Riesz transform on manifolds with quadratic curvature decay.

— We obtain Euclidean volume growth results for complete Rie- mannian manifolds satisfying a Euclidean Sobolev inequality and a spectral type condition on the Ricci curvature.. We

Brownian bridge, Cartan-Hadamard manifold, comparison theorems, Cox-Ingersoll-Ross process, heat kernel, large deviations, rank-one noncompact symmetric

Under volume growth, diameter or density of rays conditions, some results were obtained on the geometry and topology of open manifolds with nonnegative Ricci curvature... Abresch

In general, the global existence and convergence of a mean curvature flow relies on the boundedness of the second fundamental form. In the above theorem, the boundedness of the

Key-words : Martingale, Brownian motion, negative curvature, Riemannian manifold, stochastic development, escape rates, passage times, bounded harmonic

Then Int(7V) has a metric of quadratic curvature decay and slow volume growth if and only if the connected components of9N are graph manifolds.. Finally, by an argument similar to

function of physical quantities called the matter fields, which in turn are required to satisfy certain other partial differential equations.. We

Grossman’s estimate [8] of the distance from a totally geo- desic submanifold to its focal cut locus on compact manifolds of strictly positive sectional curvature