arXiv subjects

Christian Baer

Publications and source records attributed to Christian Baer.

At least 19 recordsLinked to original sources

arXiv2026

The lower mean curvature bound in Gromov's mean-of-the-mean-curvature conjecture

Gromov conjectured that for a compact Riemannian manifold $X$ with boundary, the total mean curvature $\int_{\partial X} H$ is bounded above by a constant depending only on the intrinsic geometry of $\partial X$ and a lower bound on the scalar curvature of $X$. Previous results towards this conjecture require, in addition, a lower bound on the mean curvature of the boundary. In the present paper, we investigate whether this extra assumption is necessary. In dimension $2$, we show that no lower bound on the geodesic curvature is needed. We estimate the total geodesic curvature of the boundary in terms of its length and a lower bound for the Gauss curvature of the surface. This confirms Gromov's conjecture in $2$~dimensions without any extra assumptions. In contrast, we give examples showing that a lower bound on the mean curvature is genuinely needed in dimensions $n \ge 3$.

math.DG↗

arXiv2026

Upper bound for the total mean curvature of spin fill-ins

Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in. We prove Gromov's conjecture if the manifolds are spin with a constant that also depends on a lower bound on the mean curvature $H$ (which is allowed to take negative values). If the boundary is a (not necessarily convex) hypersurface in a space form of non-negative curvature, then the constant can be made explicit in terms of the mean curvature of this model embedding. If the boundary has constant sectional curvature $\kappa>0$ and is a projective space of dimension $n\equiv 3 \mod 4$ or a sphere, then the constant can be expressed in terms of $\kappa$. If the boundary is a flat torus, then the constant can be expressed in terms of lattice data.

math.DG↗

arXiv2025

Spectral flow and the Atiyah-Patodi-Singer index theorem

We establish a formula for the spectral flow of a smooth family of twisted Dirac operators on a closed odd-dimensional Riemannian spin manifold, generalizing a result by Getzler. The spectral flow is expressed in terms of the $\hat{A}$-form of the manifold, the odd Chern character form of the family of connections, and the $\xi$-invariants of the initial and final operators. Our proof is based on a reduction to the Atiyah-Patodi-Singer index theorem for manifolds with boundary, which provides a conceptually very simple approach to the problem. As an application, we give a proof of Llarull's rigidity theorem for scalar curvature of strictly convex hypersurfaces in Euclidean space which works the same in even and odd dimensions.

math.DG↗

arXiv2025

A hitchhiker's guide to first-order elliptic boundary value problems

To empower the mathematical hitchhiker wishing to use operator methods in geometry and topology, we present this user's guide to first-order elliptic boundary value problems. Existence, regularity, and Fredholmness are discussed for general first-order elliptic operators on manifolds with compact boundary. The focus is on a very general class of elliptic boundary conditions, which contain those that are pseudo-local as a special case, yielding the relative index theorem. A new characterisation of a subclass of elliptic boundary conditions is also given.

math.AP↗

arXiv2025

The degree condition in Llarull's theorem on scalar curvature rigidity

Llarull's scalar curvature rigidity theorem states that a 1-Lipschitz map $f: M\to S^n$ from a closed connected Riemannian spin manifold $M$ with scalar curvature $\mathrm{scal}\ge n(n-1)$ to the standard sphere $S^n$ is an isometry if the degree of $f$ is nonzero. We investigate if one can replace the condition $\mathrm{deg}(f)\neq0$ by the weaker condition that $f$ is surjective. The answer turns out to be "no" for $n\ge3$ but "yes" for $n=2$. If we replace the scalar curvature by Ricci curvature, the answer is "yes" in all dimensions.

math.DG↗

arXiv2025

Contractibility of spaces of positive scalar curvature metrics with symmetry

We show the contractibility of spaces of invariant Riemannian metrics of positive scalar curvature on compact connected manifolds of dimension at least two, with and without boundary and equipped with compact Lie group actions. On manifolds without boundary, we assume that the Lie group contains a normal $S^1$-subgroup with fixed-point components of codimension two. In this situation, the existence of invariant metrics of positive scalar curvature was known previously. For the proof, we combine equivariant Morse theory with conformal deformations near unstable manifolds. On manifolds without boundary, we also use local flexibility properties of positive scalar curvature metrics and the smoothing of mean-convex singularities.

