

This paper contains 17 sections, 115 theorems, 1044 equations, 1 figure.

Table of Contents

  1. ${\operatorname{QFB}}$ structures
  2. The ${\operatorname{QFB}}$ double space
  3. ${\operatorname{QFB}}$ pseudodifferential operators
  4. The ${\operatorname{QFB}}$ triple space
  5. Composition of ${\operatorname{QFB}}$ operators and ${\operatorname{Qb}}$ operators
  6. Symbol maps of ${\operatorname{QFB}}$ operators
  7. Mapping properties of ${\operatorname{QFB}}$ operators
  8. Symbol maps for ${\operatorname{Qb}}$ operators and the edge calculus
  9. Parametrix construction of Dirac ${\operatorname{QFB}}$ operators
  10. $\mathscr{K},{\operatorname{QFB}}$ operators
  11. The $\mathscr{K},{\operatorname{QFB}}$ triple space
  12. Composition of $\mathscr{K},{\operatorname{QFB}}$ pseudodifferential operators
  13. Symbol maps
  14. Resolvent of a ${\operatorname{QFB}}$ Dirac operator in the low energy limit
  15. Inverse of a non-fully elliptic suspended Dirac ${\operatorname{QFB}}$ operator
  16. ...and 2 more sections

Key Result

Theorem 1

A fully elliptic pseudodifferential ${\operatorname{QFB}}$ operator $P\in \Psi^m_{{\operatorname{QFB}}}(M;E,F)$ induces a Fredholm operator for all multiweights $\mathfrak{t}$, where $H^{m}_{{\operatorname{QFB}}}(M;E)$ is the Sobolev space of order $m$ associated to a ${\operatorname{QFB}}$ metric $g$ with norm defined in mp.28 below. Moreover, elements in the kernel $P$ decay rapidly at infinity

Figures (1)

  • Figure 1: Picture of $H^{{\operatorname{Qb}}}_{22}$ with $H_1<H_2$ when $M$ is of depth $2$ with the dashed lines representing the intersections of $\mathfrak{D}_2^{{\operatorname{Qb}}}$ with the corresponding boundary hypersurfaces. Beware that $\mathfrak{D}_2^{{\operatorname{Qb}}}$ intersects $H^{{\operatorname{Qb}}}_{11}$, but due to the lack of dimensions, this is not represented in this picture.

Theorems & Definitions (271)

  • Theorem 1: Proposition \ref{['mp.3']} and Theorem \ref{['mp.32']} below
  • Theorem 2
  • Remark
  • Corollary 3: Corollary \ref{['do.53']} with $\delta=-\frac{1}{2}$
  • Corollary 4
  • Corollary 5: Corollary \ref{['hd.4']}
  • Corollary 6
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • ...and 261 more