Curvature flows and spectral estimates.

曲率流是一类几何激励方程，以热方程为模型。最近，研究人员开发了新的方法来研究这些方程的解的正则性，并将其应用于不同的问题，即估计依赖于薛定谔算子的较小本征值的量。该项目建立在这项研究的早期成功，并将产生一个新的理解的行为特征值，建立尖锐的估计谱量，特别是在流形曲率边界，并找到最佳条件下，非紧的解决方案曲率流是稳定的。

Curvature flows are a class of geometrically motivated equations, modelled on the heat equation. Recently, researchers have developed new methods for studying the regularity of solutions to these equations, and applied them to a different problem, that of estimating quantities depending on the smaller eigenvalues of a Schroedinger operator. This project builds on the early success of this research and will produce a new understanding of the behaviour of eigenvalues, establish sharp estimates for spectral quantities, particularly on manifolds with curvature bounds, and find optimal conditions under which non-compact solutions to curvature flows are stable.