The Morse index, topology and geometry of branched constant mean curvature surfaces.

该项目属于EPSRC职权范围内的数学科学部分。它涉及到黎曼三流形内的常平均曲率（CMC）曲面的研究。虽然这个主题是纯数学的，但CMC表面广泛地出现在自然和物理科学中，例如通过描述肥皂膜/气泡的形状，广义相对论中的事件视界和凝聚态物理学中的费米表面。具体而言，该项目将调查的有效性的莫尔斯指数界，从上面，在（分支）常数平均曲率（CMC）表面内黎曼流形的属和面积。它也将提供适当的变分界限下的分支CMC超曲面的泡紧性结果。这些结果将提供洞察力的莫尔斯指数，几何形状和拓扑结构的CMC表面之间的关系-一个话题的重大利益，由于最近的进展，存在理论的对象。特别是这个项目将有助于确定什么类型的CMC/最小表面被允许在一个给定的空间，并分类那些做。

This project lies within the mathematical sciences section of EPSRC's remit. It involves a study of constant mean curvature (CMC) surfaces inside Riemannian three-manifolds. Whilst the subject is purely mathematical, CMC surfaces appear extensively in nature and the physical sciences, for example by describing the shape of soap films/bubbles, event horizons in general relativity and Fermi surfaces from condensed matter physics. Specifically the project will investigate the validity of a Morse index bound, from above, in terms of the genus and area of (branched) constant mean curvature (CMC) surfaces inside Riemannian three-manifolds. It will also provide bubble-compactness results for branched CMC hypersurfaces under suitable variational bounds. These results will provide insights in to the relationship between the Morse index, geometry and topology of CMC surfaces - a topic of significant interest due to the recent advances in existence theory of such objects. In particular this project will help to determine what types of CMC/minimal surfaces are admitted in a given space, and to classify those that do.