# Members & Guests

## Dr. Simone Cecchini

Georg-August Universität Göttingen

E-mail: simone.cecchini(at)uni-goettingen.de
Telephone: +49 (0)551 39 7751

## Project

42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds
9Diffeomorphisms and the topology of positive scalar curvature

## Publications within SPP2026

We prove the following Lipschitz rigidity result in scalar curvature geometry. Let $M$ be a closed smooth connected spin manifold of even dimension $n$, let $g$ be a Riemannian metric of regularity $W^{1,p}$, $p > n$, on $M$ whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by $n(n-1)$, and let  $f \colon (M,g) \to \mathbb{S}^n$ be a $1$-Lipschitz continuous (not necessarily smooth) map of non-zero degree to the unit  $n$-sphere. Then $f$ is a metric isometry. This generalizes a result of Llarull (1998) and answers in the affirmative a question of Gromov (2019) in his "four lectures". Our proof is based on spectral properties of Dirac operators for  low regularity Riemannian metrics and twisted with Lipschitz bundles, and on the theory of quasiregular maps due to Reshetnyak.

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.

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.

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.

We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use the main result of the recent paper by Schoen and Yau on minimal hypersurfaces to obstruct positive scalar curvature in arbitrary dimensions. More concretely, we use this to study nonzero degree maps f from a manifold X to the product of the k-sphere with the n-k dimensional torus, with k=1,2,3. When X is a closed oriented manifold endowed with a metric g of positive scalar curvature and the map f is (possibly area) contracting, we prove inequalities relating the lower bound of the scalar curvature of g and the contracting factor of the map f.

 Journal Proc. AMS Publisher AMS Link to preprint version Link to published version

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

• 1