arXiv subjects

Rudolf Zeidler

Publications and source records attributed to Rudolf Zeidler.

At least 19 recordsLinked to original sources

arXiv2026

Scalar curvature rigidity for products of convex hypersurfaces

Let $N = N_{1} \times \dotsm \times N_{k}$, where each $N_{i} \subset \mathbb{R}^{n_i+1}$ is a closed strictly convex hypersurface. Let $M$ be a Riemannian spin manifold of dimension $n = \dim(N)$, and let $f \colon M \to N$ be an area non-increasing smooth map of non-zero degree. We show that $\mathrm{scal}_M \geq \mathrm{scal}_N \circ f$ implies $\mathrm{scal}_M = \mathrm{scal}_N \circ f$. Moreover, if $n \geq 3$ and $N$ has no circle factors, then every such map is a Riemannian isometry. In the presence of circle factors, we obtain the corresponding optimal splitting theorem for $M$ and $f$. Our results are based on an approach to the index-theoretic part of Llarull's scalar curvature rigidity theorem via Clifford-linear family index theory, which works independently of the parity of the dimension and extends naturally to products. This includes a proof of the Geroch conjecture for spin manifolds as the edge case with only circle factors.

math.DG↗

arXiv2026

Stable $2$-systoles, scalar curvature and spin$^c$ comass bounds

We prove a sharp stable $2$-systolic inequality for complex projective space under the scalar curvature lower bound of the normalized Fubini-Study metric. If $M$ is diffeomorphic to $\mathbb{C}\mathrm{P}^n$ and $\mathrm{scal}_g\ge 4n(n+1)$, then $\mathrm{sys}_2^{\mathrm{st}}(M,g)\le \pi$. Moreover, equality holds only for the Fubini-Study metric, up to biholomorphism after choosing the corresponding complex structure. The proof uses Spin$^c$ Dirac operators, a comass estimate for the curvature term in the Lichnerowicz formula, and stable norm-comass duality.

math.DG↗

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

Duality pairings with the analytic structure group

We construct a slant product $\mathrm{S}^{G\times H}_p(X\times Y)\otimes \mathrm{K}_{-q}(\bar{\mathfrak{c}}^{\mathrm{red}} Y\rtimes H)\to \mathrm{K}_{p-q}(\mathrm{C}^\ast_G X)$ on the analytic structure group of Higson and Roe and the K-theory of the stable Higson compactification taking values in the (equivariant) Roe algebra. This complements the slant products constructed in earlier work of Engel and the authors ( arXiv:1909.03777 [math.KT] ). The distinguishing feature of our new slant product is that it specializes to a duality pairing $\mathrm{S}^H_p(Y) \otimes \mathrm{K}_{-p}(\bar{\mathfrak{c}}^{\mathrm{red}} (Y)\rtimes H)\to \mathbb{Z}$ which can be used to extract numerical invariants out of elements in the analytic structure group such as rho-invariants associated to positive scalar curvature metrics.

math.KT↗

arXiv2024

Width, Largeness and Index Theory

In this note, we review some recent developments related to metric aspects of scalar curvature from the point of view of index theory for Dirac operators. In particular, we revisit index-theoretic approaches to a conjecture of Gromov on the width of Riemannian bands $M \times [-1,1]$, and on a conjecture of Rosenberg and Stolz on the non-existence of complete positive scalar curvature metrics on $M \times {\mathbb R}$. We show that there is a more general geometric statement underlying both of them implying a quantitative negative upper bound on the infimum of the scalar curvature of a complete metric on $M \times {\mathbb R}$ if the scalar curvature is positive in some neighborhood. We study ($\hat{A}$-)iso-enlargeable spin manifolds and related notions of width for Riemannian manifolds from an index-theoretic point of view. Finally, we list some open problems arising in the interplay between index theory, largeness properties and width.

