Short geodesic loops on complete Riemannian manifolds with a finite volume (Q2838109)

The author proves two theorems ensuring the abundance of geodesic loops on finite volume complete noncompact Riemannian manifolds. The first theorem implies that there are uncountably many geodesic loops of arbitrarily small length, and gives a complicated explicit estimate of how far out into the manifold you have to go (starting at any given point) before you run into such a geodesic loop. The second theorem says that, for any given loop length, once you go a certain distance away from a given point, every sphere about that point of radius larger than that given distance touches a geodesic loop of at most that loop length. (Note that geodesic loops might not be periodic geodesics, since the loop might start out pointing in a different direction than the direction it points when it closes up.) The method involves Gromov's concept of filling radius, and fillings of piecewise smooth maps of graphs into the manifold; the author provides a very clear and intuitive explanation of the method.