New methods for variational problems in Riemannian geometry

黎曼流形是曲面的多维推广。R. Thom的一个著名的开放问题是“在给定的光滑流形上，什么是最好的，或者最好的，或者最优的黎曼度规？”这让我们去寻找比所有附近的形状更不弯曲的形状。前面我们发现，每个高维流形，甚至一个球体，都允许无限多个这样的局部最优形状（=黎曼度量），它们彼此之间和与“标准”形状（例如，与圆球）非常不同。这些形状是一些鲜为人知的代数现象的几何表现（例如，存在非常短但高度非平凡的平凡群表示）。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4维），作为我们更广泛的几何和黎曼结构空间组合学研究的一部分。特别是，“最不弯曲”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）。我们知道局部最优黎曼度量对于某些泛函是存在的，但是我们想要证明它们对于其他泛函的存在性。我们想知道这些现象的一些痕迹是否存在于第三维度。******在一个不同的方向上，我们计划研究黎曼流形中的封闭最小表面（“肥皂泡”），以及它们的一维类似物：测地线网和周期测地线。测地线是流形上最直的曲线；同样，点之间的最短路径总是由测地线提供的。如果测地线平滑地闭合，它就称为周期性的。一个封闭的测地线网由有限条端点相交且在每个端点满足自然平衡条件的测地线组成。这种最小物体的存在在许多情况下都得到了证明。然而，存在性证明是非建设性的，对最自然的问题也没有什么帮助，比如“周期测地线的最小长度是多少？”一个封闭的测地线？最小曲面的最小面积是多少？延续M. Gromov和C. Croke的开创性工作，I和R. Rotman证明了许多定理，在不同的情况下回答了这些问题和类似的问题。在某些情况下，我们的上界出乎意料地涉及到很少的关于环境流形的信息，例如，只有它的体积或直径。然而，其他许多此类性质的问题仍未得到解决。它们与黎曼流形的最优循环的几何问题密切相关，以及黎曼流形上的循环和循环空间的几何问题，其中这些巨大的无限维空间的弹性在某种程度上被源于底层流形的有限维的刚性所驯服

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.**