math.DG↗

arXiv2024

Positive scalar curvature with point singularities

We show that in every dimension $n \geq 8$, there exists a smooth closed manifold $M^n$ which does not admit a smooth positive scalar curvature ("psc") metric, but $M$ admits an $\mathrm{L}^\infty$-metric which is smooth and has psc outside a singular set of codimension $\geq 8$. This provides counterexamples to a conjecture of Schoen. In fact, there are such examples of arbitrarily high dimension with only single point singularities. We also discuss related phenomena on exotic spheres and tori. In addition, we provide examples of $\mathrm{L}^\infty$-metrics on $\mathbb{R}^n$ for certain $n \geq 8$ which are smooth and have psc outside the origin, but cannot be smoothly approximated away from the origin by everywhere smooth Riemannian metrics of non-negative scalar curvature. This stands in precise contrast to established smoothing results via Ricci-DeTurck flow for singular metrics with stronger regularity assumptions. Finally, as a positive result, we describe a $\mathrm{KO}$-theoretic condition which obstructs the existence of $\mathrm{L}^\infty$-metrics that are smooth and of psc outside a finite subset. This shows that closed enlargeable spin manifolds do not carry such metrics.

math.DG↗

arXiv2024

Rigidity of spin fill-ins with non-negative scalar curvature

We establish new mean curvature rigidity theorems for spin fill-ins with non-negative scalar curvature using two different spinorial techniques. Our results address two questions by Miao and Gromov, respectively. The first technique is based on extending boundary spinors satisfying a generalized eigenvalue equation via the Fredholm alternative for an APS boundary value problem, while the second is a comparison result in the spirit of Llarull and Lott using index theory. We also show that the latter implies a new Witten-type integral inequality for the mass of an asymptotically Schwarzschild manifold, which holds even when the scalar curvature is not assumed to be non-negative.

math.DG↗

arXiv2023

Positive mass theorems for spin initial data sets with arbitrary ends and dominant energy shields

We prove a positive mass theorem for spin initial data sets $(M,g,k)$ that contain an asymptotically flat end and a shield of dominant energy (a subset of $M$ on which the dominant energy scalar $\mu-|J|$ has a positive lower bound). In a similar vein, we show that for an asymptotically flat end $\mathcal{E}$ that violates the positive mass theorem (i.e. $\mathrm{E} < |\mathrm{P}|$), there exists a constant $R>0$, depending only on $\mathcal{E}$, such that any initial data set containing $\mathcal{E}$ must violate the hypotheses of Witten's proof of the positive mass theorem in an $R$-neighborhood of $\mathcal{E}$. This implies the positive mass theorem for spin initial data sets with arbitrary ends, and we also prove a rigidity statement. Our proofs are based on a modification of Witten's approach to the positive mass theorem involving an additional independent timelike direction in the spinor bundle.

math.DG↗

arXiv2023

Nonnegative scalar curvature on manifolds with at least two ends

Let $M$ be an orientable connected $n$-dimensional manifold with $n\in\{6,7\}$ and let $Y\subset M$ be a two-sided closed connected incompressible hypersurface which does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of $M$ and $Y$ are either both spin or both non-spin. Using Gromov's $μ$-bubbles, we show that $M$ does not admit a complete metric of psc. We provide an example showing that the spin/non-spin hypothesis cannot be dropped from the statement of this result. This answers, up to dimension $7$, a question by Gromov for a large class of cases. Furthermore, we prove a related result for submanifolds of codimension two. We deduce as special cases that, if $Y$ does not admit a metric of psc and $\dim(Y) \neq 4$, then $M := Y\times\mathbb{R}$ does not carry a complete metric of psc and $N := Y \times \mathbb{R}^2$ does not carry a complete metric of uniformly psc provided that $\dim(M) \leq 7$ and $\dim(N) \leq 7$, respectively. This solves, up to dimension $7$, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.

math.DG↗

arXiv2022

Scalar and mean curvature comparison via the Dirac operator

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we thereby give refined answers to questions on metric inequalities recently proposed by Gromov. This includes optimal estimates for Riemannian bands and for the long neck problem. In the case of bands over manifolds of non-vanishing $\hat{\mathrm{A}}$-genus, we establish a rigidity result stating that any band attaining the predicted upper bound is isometric to a particular warped product over some spin manifold admitting a parallel spinor. Furthermore, we establish scalar- and mean curvature extremality results for certain log-concave warped products. The latter includes annuli in all simply-connected space forms. On a technical level, our proofs are based on new spectral estimates for the Dirac operator augmented by a Lipschitz potential together with local boundary conditions.

