Foliations of Bubblesheet Singularities: A Spectral Geometric Approach

This seminar talk from the "Spectral Geometry in the clouds" series features Jean Lagacé from King's College London exploring foliations of bubblesheet singularities through a spectral geometric approach. Delve into the fundamental concept of differential geometry where important features of a space should remain invariant under coordinate changes, while examining how spaces with special structure may benefit from preferred coordinate systems. Learn about distinguished parameterisations found by identifying foliations of space by submanifolds determined by its geometry, with particular focus on constant mean curvature (CMC) hypersurfaces and their applications in parameterising ends of asymptotically flat manifolds, defining center of mass for isolated gravitating systems, and their role in proving the stability of Minkowski spacetime. Discover why the codimension n ≥ 2 setting presents more complications, where Parallel Mean Curvature (PMC) faces generic geometric obstructions. Explore the innovative "Quasi-Parallel Mean Curvature" (QPMC) condition introduced by the speaker, and understand how bubblesheet singularities can be foliated by QPMC embedded spheres. The presentation covers this curvature condition built on eigenspaces of the connection Laplacian, the construction of the foliation, examples demonstrating the necessity of such a condition, and potential applications to Mean Curvature Flow.