We study the limit of quasilocal mass defined in [4] and [5] for a family of spacelike 2-surfaces in spacetime. In particular, we show the limit coincides with the ADM mass at spatial infinity. The limit for coordinate spheres of a boosted slice of the Schwarzchild solution is computed explicitly and shown to give the expected energymomentum four-vector

Compact locally maximal hyperbolic sets are studied via geometrically defined functional spaces that take advantage of the smoothness of the map in a neighborhood of the hyperbolic set. This provides a self-contained theory that not only reproduces all the known classical results, but also gives new insights on the statistical properties of these systems.

We define the quantum correction of the Teichmller space \mathcal {T} of Calabi-Yau manifolds. Under the assumption of no weak quantum correction, we prove that the Teichmller space \mathcal {T} is a locally symmetric space with the Weil-Petersson metric. For Calabi-Yau threefolds, we show that no strong quantum correction is equivalent to that, with the Hodge metric, the image \mathcal {T} of the Teichmller space \mathcal {T} under the period map \mathcal {T} is an open submanifold of a globally Hermitian symmetric space \mathcal {T} of the same dimension as \mathcal {T} . Finally, for Hyperkhler manifold of dimension \mathcal {T} , we find both locally and globally defined families of \mathcal {T} and \mathcal {T} -classes over the Teichmller space of polarized Hyperkhler manifolds.

In this paper we prove that given a point p ∈ Mn, where Mn is a closed Riemannian manifold of dimension n, the length of a shortest geodesic loop lp(Mn) at this point is bounded above by 2nd, where d is the diameter of Mn. Moreover, we show that on a closed simply connected Riemannian manifold Mn with a nontrivial second homotopy group there either exist at least three geodesic loops of length less than or equal to 2d at each point of Mn, or the length of a shortest closed geodesic on Mn is bounded from above by 4d.

1 Preliminaries 3 1.1 Linear algebra 3 1.1. 1 Vectors and matrices 3 1.1. 2 Symmetric bilinear forms 5 1.1. 3 Vector subspaces 6 1.1. 4 Linear maps from Rn to Rn 7 1.1. 5 A convention 9 1.2 Vector calculus 9