Alexandrov spaces are defined via axioms similar to those given by Euclid. The Alexandrov
axioms replace certain equalities with inequalities. Depending on the signs of the …

A sharp lower bound for the injectivity radius in noncompact nonnegatively curved
Riemannian manifolds involving their soul goes back to Šarafutdinov. We generalize this …

In this paper we develop a Morse theory for the uniform energy. We use the one-sided
directional derivative of the distance function to study the minimizing properties of variations …

We prove that the sub-Riemannian exponential map is not injective in any neighbourhood of
certain critical points. Namely that it does not behave like the injective map of reals given by f …

In this work we study the issue of geodesic extendibility on complete and locally compact
metric length spaces. We focus on the geometric structure of the space (Σ (X), d H) of …

Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have
Finitely Generated Fundamental Groups Page 1 Canad. Math. Bull. Vol. ( ), pp. – http://dx.doi.org …

We give an optimal estimate for the norm of any submanifold's second fundamental form in
terms of its focal radius and the lower sectional curvature bound of the ambient manifold …

In this work we study the geodesic structure of the space $\Sigma (X) $ of compact balls of a
complete and locally compact metric length space endowed with the Hausdorff distance …

I completed my dissertation with Jeff Cheeger at Courant in 1996 and then had a one year
postdoc with Shing-Tung Yau at Harvard. At this time I focused on complete noncompact …