math.DG↗

arXiv2021

The positive mass theorem and distance estimates in the spin setting

Let $\mathcal{E}$ be an asymptotically Euclidean end in an otherwise arbitrary complete and connected Riemannian spin manifold $(M,g)$. We show that if $\mathcal{E}$ has negative ADM-mass, then there exists a constant $R > 0$, depending only on $\mathcal{E}$, such that $M$ must become incomplete or have a point of negative scalar curvature in the $R$-neighborhood around $\mathcal{E}$ in $M$. This gives a quantitative answer to Schoen and Yau's question on the positive mass theorem with arbitrary ends for spin manifolds. Similar results have recently been obtained by Lesourd, Unger and Yau without the spin condition in dimensions $\leq 7$ assuming Schwarzschild asymptotics on the end $\mathcal{E}$. We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end $\mathcal{E}$. Here we obtain refined constants reminiscent of Gromov's metric inequalities with scalar curvature.

math.DG↗

arXiv2021

On the range of the relative higher index and the higher rho-invariant for positive scalar curvature

Let $M$ be a closed spin manifold which supports a positive scalar curvature metric. The set of concordance classes of positive scalar curvature metrics on $M$ forms an abelian group $P(M)$ after fixing a positive scalar curvature metric. The group $P(M)$ measures the size of the space of positive scalar curvature metrics on $M$. Weinberger and Yu gave a lower bound of the rank of $P(M)$ in terms of the number of torsion elements of $π_1(M)$. In this paper, we give a sharper lower bound of the rank of $P(M)$ by studying the image of the relative higher index map from $P(M)$ to the real K-theory of the group $\mathrm{C}^\ast$-algebra $\mathrm{C}^\ast_{\mathrm{r}}(π_1(M))$. We show that it rationally contains the image of the Baum-Connes assembly map up to a certain homological degree depending on the dimension of $M$. At the same time we obtain lower bounds for the positive scalar curvature bordism group by applying the higher rho-invariant.

math.KT↗

arXiv2020

Slant products on the Higson-Roe exact sequence

We construct a slant product $/ \colon \mathrm{S}_p(X \times Y) \times \mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X)$ on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer. The latter is the domain of the co-assembly map $μ^\ast \colon \mathrm{K}_{1-\ast}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{K}^\ast(Y)$. We obtain such products on the entire Higson--Roe sequence. They imply injectivity results for external product maps. Our results apply to products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the class of manifolds where this method applies, we say that a complete $\mathrm{spin}^{\mathrm{c}}$-manifold is Higson-essential if its fundamental class is detected by the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. We draw conclusions for positive scalar curvature metrics on product spaces, particularly on non-compact manifolds. We also obtain equivariant versions of our constructions and discuss related problems of exactness and amenability of the stable Higson corona.

math.KT↗

arXiv2020

Band width estimates via the Dirac operator

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $σ> 0$, the distance between the boundary components of $V$ is at most $C_n/\sqrtσ$, where $C_n = \sqrt{(n-1)/{n}} \cdot C$ with $C < 8(1+\sqrt{2})$ being a universal constant. This verifies a conjecture of Gromov for such manifolds. In particular, our result applies to all high-dimensional closed simply connected manifolds $M$ which do not admit a metric of positive scalar curvature. We also establish a quadratic decay estimate for the scalar curvature of complete metrics on manifolds, such as $M \times \mathbb{R}^2$, which contain $M$ as a codimension two submanifold in a suitable way. Furthermore, we introduce the "$\mathcal{KO}$-width" of a closed manifold and deduce that infinite $\mathcal{KO}$-width is an obstruction to positive scalar curvature.

