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

该项目属于EPSRC的数学科学部分。它涉及到黎曼三流形内的等平均曲率曲面的研究。虽然这门学科是纯数学的，但CMC表面在自然界和物理科学中广泛出现，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。具体来说，该项目将从上面研究莫尔斯指数界的有效性，根据黎曼三流形内（分支）恒定平均曲率（CMC）表面的属和面积。在适当的变分界下，给出了分支CMC超曲面的气泡紧性结果。这些结果将为CMC表面的莫尔斯指数，几何和拓扑之间的关系提供见解-由于此类物体存在理论的最新进展，这是一个非常感兴趣的主题。特别是这个项目将有助于确定在给定空间中允许使用什么类型的CMC/最小表面，并对这些表面进行分类。

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.