New methods for variational problems in Riemannian geometry

黎曼流形是曲面的多维推广。R.Thom的一个著名的公开问题：在给定的光滑流形上，什么是最好的，最好的，或最优的黎曼度量？邀请我们寻找比附近的所有形状更少弯曲的形状。早些时候我们发现，每个高维流形，甚至一个球体，都可以容纳无限多个这样的局部最优形状(=黎曼度量)，这些形状彼此之间以及与“标准”形状(例如，来自圆形球体)非常不同。这些形状是一些鲜为人知的代数现象的几何表示(例如，平凡群存在非常简短但非常非平凡的表示)。我们计划继续研究这些局部最优形状(特别是与量子引力相关的维度4)，作为我们更广泛的几何研究的一部分，以及具有各种几何界限的黎曼结构空间的组合学。具体地说，“最不弯曲的”可以用许多自然但不同的方式来理解(对应于不同的黎曼泛函)。我们知道，对于这些泛函中的某些泛函，存在局部最优的黎曼度量，但对于另一些泛函，我们想证明它们的存在性。我们想要找出这些现象的一些痕迹是否存在于3维空间。 在不同的方向上，我们计划研究黎曼流形中的闭极小曲面(‘肥皂泡’)及其一维类似物：测地线网和周期测地线。测地线是流形上最直的可能曲线；而且，点与点之间的最短路径总是由测地线提供。如果一条测地线平滑地闭合它自己，它被称为周期性的。一个封闭的测地网由有限多条在其端点处相交并在每个端点处满足自然平衡条件的测地线组成。这种极小物体的存在在许多情况下都得到了证明。然而，存在的证明是非建设性的，并且几乎没有阐明最自然的问题，例如``周期测地线的最小长度是多少？一个封闭的测地网？最小曲面的最小面积是多少？“延续了M.Gromov和C.Croke的开创性工作，我和R.罗特曼在不同的情况下证明了许多回答这些和类似问题的定理。在某些情况下，我们的上界出人意料地只涉及到关于环境流形的很少信息，例如，只有它的体积或直径。然而，许多其他类似性质的问题仍未得到解决。它们与黎曼流形按圈最优扫除的几何问题密切相关，与黎曼流形上的圈和圈空间的几何密切相关，其中这些巨大的无限维空间的柔软性在某种程度上被基础流形的有限维刚性所驯服。

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.