Embedded constant mean curvature hypersurfaces

研究领域：主要研究领域：几何和拓扑学。次要研究领域：数学分析最近的研究表明，在任何三维或更高维的紧致黎曼流形中，给定任何非零数a，存在一个封闭的常数平均曲率超曲面，其标量平均曲率等于a。超曲面可以有一个余维为7的奇异集合，并且是准嵌入的，即它是浸入的图像，并且只有在其周围的结构由恰好两个嵌入圆盘的并集给出的点处才不是嵌入，这两个嵌入圆盘位于彼此的一侧并且相切相交。该项目的范围是研究黎曼度量的假设，在该假设下，可以找到一个如上所述的常平均曲率超曲面，该超曲面另外嵌入（而不是准嵌入）远离余维7的奇异集。这对于具有正Ricci曲率的度量和在低维中对于凹凸度量（光滑极小超曲面的第二变分具有平凡零性的度量的一般集合）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).