New methods for variational problems in Riemannian geometry

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Riemannian manifolds are multidimensional generalizations of surfaces. A well-known open question of R. Thom "What is the best, or the nicest, or the optimal Riemannian metric on a given smooth manifold?" invites us to look for shapes that are less curved that all nearby shapes. Earlier we discovered that each high dimensional manifold, even a sphere, admits infinitely many such locally optimal shapes (=Riemannian metrics) that are very different from each other and from a ``standard" shape (e.g. from a round sphere). These shapes are geometric manifestations of some poorly understood algebraic phenomena (e.g. the existence of very short but highly non-trivial presentations of the trivial group). We plan to continue investigating these locally optimal shapes (especially, in dimension 4 which is relevant for Quantum Gravity) as a part of our broader study of geometry and combinatorics of spaces of Riemannian structures with various bounds on geometry. In particular, ``the least curved" can be understood in a number of natural but different ways (corresponding to different Riemannian functionals). We know that the locally optimal Riemannian metrics exist for some of these functionals, but would like to prove their existence for some others. We would like to find out if some vestiges of these phenomena exist in dimension 3. ******In a different direction we plan to study closed minimal surfaces (``soap bubbles") in Riemannian manifolds, and their one-dimensional analogs: geodesic nets and periodic geodesics. A geodesic is a straightest possible curve on a manifold; also, the shortest way to travel between points is always provided by a geodesic. If a geodesic smoothly closes upon itself, it is called periodic. A closed geodesic net consists of finitely many geodesics meeting at their endpoints and satisfying a natural equilibrium condition at every endpoint. The existence of such minimal objects was proven in many situations. Yet the existence proofs are non-constructive and shed little light on the most natural questions such as ``What is the smallest length of a periodic geodesic? a closed geodesic net? What is the smallest area of a minimal surface?". Continuing a pioneering work of M. Gromov and C. Croke, I and R. Rotman proved many theorems answering these and similar questions in different situations. In some cases our upper bounds unexpectedly involve surprisingly little information about the ambient manifold, e.g. only its volume or diameter. Yet many other questions of such nature remain unsolved. They are closely related to questions about geometry of optimal sweep-outs of Riemannian manifolds by cycles, and geometry of spaces of loops and cycles on Riemannian manifolds, where the flabbiness of these huge infinite-dimensional spaces is somewhat tamed by the rigidity stemming from finite-dimensionality of the underlying manifold.**