05

Index theory on Lorentzian manifolds

Index theory for hyperbolic equations on Lorentzian manifolds has recently been initiated and has already had applications in quantum field theory. The index theorem of C. Bär and A. Strohmaier applies to Lorentzian manifolds with compact Cauchy hypersurfaces.

In this project, Lorentzian index theory will be developed further and, in particular, the assumption on spatial compactness will be relaxed. This will allow for further physical applications.

The work program can be structured as follows.

1. Determine the class of admissible boundary conditions in the spatially compact case.
2. Relative index theory.
3. Spatial preiodicity.
4. Spatially bounded geometry.
5. Decay conditions at spatial infinity.

## Publications

In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these methods as well as their interplay. This survey is succinct rather than comprehensive, and its aim is to inspire geometers and analysts alike to study these methods so that they can be adapted and potentially applied more widely.

Related project(s):
5Index theory on Lorentzian manifolds

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.

Related project(s):
5Index theory on Lorentzian manifolds

We show that local deformations of solutions to open partial differential relations near suitable subsets can be extended to global solutions, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with ordinary differential inequalities, convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry. The main application concerns Riemannian metrics. We prove an approximation result which implies, for instance, that every compact surface carries a C^{1,1}-metric with Gauss curvature ≥1 a.e. on a dense open subset. For C^2-metrics this is, of course, impossible if the genus of the surface is positive.

We review some recent results on geometric equations on Lorentzian manifolds such as the wave and Dirac equations. This includes well-posedness and stability for various initial value problems, as well as results on the structure of these equations on black-hole spacetimes (in particular, on the Kerr solution), the index theorem for hyperbolic Dirac operators and properties of the class of Green-hyperbolic operators.

 Publisher de Gruyter Book J. Brüning, M. Staudacher (Eds.): Space - Time - Matter Pages 324-348 Link to preprint version Link to published version

Related project(s):
5Index theory on Lorentzian manifolds

On a compact globally hyperbolic Lorentzian spin manifold with smooth spacelike Cauchy boundary the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah-Patodi-Singer boundary conditions are imposed. In this paper we investigate to what extent these boundary conditions can be replaced by more general ones and how the index then changes. There are some differences to the classical case of the elliptic Dirac operator on a Riemannian manifold with boundary.

 Journal tba Link to preprint version

Related project(s):
5Index theory on Lorentzian manifolds

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are H\"older continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.

Related project(s):
5Index theory on Lorentzian manifolds

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${\mathcal B} \to {\mathrm{D}}_{\mathcal B}(1 + {\mathrm{D}}_{\mathcal B}^2)^{-\frac{1}{2}}$ depends on Lipschitz smoothness and ellipticity of ${\mathcal B}$ and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.

Related project(s):
5Index theory on Lorentzian manifolds

We prove that the Atiyah-Singer Dirac operator ${\mathrm D}_{\mathrm g}$ in ${\mathrm L}^2$ depends Riesz continuously on ${\mathrm L}^{\infty}$ perturbations of complete metrics ${\mathrm g}$ on a smooth manifold. The Lipschitz bound for the map ${\mathrm g} \to {\mathrm D}_{\mathrm g}(1 + {\mathrm D}_{\mathrm g}^2)^{-\frac{1}{2}}$ depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\'on's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.

Related project(s):
5Index theory on Lorentzian manifolds

We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the η-invariant of the Cauchy hypersurfaces.

 Journal Commun. Math. Phys. Publisher Springer Volume 347 Pages 703-721 Link to preprint version Link to published version

Related project(s):
5Index theory on Lorentzian manifolds

We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed.

 Journal Amer. J. Math (to appear) Publisher John Hopkins Univ. Press Link to preprint version

Related project(s):
5Index theory on Lorentzian manifolds

• 1

## Team Members

Dr. Lashi Bandara
Researcher
Universität Potsdam
lashi.bandara(at)uni-potsdam.de

Prof. Dr. Christian Bär
Universität Potsdam
baer(at)math.uni-potsdam.de

Sebastian Hannes
Doctoral student
Universität Potsdam
shannes(at)math.uni-potsdam.de

PD Dr. habil. Olaf Müller
Researcher
Humboldt-Universität zu Berlin
mullerol(at)math.hu-berlin.de