New methods for variational problems in Riemannian geometry

黎曼流形是曲面的多维推广。 R.在给定的光滑流形上，什么是最好的，或者最好的，或者最优的黎曼度量？“邀请我们去寻找比附近所有形状都更少弯曲的形状。早些时候我们发现，每个高维流形，甚至一个球，承认无限多个这样的局部最优形状（=黎曼度量）是非常不同的彼此和从一个“标准”的形状（例如，从一个圆球）。这些形状是一些很少被理解的代数现象的几何表现（例如，平凡群的非常短但高度非平凡的表示的存在）。我们计划继续研究这些局部最优形状（特别是在与量子引力相关的4维中），作为我们更广泛研究具有各种几何边界的黎曼结构空间的几何和组合学的一部分。特别是，“最小弯曲”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）。我们知道局部最优的黎曼度量存在于这些泛函中的一些，但想证明它们存在于其他一些。我们想知道这些现象是否存在于第三维度。

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.