Browsing by Author "Cecchini, Simone"

We develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a “long neck principle” for a compact Riemannian spin n-manifold with boundary X, stating that if scal(X)≥n(n−1) and there is a nonzero degree map into the sphere f:X→Sn which is strictly area decreasing, then the distance between the support of df and the boundary of X is at most π/n. This answers, in the spin setting and for strictly area decreasing maps, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold X obtained by removing k pairwise disjoint embedded n-balls from a closed spin n-manifold Y. We show that if scal(X)>σ>0 and Y satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of ∂X is at most π(n−1)/(nσ)−−−−−−−−−−√. Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to N×[−1,1], with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if scal(V)≥σ>0, then the distance between the boundary components of V is at most 2π(n−1)/(nσ)−−−−−−−−−−√. This last constant is sharp by an argument due to Gromov.

Let be an orientable connected -dimensional manifold with and let be a two-sided closed connected incompressible hypersurface that does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of and are either both spin or both nonspin. Using Gromov's -bubbles, we show that does not admit a complete metric of psc. We provide an example showing that the spin/nonspin 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 2. We deduce as special cases that, if does not admit a metric of psc and , then does not carry a complete metric of psc and does not carry a complete metric of uniformly psc, provided that and , respectively. This solves, up to dimension 7, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.

Abstract 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., $\textrm{E} < |\textrm{P}|$), 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.

Let E \mathcal {E} be an asymptotically Euclidean end in an otherwise arbitrary connected Riemannian spin manifold ( M , g ) (M,g) . We show that if E \mathcal {E} has negative ADM-mass, then there exists a constant R > 0 R > 0 , depending only on E \mathcal {E} , such that M M must become incomplete or have a point of negative scalar curvature in the R R -neighborhood around E \mathcal {E} in M 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 [ Positive scalar curvature on noncompact manifolds and the liouville theorem , 2020; The positive mass theorem with arbitrary ends , 2021] without the spin condition in dimensions ≤ 7 \leq 7 assuming Schwarzschild asymptotics on the end E \mathcal {E} . We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end E \mathcal {E} . Here we obtain refined constants reminiscent of Gromov’s metric inequalities with scalar curvature.