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.
We describe the action of the fundamental group of a closed Finsler surface of negative curvature on the geodesics in the universal covering in terms of a flat symplectic connection and consider the first order deformation theory about a hyperbolic metric. A construction of O.Biquard yields a family of metrics which give nontrivial deformations of the holonomy, extending the representation of the fundamental group from SL(2,R) into the group of Hamiltonian diffeomorphisms of S^1 x R, and producing an infinite-dimensional version of Teichmuller space which contains the classical one.
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.
Let $nge 2$ and $kge 1$ be two integers. Let $M$ be an isometrically immersed closed $n$-submanifold of co-dimension $k$ that is homotopic to a point in a complete manifold $N$, where the sectional curvature of $N$ is no more than $delta<0$. We prove that the total squared mean curvature of $M$ in $N$ and the first non-zero eigenvalue $lambda_1(M)$ of $M$ satisfies $$lambda_1(M)le nleft(delta +frac{1}{operatorname{Vol} M}int_M |H|^2 operatorname{dvol}right).$$ The equality implies that $M$ is minimally immersed in a metric sphere after lifted to the universal cover of $N$. This completely settles an open problem raised by E. Heintze in 1988.
We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold $N$ with sectional curvatures bounded from above by a negative quantity $K_{N}leq b<0$