Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators

Sebastian Boldt; Batu Güneysu

Paper · Mar 23, 2020

Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators

Sebastian Boldt, Batu Güneysu

math.DGarXiv:2003.10204

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Abstract

Given a noncompact spin manifold $M$ with a fixed topological spin structure and two complete Riemannian metrics $g$ and $h$ on $M$ with bounded sectional curvatures, we prove a criterion for the existence and completeness of the wave operators $W_{\pm} (D_{h}, D_{g}, I_{g, h})$ and $W_{\pm} (D_{h}^{2}, D_{g}^{2}, I_{g, h})$ , where $I_{g, h}$ is the canonically given unitary map between the underlying $L^{2}$ -spaces of spinors. This criterion does not involve any injectivity radius assumptions and leads to a criterion for the stability of the absolutely continuous spectrum of a Dirac operator and its square under a Ricci flow.

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Submitted: Mar 23, 2020
Subjects: math.DG, math-ph, math.AP

Ricci flow Stability & uniqueness Boundaries & non-compact Eigenvalues & spectrum

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.
11A.1
The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles
A heat equation for the metric: spheres, cylinders and scaling.

[1] 5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

[2] 10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

[3] 11A.1
The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles
A heat equation for the metric: spheres, cylinders and scaling.

Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators

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