NEW METHODS FOR VARIATIONAL PROBLEMS IN RIEMANNIAN GEOMETRY

黎曼流形是欧氏空间中曲面的多维推广。 我们想要研究各种流形(如 例如，圆形球体)。我们正在开发一种新的方法来构建他们的基础 关于数理逻辑。我们还对一般黎曼流形上的极值对象感兴趣， 例如像肥皂膜一样的极小曲面，或者类似于赤道的周期测地线 一个圆形球体。以前已知的用于证明这种对象的存在的拓扑方法 几乎没有关于它们的性质的信息。因此，我们正在开发新的方法，可以 用于获取有关这些极端物体的复杂性的新信息。我们的方法包括 看黎曼流形分成闭曲线的复杂性和复杂性 关于将黎曼流形上的闭合曲线或球面收缩到一点。这些问题是 对几何学家来说也是一个独立的兴趣。

Riemannian manifolds are multidimensional generalizations of surfaces in the Euclidean spaces. We would like to study ``optimal" and ``locally optimal" shapes of various manifolds (such as as, for example, the round sphere). We are developing a new approach to their construction based on mathematical logic. We are also interested in extremal objects on general Riemannian manifolds, such as the soap-film like minimal surfaces, or periodic geodesics that resemble the equator of a round sphere. Previously known topological methods used to prove the existence of such objects yield little information about their properties. Therefore we are developing new methods that can be used to obtain new information about complexity of these extremal objects. Our methods involve looking at the complexities of slicings of Riemannian manifolds into closed curves and complexities of contracting closed curves or spheres on a Riemannian manifold to a point. These problems are also of an independent interest to geometers.