NEW METHODS FOR VARIATIONAL PROBLEMS IN RIEMANNIAN GEOMETRY

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Riemannian manifolds are multidimensional generalizations of surfaces in the Euclidean spaces. We would like to study ``optimal" and ``locally optimal" shapes of various manifolds (such as as, for example, the round sphere). We are developing a new approach to their construction based on mathematical logic. We are also interested in extremal objects on general Riemannian manifolds, such as the soap-film like minimal surfaces, or periodic geodesics that resemble the equator of a round sphere. Previously known topological methods used to prove the existence of such objects yield little information about their properties. Therefore we are developing new methods that can be used to obtain new information about complexity of these extremal objects. Our methods involve looking at the complexities of slicings of Riemannian manifolds into closed curves and complexities of contracting closed curves or spheres on a Riemannian manifold to a point. These problems are also of an independent interest to geometers.