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。

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.