几何分析系列报告之25【James Alexander McCoy】

Abstract: Analogous to the parabolic case considered by Smoczyk in 2005, we consider linear hyperbolic curvature flow of convex hypersurfaces.  Solutions may be written down explicitly as Fourier series of spherical harmonics, that, given appropriate conditions on the initial data, represent convex hypersurfaces for all time.  In the unconstrained case, solutions expand indefinitely, becoming asymptotically round.  For constrained flows, solutions converge to round spheres without rescaling.  Finally, we consider a hyperbolic approach to the Yau problem of when one can flow one convex hypersurface to another, in this setting by a nonhomogeneous linear curvature flow.  In so doing we can answer the Christoffel problem of convex geometry if we have an initial hypersurface whose sum of principal radii of curvature is sufficiently close to a target principal radii of curvature.

Speaker Introduction：James McCoy is a Professor at the University of Newcastle, Australia, and is a 'geometric analyst' whose main research interests are questions related to curvature driven heat-type flows of hypersurfaces. He received his Ph.D. from Monash University in 2002 under the supervision of Klaus Ecker.  After completing post doctoral position at the Australian National University with Ben Andrews and Neil Trudinger, he spent thirteen years at the University of Wollongong. In 2018 he moved to the University of Newcastle, and in July that year he became Head of Mathematics Discipline at Newcastle. He relinquished this role in 2020 to take up a secondment to the Vice Chancellor's Academic Excellence team. James McCoy is the Deputy Director of the Australian Mathematical Sciences Institute (AMSI), and Deputy Editor of the Journal of the Australian Mathematical Society.