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Adiabatic Limit of Calderon Projector on Manifold with Cylindrical End

Kunal Sharma

TL;DR

This work analyzes the Calderón projector for a Dirac-type operator on a manifold with a cylindrical end under generalized APS boundary conditions. It develops a resolvent-based parametrix within Melrose's $b$-calculus to construct the Calderón projector without relying on an invertible double, and proves that the projector converges in the adiabatic limit to the generalized APS projector $ ext{P}_{ ext{APS}}$, with explicit convergence rates. The nonresonance condition ensures a clean limit, extending Nicolaescu's adiabatic results to asymptotically cylindrical geometries and to a broader symbolic framework. The results provide a robust approach to boundary data problems on singular spaces, linking the adiabatic behavior of Cauchy data to explicit boundary projectors with precise operator-theoretic control.

Abstract

For a Riemannian manifold with a cylindrical end, consider a Dirac-type operator that is asymptotically product type with the generalized Atiyah-Patodi-Singer boundary condition on any finite portion of the cylinder. In the present work we consider the problem of constructing the Calderon projector in this setting and studying the adiabatic limit of it along the cylindrical end. As a consequence, we extend a result of Nicolaescu on adiabatic limits of Cauchy data spaces. The proof leverages resolvent and its estimates in the framework of the b-calculus needed for the construction of the Calderon projector corresponding to our Dirac-type operator.

Adiabatic Limit of Calderon Projector on Manifold with Cylindrical End

TL;DR

This work analyzes the Calderón projector for a Dirac-type operator on a manifold with a cylindrical end under generalized APS boundary conditions. It develops a resolvent-based parametrix within Melrose's -calculus to construct the Calderón projector without relying on an invertible double, and proves that the projector converges in the adiabatic limit to the generalized APS projector , with explicit convergence rates. The nonresonance condition ensures a clean limit, extending Nicolaescu's adiabatic results to asymptotically cylindrical geometries and to a broader symbolic framework. The results provide a robust approach to boundary data problems on singular spaces, linking the adiabatic behavior of Cauchy data to explicit boundary projectors with precise operator-theoretic control.

Abstract

For a Riemannian manifold with a cylindrical end, consider a Dirac-type operator that is asymptotically product type with the generalized Atiyah-Patodi-Singer boundary condition on any finite portion of the cylinder. In the present work we consider the problem of constructing the Calderon projector in this setting and studying the adiabatic limit of it along the cylindrical end. As a consequence, we extend a result of Nicolaescu on adiabatic limits of Cauchy data spaces. The proof leverages resolvent and its estimates in the framework of the b-calculus needed for the construction of the Calderon projector corresponding to our Dirac-type operator.
Paper Structure (9 sections, 7 theorems, 35 equations, 2 figures)

This paper contains 9 sections, 7 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Background and Notation
  4. Blow-up
  5. b-pseudodifferential operators
  6. Construction of the Calderón projector
  7. Parametrix of $\mathcal{D}(r)$
  8. Adiabatic limit of the projector
  9. Consequences

Key Result

Lemma 1

We have $P\in \Psi^m_b(X)$ if and only if $K_P$ vanishes to infinite order (in Taylor series) to the left and right boundaries of $X^2_b$ while when supported on a patch near the ff $\cap \bigtriangleup_b$, $K_P = \int e^{i \langle z, \xi \rangle}a(r,y,\xi)d\xi, \text{ } a\in S^m([0,1) \times \mathb

Figures (2)

  • Figure 1: Manifold with cylindrical end
  • Figure 2: Blow-up of $X^2$

Theorems & Definitions (17)

  • Definition
  • Definition
  • Definition
  • Definition
  • Definition
  • Definition
  • Lemma 1
  • Definition
  • Theorem 1
  • proof
  • ...and 7 more