Members & Guests

Dr. Rudolf Zeidler

Project leader


Westfälischen Wilhelms-Universität Münster

E-mail: rudolf.zeidler(at)uni-muenster.de
Homepage: https://www.rzeidler.eu/

Project

78Duality and the coarse assembly map II

Publications within SPP2026

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 \(\mu\)-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.

 

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

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.

 

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

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 \(\widehat{\mathrm{A}}\)-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.

 

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

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