IPDF Progress Seminar
Speaker: Dr. Utsav Dewan, IIT Bombay
Host: Mayukh Mukherjee
Title: Boundary behaviour of eigenfunctions and superharmonic functions on harmonic manifolds of purely exponential volume growth
Time, day and date: 3:00:00 PM – 4:00:00 PM, Wednesday, September 16
Venue: Ramanujan Hall
Abstract: On X, a non-positively curved harmonic manifold of purely exponential volume growth, of dimension at least 3, we study certain quantitative aspects of the boundary behaviour of eigenfunctions and superharmonic functions. We first focus on complex-valued eigenfunctions with eigenvalues lying outside the L^2-spectrum of the Laplace-Beltrami operator and obtain the almost everywhere existence of weighted non-tangential limits, sharp Hausdorff dimension and Hausdorff measure estimates of the boundary exceptional sets for radial limits. Then in the second part, we shift our attention to non-tangential and tangential boundary behaviour of positive superharmonic functions. Most of our results are new even for the homogeneous setting of rank one Riemannian symmetric spaces of non-compact type and Damek-Ricci spaces. Our arguments are based on potential theory adapted to the intrinsic Gromov hyperbolic geometry of X.