The use of harmonic maps has been spectacularly successful in proving rigidity (and superrigidity) results for non-positively curved Riemannian manifolds. This is witnessed for example by results of Sui [31], Sampson [26], Corlette [6], Gromov and Schoen [18], Jost and Yau [22], and Mok, Sui and Yeung [23]. All of which are based on the pioneering existence theorem of Eells and Sampson [13] and...

This paper describes an approach that uses flat-spacetime dimension estimators to estimate the manifold dimension of causal sets that can be faithfully embedded into curved spacetimes. The approach is invariant under coarse graining and can be implemented independently of any specific curved spacetime. Results are given based on causal sets generated by random sprinklings into conformally flat ...

The space of positive definite matrices P(n) is a Riemannian manifold with variable nonpositive curvature. It includes Euclidean space and hyperbolic space as submanifolds, and poses significant challenges for the design of algorithms for data analysis. In this paper, we develop foundational geometric structures and algorithms for analyzing collections of such matrices. A key technical contribu...

Let X be a compact complex manifold, L → X an ample line bundle over X, and H the space of all positively curved metrics on L. We show that a pair (h0, T ) consisting of a point h0 ∈ H and a test configuration T = (L → X → C), canonically determines a weak geodesic ray R(h0, T ) in H which emanates from h0. Thus a test configuration behaves like a vector field on the space of Kähler potentials ...

If a class of nitely generated groups G is closed under isometric amalgamations along free subgroups, then every G 2 G can be quasi-isometrically embedded in a group Ĝ 2 G that has no proper subgroups of nite index. Every compact, connected, non-positively curved space X admits an isometric embedding into a compact, connected, non-positively curved space X such that X has no non-trivial nite-sh...

We give a new simple construction of the maximal entropy invariant measure (the Bowen{Margulis measure) for the geodesic ow on a compact negatively curved manifold. This construction directly connects invariant`conformal density" arising from the Patterson measure on the sphere at innnity of the universal covering space with the maximal entropy measureof the geodesic ow. 0. Introduction Let M b...

In this paper we announce the following result: “Every manifold of dimension ≥ 3 admits a complete negatively Ricci curved metric.” Furthermore we describe some sharper results and sketch proofs.