Embedded constant mean curvature hypersurfaces

主要研究方向：几何与拓扑学。次要研究领域：数学分析表明，在任何3维以上的紧致黎曼流形中，给定任意非零数a，存在一个标量平均曲率等于a的闭常平均曲率超曲面，该超曲面可以有一个奇异的余维数集7，并且是拟嵌入的。也就是说，它是一个浸入的图像，而不是一个嵌入，只有在周围的点上，结构是由两个嵌入的圆盘的并集给出的，这两个圆盘彼此相接，切线相交。这个项目的范围是研究黎曼度规上的假设，在这个假设下，有可能找到一个常数平均曲率超曲面，它是额外嵌入（而不是准嵌入）远离7维的奇异集。对于具有正里奇曲率的度量，以及在低维中，对于凹凸不平的度量（光滑极小超曲面的第二种变化具有平凡零的度量的一般集合），这可能是正确的。

Reserach Areas: Primary research area: Geometry and Topology.Secondary research area: Mathematical AnalysisRecent has shown that in any compact Riemannian manifold of dimension 3 or higher, and given any non-zero number a, there exists a closed constant mean curvature hypersurface with scalar mean curvature equal to a. The hypersurface may have a singular set of codimension 7 and is quasi-embedded, namely it is the image of an immersion and fails to be an embedding only at points around which the structure is given by the union of exactly two embedded disks that lie on one side of each other and intersect tangentially. The scope of this project is to investigate assumptions on the Riemannian metric under which it is possible to find a constant mean curvature hypersurface as above that is additionally embedded (rather than quasi-embedded) away from a singular set of codimension 7. This may be true for metrics with positive Ricci curvature and, in low dimensions, for bumpy metrics (a generic set of metrics for which the second variation of smooth minimal hypersurfaces has trivial nullity).