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We prove a sharp upper bound for the bottom spectrum of Laplacian on geometrically contractible manifolds with scalar curvature lower bound, and characterize the distribution of scalar curvature when equality holds. Moreover, we prove a scalar curvature rigidity theorem if the manifold is the universal cover of a closed hyperbolic manifold., Comment: 31 pages

We prove the scalar curvature rigidity for $L^\infty$ metrics on $\mathbb S^n\backslash\Sigma$, where $\mathbb S^n$ is the $n$-dimensional sphere with $n\geq 3$ and $\Sigma$ is a closed subset of $\mathbb S^n$ of codimension at least $\frac{n}{2}+1$ that satisfies the wrapping property. The notion of wrapping property was introduced by the second author for studying related scalar curvature rigidity problems on spheres. For example, any closed subset of $\mathbb S^n$ contained in a hemisphere and any finite subset of $\mathbb S^n$ satisfy the wrapping property. The same techniques also apply to prove an analogous scalar rigidity result for $L^\infty$ metrics on tori that are smooth away from certain subsets of codimension at least $\frac{n}{2}+1$., Comment: 22 pages

Let $(M,g)$ be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by $n(n-1)$. In this paper, we prove that if $f$ is a smooth map of non-zero degree from $(M, g)$ to the unit four-sphere, then $f$ is an isometry. Following ideas of Gromov, we use $\mu$-bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow., Comment: Improved exposition. Comments are welcome!

We prove a quantitative upper bound on the filling radius of complete, spin manifolds with uniformly positive scalar curvature using the quantitative operator $K$-theory and index theory., Comment: minor revision

In this paper we prove the scalar curvature extremality and rigidity for a class of warped product spaces that are possibly degenerate at the two ends. The leaves of these warped product spaces can be any closed Riemannian manifolds with nonnegative curvature operators and nonvanishing Euler characteristics, flat tori, round spheres and their direct products. In particular, we obtain the scalar curvature extremality and rigidity for certain degenerate toric bands and also for round spheres with two antipodal points removed. This answers positively the corresponding questions of Gromov in all dimensions., Comment: 39 pages. Some proofs in Section 3 are revised and simplified by using the arguments from Section 4

In this paper, we prove a dihedral extremality and rigidity theorem for a large class of codimension zero submanifolds with polyhedral boundary in warped product manifolds. We remark that the spaces considered in this paper are not necessarily warped product manifolds themselves. In particular, the results of this paper are applicable to submanifolds (of warped product manifolds) with faces that are neither orthogonal nor parallel to the radial direction of the warped product metric. Generally speaking, the dihedral rigidity results require the leaf of the underlying warped space to have positive Ricci curvature and the warping function to be strictly log-concave. Nevertheless, we prove a dihedral rigidity theorem for a large class of hyperbolic polyhedra, where the leaf of the underlying warped product space is flat and the warping function is not strictly log-concave., Comment: 36 pages, minor changes

We investigate how the positive scalar curvature controls the size of a Ricci limit space when it comes from a sequence of $n$-manifolds with non-negative Ricci curvature and strictly positive scalar curvature lower bound. We prove such a limit space can split off $\mathbb{R}^{n-2}$ at most, and when the maximal splitting happens, the other non-splitting factor has an explicit uniform diameter upper bound. Besides, we study some other consequences of having positive scalar curvature for manifolds using Ricci limit spaces techniques, for instance volume gap estimates and volume growth order estimates., Comment: 22 pages. Some conditions added to the theorem 1.1 due to a gap in the original proof, and the proof is slightly changed accordingly. A corollary about the first Betti number is added

In this paper, we prove a rigidity theorem for smooth strictly convex domains in Euclidean spaces., Comment: 7 pages. merged with arXiv:2203.09511

In this paper, we study $l^1$-higher index theory and its pairing with cyclic cohomology for both closed manifolds and compact manifolds with boundary. We first give a sufficient geometric condition for the vanishing of the $l^1$-higher indices of Dirac-type operators on closed manifolds. This leads us to define an $l^1$-version of higher rho invariants. We prove a product formula for these $l^1$-higher rho invariants. A main novelty of our product formula is that it works in the general Banach algebra setting, in particular, the $l^1$-setting. On compact spin manifolds with boundary, we also give a sufficient geometric condition for Dirac operators to have well-defined $l^1$-higher indices. More precisely, we show that, on a compact spin manifold $M$ with boundary equipped with a Riemannian metric which has product structure near the boundary, if the scalar curvature on the boundary is sufficiently large, then the $l^1$-higher index of its Dirac operator $D_M$ is well-defined and lies in the $K$-theory of the $l^1$-algebra of the fundamental group. As an immediate corollary, we see that if the Bost conjecture holds for the fundamental group of $M$, then the $C^\ast$-algebraic higher index of $D_M$ lies in the image of the Baum-Connes assembly map. By pairing the above $K$-theoretic $l^1$-index results with cyclic cocycles, we prove an $l^1$-version of the higher Atiyah-Patodi-Singer index theorem for manifolds with boundary. A key ingredient of its proof is the product formula for $l^1$-higher rho invariants mentioned above., Comment: 49 pages

