We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on scalar curvature in interior and mean of boundary.In situations we consider, thereby give refined answers questions metric inequalities recently proposed by Gromov.This includes optimal Riemannian bands long neck problem.In case over non-vanishing A-genus, a rigidity result stating that any band attaining predicted upper bound is isometric particular warped product some...

We introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature (upsc) outside a prescribed subset on spin manifold. These can be used distinguish such up concordance relative the subset. exhibit general external product formula for invariants, from we deduce formulas higher ρ-invariant of metric with upsc as well index two upsc. Our methods yield new conceptual proof partitioned manifold theorem and refined version...

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove for any Riemannian metric on $V=M \times [-1,1]$ with scalar curvature bounded below by $\sigma \gt 0$, the distance between boundary components of $V$ is at most $C_n / \sqrt{\sigma}$, where = \sqrt{(n-1)/n} \cdot C$ $C \lt 8(1+\sqrt{2})$ being universal constant. This verifies conjecture Gromov manifolds. In particular, our result applies to all high-dimensional...

Abstract We prove a positive mass theorem for spin initial data sets $(M,g,k)$ that contain an asymptotically flat end and shield of dominant energy (a subset $M$ on which the scalar $\mu -|J|$ has lower bound). In similar vein, we show $\mathcal{E}$ violates (i.e., $\textrm{E} < |\textrm{P}|$), there exists constant $R>0$, depending only $\mathcal{E}$, such any set containing must violate hypotheses Witten’s proof in $R$-neighborhood $\mathcal{E}$. This implies with arbitrary...

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

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on scalar curvature in interior and mean of boundary. In situations we consider, thereby give refined answers questions metric inequalities recently proposed by Gromov. This includes optimal Riemannian bands long neck problem. case over non-vanishing $\hat{\mathrm{A}}$-genus, a rigidity result stating that any band attaining predicted upper bound is isometric particular...

Let $M$ be an orientable connected $n$-dimensional manifold with $n\in\{6,7\}$ and let $Y\subset M$ a two-sided closed incompressible hypersurface which does not admit metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers $Y$ are either both spin or non-spin. Using Gromov's $\mu$-bubbles, we show complete psc. We provide example showing spin/non-spin hypothesis cannot dropped from statement this result. This answers, up to dimension $7$,...

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

We exhibit geometric situations, where higher indices of the spinor Dirac operator on a spin manifold N are obstructions to positive scalar curvature an ambient M that contains as submanifold.In main result this note, we show Rosenberg index is obstruction if → B fiber bundle manifolds with aspherical and π 1 (B) finite asymptotic dimension.The proof based new variant multi-partitioned theorem which might be independent interest.Moreover, present analogous statement for codimension one...

Let N \subset M be a submanifold embedding of spin manifolds some codimension k \geq 1 . A classical result Gromov and Lawson, refined by Hanke, Pape Schick, states that does not admit metric positive scalar curvature if = 2 the Dirac operator has non-trivial index, provided suitable geometric conditions on are satisfied. In cases k=1 k=2 , Zeidler Kubota, respectively, established more systematic results: There exists transfer \text{KO}_\ast(\text{C}^{\ast} \pi_1 M)\to \text{KO}_{\ast -...

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $σ> 0$, the distance between boundary components of $V$ is at most $C_n/\sqrtσ$, where $C_n \sqrt{(n-1)/{n}} \cdot C$ $C < 8(1+\sqrt{2})$ being universal constant. This verifies conjecture Gromov manifolds. In particular, our result applies to all high-dimensional simply...

Let $\mathcal{E}$ be an asymptotically Euclidean end in otherwise arbitrary complete and connected Riemannian spin manifold $(M,g)$. We show that if has negative ADM-mass, then there exists a constant $R > 0$, depending only on $\mathcal{E}$, such $M$ must become incomplete or have point of scalar curvature the $R$-neighborhood around $M$. This gives quantitative answer to Schoen Yau's question positive mass theorem with ends for manifolds. Similar results recently been obtained by...

We establish new mean curvature rigidity theorems of spin fill-ins with non-negative scalar using two different spinorial techniques. Our results address 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 value problem, while second comparison result in spirit Llarull Lott index theory. also show that latter implies Witten-type integral inequality mass...

We show that in every dimension $n \geq 8$, there exists a smooth closed manifold $M^n$ which does not admit positive scalar curvature ("psc") metric, but $M$ admits an $\mathrm{L}^\infty$-metric is and has psc outside singular set of codimension $\geq 8$. This provides counterexamples to conjecture Schoen. In fact, are such examples arbitrarily high with only single point singularities. addition, we provide $\mathrm{L}^\infty$-metrics on $\mathbb{R}^n$ for certain 8$ have the origin, cannot...

Let $N \subset M$ be a submanifold embedding of spin manifolds some codimension $k \geq 1$. A classical result Gromov and Lawson, refined by Hanke, Pape Schick, states that $M$ does not admit metric positive scalar curvature if = 2$ the Dirac operator $N$ has non-trivial index, provided suitable conditions are satisfied. In cases $k=1$ $k=2$, Zeidler Kubota, respectively, established more systematic results: There exists transfer $\mathrm{KO}_\ast(\mathrm{C}^{\ast} \pi_1 M)\to...

We construct a slant product $/ \colon \mathrm{S}_p(X \times Y) \mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X)$ on the analytic structure group of Higson and Roe K-theory stable corona Emerson Meyer. The latter is domain co-assembly map $μ^\ast \mathrm{K}_{1-\ast}(\mathfrak{c}^{\mathrm{red}}Y) \mathrm{K}^\ast(Y)$. obtain such products entire Higson--Roe sequence. They imply injectivity results for external maps. Our apply to with aspherical manifolds whose fundamental...