math.DG↗

arXiv2024

Dirac eigenvalues and the hyperspherical radius

For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number of geometric consequences. Some are known and include Llarull's scalar curvature rigidity of the standard metric on the sphere, Geroch's conjecture on the impossibility of positive scalar curvature on tori and a mean curvature estimate for spin fill-ins with nonnegative scalar curvature due to Gromov, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler. New applications provide a comparison of the hyperspherical radius with the Yamabe constant and improved estimates of the hyperspherical radius for K\"ahler manifolds, K\"ahler-Einstein manifolds, quaternionic K\"ahler manifolds and manifolds with a harmonic 1-form of constant length.

math.DG↗

arXiv2024

First-order elliptic boundary value problems on manifolds with non-compact boundary

We consider first-order elliptic differential operators acting on vector bundles over smooth manifolds with smooth boundary, which is permitted to be noncompact. Under very mild assumptions, we obtain a regularity theory for sections in the maximal domain. Under additional geometric assumptions, and assumptions on an adapted boundary operator, we obtain a trace theorem on the maximal domain. This allows us to systematically study both local and nonlocal boundary conditions. In particular, the Atiyah-Patodi-Singer boundary condition occurs as a special case. Furthermore, we study contexts which induce semi-Fredholm and Fredholm extensions.

math.DG↗

arXiv2023

K-cowaist of manifolds with boundary

We extend the K-cowaist inequality to generalized Dirac operators in the sense of Gromov and Lawson and study applications to manifolds with boundary.

math.DG↗

arXiv2023

Scalar curvature rigidity of warped product metrics

We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped with strictly log-concave warping functions. This generalizes earlier results of Cecchini-Zeidler to all dimensions. Moreover, we show scalar curvature rigidity of round spheres of dimension at least 3 with two antipodal points removed. This resolves a problem in Gromov's ''Four Lectures'' in all dimensions. Our arguments are based on spin geometry.

math.DG↗

arXiv2023

Rigidity results for initial data sets satisfying the dominant energy condition

Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes we use approximations by manifolds with smooth boundary.

math.DG↗

arXiv2022

The Cauchy problem for Lorentzian Dirac operators under non-local boundary conditions

Non-local boundary conditions, such as the Atiyah-Patodi-Singer (APS) conditions, for Dirac operators on Riemannian manifolds are well understood while not much is known for such operators on spacetimes with timelike boundary. We define a class of Lorentzian boundary conditions that are local in time and non-local in the spatial directions and show that they lead to a well-posed Cauchy problem for the Dirac operator. This applies in particular to the APS conditions imposed on each level set of a given Cauchy temporal function.

math.DG↗

arXiv2022

Remarks on the paper "On Gromov's dihedral extremality and rigidity conjectures" by Jinmin Wang, Zhizhang Xie and Guoliang Yu

Version 2 of the article "On Gromov's dihedral extremality and rigidity conjectures" by Jinmin Wang, Zhizhang Xie and Guoliang Yu makes a number of claims for self-adjoint extensions of Dirac type operators on manifolds with corners under local boundary conditions. We construct a counterexample to an index computation in that paper which affects the proof of its main result stating a generalisation of Gromov's dihedral extremality conjecture.

math.DG↗

arXiv2021

The Faddeev-LeVerrier algorithm and the Pfaffian

We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost $O(n^{β+1})$ where $n$ is the size of the matrix and $O(n^β)$ is the cost of multiplying $n\times n$-matrices, $β\in[2,2.37286)$. We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.

math.CO↗

arXiv2021

Manifolds with many Rarita-Schwinger fields

The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kähler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension.

math.DG↗

arXiv2021

Boundary value problems for general first-order elliptic differential operators

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local. We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. We provide examples which are conveniently treated by our methods.

math.DG↗

arXiv2020

Boundary conditions for scalar curvature

Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites a la Miao and the deformation of boundary conditions a la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.

math.DG↗

arXiv2020

Local Index Theory for Lorentzian Manifolds

Index theory for Lorentzian Dirac operators is a young subject with significant differences to elliptic index theory. It is based on microlocal analysis instead of standard elliptic theory and one of the main features is that a nontrivial index is caused by topologically nontrivial dynamics rather than nontrivial topology of the base manifold. In this paper we establish a local index formula for Lorentzian Dirac-type operators on globally hyperbolic spacetimes. This local formula implies an index theorem for general Dirac-type operators on spatially compact spacetimes with Atiyah-Patodi-Singer boundary conditions on Cauchy hypersurfaces. This is significantly more general than the previously known theorems that require the compatibility of the connection with Clifford multiplication and the spatial Dirac operator on the Cauchy hypersurface to be selfadjoint with respect to a positive definite inner product.

math.DG↗