Related papers: Essential minimal volume of Einstein 4-manifolds

We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of certain characteristic numbers of a Riemannian manifold, including all Pontryagin and Chern numbers, is bounded proportionally to the…

A weakly Einstein manifold is a generalization of a 4-dimensional Einstein manifold, which is defined as an application of a curvature identity derived from the generalized Gauss-Bonnet formula for a 4-dimensional compact oriented…

We show that if a closed manifold M admits an F-structure (possibly of rank 0) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial circle action. As a corollary we obtain that the simplicial volume…

Let \Sigma be a compact oriented surface immersed in a four dimensional K\"ahler-Einstein manifold M. We consider the evolution of \Sigma in the direction of its mean curvature vector. It is proved that being symplectic is preserved along…

We produce some explicit examples of conformally compact Einstein manifolds, whose conformal compactifications are foliated by Riemannian products of a closed Einstein manifold with the total space of a principal circle bundle over products…

We show that a complete $3$-dimensional Riemannian manifold $M$ with finitely generated first homology has macroscopic dimension $1$ if it satisfies the following "macroscopic curvature" assumptions: every ball of radius $10$ in $M$ has…

Given an Einstein structure with positive scalar curvature on a four-dimensional Riemannian manifolds, that is $Ric=\lambda g$ for some positive constant $\lambda$. For convenience, the Ricci curvature is always normalized to $Ric=1$. A…

Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the…

In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very…

We prove that on any compact manifold $M^n$ with boundary, there exist a conformal class $C$ such that for any riemannian metric $g\in C$, $\lambda_1(M^n,g)Vol(M^n,g)^{2/n}< n.Vol(S^n,g_{\textrm{can}})^{2/n}$ and $\sigma_1(M,g,\rho)\mathcal…

We observe inequalities involving the Herzlich volume of a 4-dimensional asymptotically complex hyperbolic Einstein manifold and its Euler characteristic provided the metrics is either Kaehler or selfdual. In the selfdual case we have to…

In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume ${\rm Vol}^{(2,2)}$ for 5-dimensional and 6-dimensional spin manifolds with boundary and we also get the…

We show that the minimal volume of surfaces of log general type, with non-empty non-klt locus on the ample model, is $\frac{1}{825}$. Furthermore, the ample model $V$ achieving the minimal volume is determined uniquely up to isomorphism.…

We show that the simplicial volume of a contractible 3-manifold not homeomorphic to $\mathbb{R}^3$ is infinite. As a consequence, the Euclidean space may be characterized as the unique contractible $3$-manifold with vanishing minimal…

In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as…

We derive an integral inequality between the mean curvature and the scalar curvature of the boundary of any scalar flat conformal compactifications of Poincar{\'e}-Einstein manifolds. As a first consequence , we obtain a sharp lower bound…

Given a closed Riemannian manifold $(M, g_M)$ of dimension $n \geq 3$, we prove the existence of a conformally compact Einstein metric $g_{+}$ defined on a collar neighborhood $M \times (0,1]$ whose conformal infinity is $[g_M]$.

We simplify V\'etois' Obata-type argument and use it to identify a closed interval $I_n$, $n \geq 3$, containing zero such that if $a \in I_n$ and $(M^n,g)$ is a closed conformally Einstein manifold with nonnegative scalar curvature and…

The Hausmann-Weinberger invariant of a group G is the minimal Euler characteristic of a closed orientable 4-manifold M with fundamental group G. We compute this invariant for finitely generated free abelian groups and estimate the invariant…

We prove that for manifolds with negative curvature bounded away from $0$ of infinite volume and bounded geometry, the bounded fundamental class, defined via integration of the volume form over straight top-dimensional simplices, vanishes…