We prove for $n\in\{3,4,5\}$ that the connected sum of a closed aspherical $n$-manifold with an arbitrary non-compact manifold does not admit a complete metric with nonnegative scalar curvature. In particular, a special case of our result answers a question of Gromov.
More generally, we generalize the partial classification result of Chodosh, Li, and Liokumovich to the non-compact domination case with our newly-developed technique.
Our result unifies all previous results of this type, and confirms the validity of Gromov's non-compact domination conjecture for closed aspherical manifolds of dimensions 3, 4, and 5.
We exhibit a closed aspherical 5-manifold of nonpositive curvature that fibers over a circle whose fundamental group is hyperbolic relative to abelian subgroups such that the fiber is a closed aspherical 4-manifold whose fundamental group is not hyperbolic relative to abelian subgroups.
In this paper we study invariant almost Hermitian geometry on generalized flag manifolds which the isotropy representation decompose into two or three irreducible components. We will provide a classification of such flag manifolds admitting Kähler like scalar curvature metric, that is, almost Hermitian structures $(g,J)$ satisfying $s=2s_C$ where $s$ is Riemannian scalar curvature and $s_C$ is the Chern scalar curvature.
A metric space $X$ is called uniformly acyclic if there there exists an {\it acyclicty control function} $R=R(r)=R_X(r)\geq r $, $0\leq r <\infty$, such that the homology inclusion homomorphisms between the balls around all points $x\in X$, $$H_i(B_x(r))\to H_i(B_x(R))$$ vanish for all $i=1,2,\ldots$. We show that if a complete orientable $m$-dimensional manifold $\tilde X$ of dimension $m\leq 5$ admits a proper (infinity goes to infinity) distance decreasing map to a complete $m$-dimensional uniformly acyclic manifold, then the scalar curvature of $\tilde X$ can't be uniformly positive, $$\in…
Up to dimension five, we can prove that given any closed Riemannian manifold with nonnegative scalar curvature, of which the universal covering has vanishing homology group $H_k$ for all $k\geq 3$, either it is flat or it has Gauss-Bonnet quantity (defined by (1.3)) no greater than $8π$. In the second case, the equality for Gauss-Bonnet quantity yields that the universal covering splits as the Riemannian product of a $2$-sphere with non-negative sectional curvature and the Euclidean space. We also establish a dominated version of this result and its application to homotopical $2$-systole estim…
We show that if a sequence $M_n$ of closed aspherical $d$-dimensional Riemannian manifolds with Ricci curvature uniformly bounded below and diameter uniformly bounded above collapses, then for all large enough $n$, the fundamental groups $π_1(M_n)$ have non-trivial finitely generated abelian normal subgroups. In particular, the groups $π_1(M_n)$ cannot be non-elementary hyperbolic.
We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, 3) a conjectural bound of $L^2$-Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of $1$-balls in the universal cover.
We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle-Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimensions at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, inc…
Gromov's Simplicial Volume Vanishing Conjecture for Positive Scalar Curvature
Department of Mathematics, Texas A&M University（德州农工大学数学系）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
Inspired by Gromov's work on 'Metric inequalities with scalar curvature' we establish band width inequalities for Riemannian bands of the form $(V=M\times[0,1],g)$, where $M^{n-1}$ is a closed manifold. We introduce a new class of orientable manifolds we call filling enlargeable and prove: If $M$ is filling enlargeable and all unit balls in the universal cover of $(V,g)$ have volume less than a constant $\frac{1}{2}\varepsilon_n$, then $width(V,g)\leq1$. We show that if a closed orientable manifold is enlargeable or aspherical, then it is filling enlargeable. Furthermore we establish that whet…