In this paper, we prove Gromov's flat corner domination conjecture in all dimensions. As a consequence, we answer positively the Stoker conjecture for convex Euclidean polyhedra in all dimensions. By applying the same techniques, we also prove a rigidity theorem for strictly convex domains in Euclidean spaces., Comment: 27 pages. arXiv:2207.05731 has been merged into this paper. arXiv admin note: text overlap with arXiv:2112.01510

In this paper, we develop a new index theory for manifolds with polyhedral boundary. As an application, we prove Gromov's dihedral extremality conjecture regarding comparisons of scalar curvatures, mean curvatures and dihedral angles between two compact manifolds with polyhedral boundary in all dimensions. We also prove Gromov's dihedral rigidity conjecture for a class of positively curved manifolds with polyhedral boundary in all dimensions., Comment: 116 pages. Details for the index theorem of manifolds with polyhedral boundary have been provided

Gromov proved a cube inequality on the bound of distances between opposite faces of a cube equipped with a positive scalar curvature metric in dimension $\leq 8$ using minimal surface method. He conjectured that the cube inequality also holds in dimension $\geq 9$. In this paper, we prove Gromov's cube inequality in all dimensions with the optimal constant via Dirac operator method. In fact, our proof yields a strengthened version of Gromov's cube inequality, which does not seem to be accessible by minimal surface method., Comment: 21 pages. v3 to v4: the proof for the strict inequality has been revised

Gromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly. This shows that our result is the best possible. We prove our result by studying the index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control. A key technical ingredient for the proof of our main result is a Lipschitz control for the topological $K$-theory of finite dimensional simplicial complexes., Comment: 66 pages. minor revison

For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose $M$ is a closed smooth spin manifold and $\widetilde M$ is a $\Gamma$-regular covering space of $M$. Let $\langle \alpha \rangle$ be the conjugacy class of a non-identity element $\alpha\in \Gamma$. Suppose $\{\Gamma_i\}$ is a sequence of finite-index normal subgroups of $\Gamma$ that distinguishes $\langle \alpha \rangle$. Let $\pi_{\Gamma_i}$ be the quotient map from $\Gamma$ to $\Gamma/\Gamma_i$ and $\langle \pi_{\Gamma_i}(\alpha) \rangle$ the conjugacy class of $\pi_{\Gamma_i}(\alpha)$ in $\Gamma/\Gamma_i$. If the scalar curvature on $M$ is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of $\widetilde M$ at the conjugacy class $\langle \alpha \rangle$ is equal to the limit of the delocalized eta invariants for the Dirac operators of $M_{\Gamma_i}$ at the conjugacy class $\langle \pi_{\Gamma_i}(\alpha) \rangle$, where $M_{\Gamma_i}= \widetilde M/\Gamma_i$ is the finite-sheeted covering space of $M$ determined by $\Gamma_i$. In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of $M_{\Gamma_i}$ at the conjugacy class $\langle \pi_{\Gamma_i}(\alpha) \rangle$ converges, under the assumption that the rational maximal Baum-Connes conjecture holds for $\Gamma$., Comment: 41 pages. Minor revision and updated reference

Higher index of signature operator is a far reaching generalization of signature of a closed oriented manifold. When two closed oriented manifolds are homotopy equivalent, one can define a secondary invariant of the relative signature operator called higher rho invariant. The higher rho invariant detects the topological nonrigidity of a manifold. In this paper, we prove product formulas for higher index and higher rho invariant of signature operator on fibered manifolds. Our result implies the classical product formula for numerical signature of fiber manifolds obtained by Chern, Hirzebruch, and Serre in "On the index of a fibered manifold". We also give a new proof of the product formula for higher rho invariant of signature operator on product manifolds, which is parallel to the product formula for higher rho invariant of Dirac operator on product manifolds obtained by Xie and Yu in "Positive scalar curvature, higher rho invariants and localization algebras" and Zeidler in "Positive scalar curvature and product formulas for secondary index invariants"., Comment: 30 pages

The first main result of this paper is to prove that the convergence of Lott's delocalized eta invariant holds for all differential operators with a sufficiently large spectral gap at zero. Furthermore, to each delocalized cyclic cocycle, we define a higher analogue of Lott's delocalized eta invariant and prove its convergence when the delocalized cyclic cocycle has at most exponential growth. Our second main result is to obtain an explicit formula of the delocalized Connes-Chern character of all $C^\ast$-algebraic secondary invariants for word hyperbolic groups. Equivalently, we give an explicit formula for the pairing between $C^\ast$-algebraic secondary invariants and delocalized cyclic cocycles of the group algebra. When the $C^\ast$-algebraic secondary invariant is a $K$-theoretic higher rho invariant of an invertible differential operator, we show this pairing is precisely the higher analogue of Lott's delocalized eta invariant alluded to above. Our work uses Puschnigg's smooth dense subalgebra for word hyperbolic groups in an essential way. We emphasize that our construction of the delocalized Connes-Chern character is at $C^\ast$-algebra $K$-theory level. This is of essential importance for applications to geometry and topology. As a consequence, we compute the paring between delocalized cyclic cocycles and $C^\ast$-algebraic Atiyah-Patodi-Singer index classes for manifolds with boundary, when the fundamental group of the given manifold is hyperbolic., Comment: This is a revised version. Several extra results have been added. arXiv admin note: text overlap with arXiv:1804.09026 by other authors