An assortment of negatively curved ends

Motivated by a recent groundbreaking work of Ontaneda, we describe a sizable class of closed manifolds such that the product of each manifold in the class with the real line admits a complete metric of bounded negative sectional curvature which is an exponentially warped near one end and has finite volume near the other end.

Almost non-negatively curved 4-manifolds with torus symmetry

We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank, giving a classification of closed, smooth, simply-connected 4-manifolds of almost non-negative curvature under the assumption of torus symmetry.

Graphic Bernstein Results in Curved Pseudo-Riemannian Manifolds

We generalize a Bernstein-type result due to Albujer and Alias, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $Sigma_1times mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete spacelike graphic submanifold with parallel mean curvature, defined by a map $f: Sigma_1to Sigma_2$ between two Riemannian manifolds $(Sigma_1^m, g_1)$ and $(Sigma^n_2, g_2)$ of sectional curvatures $K_1$ and $K_2$, respectively. We take on $Sigma_1times Sigma_2$ the pseudo-Riemannian product metric $g_1-g_2$. Under the curvature conditions, $mathrm{Ricci}_1 geq 0$ and $K_1geq K_2$, we prove that, if the second fundamental form of $M$ satisfies an integrability condition, then $M$ is totally geodesic, and it is a slice if $mathrm{Ricci}_1(p)>0$ at some point. For bounded $K_1$, $K_2$ and hyperbolic angle $theta$, we conclude $M$ must be maximal. If $M$ is a maximal surface and $K_1geq K_2^+$, we show $M$ is totally geodesic with no need for further assumptions. Furthermore, $M$ is a slice if at some point $pin Sigma_1$, $K_1(p)> 0$, and if $Sigma_1$ is flat and $K_2<0$ at some point $f(p)$, then the image of $f$ lies on a geodesic of $Sigma_2$.

2-Riemannian manifolds

A {em 2-Riemannian manifold} is a differentiable manifold exhibiting a 2-inner product on each tangent space. We first study lower dimensional 2-Riemannian manifolds by giving necessary and sufficient conditions for flatness. Afterward we associate to each 2-Riemannian manifold a unique torsion free compatible pseudoconnection. Using it we define a curvature for 2-Riemannian manifolds and study its properties. We also prove that 2-Riemannian pseudoconnections do not have Koszul derivatives. Moreover, we define stationary vector field with respect to a 2-Riemannian metric and prove that the stationary vector fields in $mathbb{R}^2$ with respect to the 2-Riemannian metric induced by the Euclidean product are the divergence free ones.

Affine images of Riemannian manifolds

We describe all affine maps from a Riemannian manifold to a metric space and all possible image spaces.