math.DG↗

arXiv2019

Transfer maps in generalized group homology via submanifolds

Let $N \subset M$ be a submanifold embedding of spin manifolds of some codimension $k \geq 1$. A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that $M$ does not admit a metric of positive scalar curvature if $k = 2$ and the Dirac operator of $N$ has non-trivial index, provided that suitable conditions are satisfied. In the cases $k=1$ and $k=2$, Zeidler and Kubota, respectively, established more systematic results: There exists a transfer $\mathrm{KO}_\ast(\mathrm{C}^{\ast} π_1 M)\to \mathrm{KO}_{\ast - k}(\mathrm{C}^\ast π_1 N)$ which maps the index class of $M$ to the index class of $N$. The main goal of this article is to construct analogous transfer maps $E_\ast(\mathrm{B}π_1M) \to E_{\ast-k}(\mathrm{B}π_1N)$ for different generalized homology theories $E$ and suitable submanifold embeddings. The design criterion is that it is compatible with the transfer $E_\ast(M) \to E_{\ast-k}(N)$ induced by the inclusion $N \subset M$ for a chosen orientation on the normal bundle. Under varying restrictions on homotopy groups and the normal bundle, we construct transfers in the following cases in particular: In ordinary homology, it works for all codimensions. This slightly generalizes a result of Engel and simplifies his proof. In complex K-homology, we achieve it for $k \leq 3$. For $k \leq 2$, we have a transfer on the equivariant KO-homology of the classifying space for proper actions.

math.AT↗

arXiv2018

Positive scalar curvature and low-degree group homology

Let $Γ$ be a discrete group. Assuming rational injectivity of the Baum-Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz' positive scalar curvature sequence for $\mathrm{B} Γ$. The lower bounds are formulated in terms of the part of degree up to $2$ in the group homology of $Γ$ with coefficients in the $\mathbb{C}Γ$-module generated by finite order elements. Our results use and extend work of Botvinnik and Gilkey which treated the case of finite groups. Further crucial ingredients are a real counterpart to the delocalized equivariant Chern character and Matthey's work on explicitly inverting this Chern character in low homological degrees.

math.KT↗

arXiv2017

An index obstruction to positive scalar curvature on fiber bundles over aspherical manifolds

We exhibit geometric situations, where higher indices of the spinor Dirac operator on a spin manifold $N$ are obstructions to positive scalar curvature on an ambient manifold $M$ that contains $N$ as a submanifold. In the main result of this note, we show that the Rosenberg index of $N$ is an obstruction to positive scalar curvature on $M$ if $N \hookrightarrow M \twoheadrightarrow B$ is a fiber bundle of spin manifolds with $B$ aspherical and $π_1(B)$ of finite asymptotic dimension. The proof is based on a new variant of the multi-partitioned manifold index theorem which might be of independent interest. Moreover, we present an analogous statement for codimension one submanifolds. We also discuss some elementary obstructions using the $\hat{A}$-genus of certain submanifolds.

math.KT↗

arXiv2015

Positive scalar curvature and product formulas for secondary index invariants

We introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature outside a prescribed subset on a spin manifold. These can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external product formula for partial secondary invariants, from which we deduce product formulas for the higher rho-invariant of a metric with uniformly positive scalar curvature as well as for the higher relative index of two metrics with uniformly positive scalar curvature. Our methods yield a new conceptual proof of the secondary partitioned manifold index theorem and a refined version of the delocalized APS-index theorem of Piazza-Schick for the spinor Dirac operator in all dimensions. We establish a partitioned manifold index theorem for the higher relative index. We also show that secondary invariants are stable with respect to direct products with aspherical manifolds that have fundamental groups of finite asymptotic dimension. Moreover, we construct examples of complete metrics with uniformly positive scalar curvature on non-compact spin manifolds which can be distinguished up to concordance relative to subsets which are coarsely negligible in a certain sense. A technical novelty in this paper is that we use Yu's localization algebras in combination with the description of K-theory for graded C*-algebras due to Trout. This formalism allows direct definitions of all the invariants we consider in terms of the functional calculus of the Dirac operator and enables us to give concise proofs of the product formulas.

math.KT↗