Sharp Resolvent Estimates on Non-positively Curved Asymptotically Hyperbolic Manifolds

We study the high energy estimate for the resolvent $R(\lambda )$ of the Laplacian on non-trapping asymptotically hyperbolic manifolds (AHMs). In the literature, estimates of $R(\lambda )$ on weighted Sobolev spaces of the order $O(|\lambda |^{N})$ were established for some $N> -1$, $|\lambda |$ large, and $\lambda \in{{\mathbb{C}}}$ in strips where $R(\lambda )$ is holomorphic. In this work, we prove the optimal bound $O(|\lambda |^{-1})$ under the non-positive sectional curvature assumption by taking into account the oscillatory behavior of the Schwartz kernel of the resolvent.

On the δ-Pinching Function of the Sectional Curvature of a Compact Connected Lie Group G with a Bi-Invariant Riemannian Metric and a Vectorial Torsion Connection

One of the important problems of Riemannian geometry is the problem of establishing connections between curvature and the topology of a Riemannian manifold, and, in particular, the influence of the sign of sectional curvature on the topological structure of a Riemannian manifold. Of particular importance in these studies is the question of the influence of d-pinching of Riemannian metrics of positive sectional curvature on the geometric and topological structure of the Riemannian manifold. This question is most studied for the homogeneous Riemannian case. In this direction, the classification of homogeneous Riemannian manifolds of positive sectional curvature, obtained by M. Berger, N. Wallach, L. Bergeri, as well as a number of results on d- pinching of homogeneous Riemannian metrics of positive sectional curvature, is well known. In this paper, we investigate Riemannian manifolds with metric connection being a connection with vectorial torsion. The Levi-Civita connection falls into this class of connections. Although the curvature tensor of these connections does not possess the symmetries of the Levi-Civita connection curvature tensor, it seems possible to determine sectional curvature. This paper studies the d-pinch function of the sectional curvature of a compact connected Lie group G with a biinvariant Riemannian metric and a connection with vectorial torsion. It is proved that it takes the values d(||V ||)∈(0,1].

Open manifolds with non-homeomorphic positively curved souls

We extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic simply connected and positively-curved souls. Examples of such pairs are given by explicit pairs of Eschenburg spaces. To deduce the second statement from the first, we extend our earlier work on the stable converse soul question and show that it has a positive answer for a class of spaces that includes all Eschenburg spaces.

Positive curvature and symmetry in small dimensions

Extending existing work in small dimensions, Dessai computed the Euler characteristic, signature, and elliptic genus for [Formula: see text]-manifolds of positive sectional curvature in the presence of torus symmetry. He also computes the diffeomorphism type by restricting his results to classes of manifolds known to admit non-negative curvature, such as biquotients. The first part of this paper extends Dessai’s calculations to even dimensions up to [Formula: see text]. In particular, we obtain a first characterization of the Cayley plane in such a setting. The second part studies a closely related family of manifolds called positively elliptic manifolds, and we prove a conjecture of Halperin in this context for dimensions up to [Formula: see text] or Euler characteristics up to [Formula: see text].

Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space

A Riemannian manifold is said to be almost positively curved if the set of points for which all two-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented two-planes in $\mathbb{R}^{7}$ admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1. The construction and verification rely on the Lie group $\mathbf{G}_{2}$ and the octonions, so do not obviously generalize to any other Grassmannians.

Lower Bound for the First Steklov Eigenvalue

In this paper we find lower bounds for the first Steklov eigenvalue in Riemannian n-manifolds, n = 2, 3, with non-positive sectional curvature.

On a generalized conjecture of Hopf with symmetry

A famous conjecture of Hopf states that$\mathbb{S}^{2}\times \mathbb{S}^{2}$does not admit a Riemannian metric with positive sectional curvature. In this article, we prove that no manifold product$N\times N$can carry a metric of positive sectional curvature admitting a certain degree of torus symmetry.