Laplacian on Riemannian manifolds

In this lecture, we investigate the function theory on Riemannian manifolds. It is divided into three parts. The first part is devoted to the local properties of harmonic functions and eigenfunctions on Euclidean spaces and Riemannian manifolds. This involves detailed analysis of the local behaviors of solutions to elliptic PDEs. Our main interests lie on the study of nodal sets of harmonic functions and eigenfunctions on manifolds [4-7]. The second part of the lecture is about the global properties of harmonic functions on Riemannian manifolds, e.g. the asymptotical behaviors of harmonic functions on noncompact manifolds such as Liouville theorem of harmonic functions by Cheng-Yau gradient estimate. We will introduce the solution of Yau's conjecture on polynomial growth harmonic functions on manifolds with nonnegative Ricci curvature by Colding-Minicozzi [2-3]. The third part consists of the heat kernel estimates, the wave kernel estimates and their applications. The heat kernel is widely used in the study of spectral theory on Riemannian manifolds, traced back to the classical works by Li-Yau and Grigor'yan. The classical wave kernel method by Cheeger-Gromov-Taylor [1] and its geometric applications will be explained in detail.

References

1. Cheeger-Gromov-Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom. (1982).
2. Colding-Minicozzi, Harmonic functions with polynomial growth, J. Diff. Geom. (1997).
3. Colding-Minicozzi, Harmonic functions on manifolds, Ann. of Math. (2) (1997).
4. Colding-Minicozzi, Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys. (2011).
5. Han-Lin, Nodal sets of solutions of elliptic differential equations, preprint.
6. Hardt-Hoffmann-Ostenhop-Hoffmann-Ostenhop-Nadirashvili, Critical sets of solutions to elliptic equations, J. Diff. Geom. (1999).
7. Hardt-Simon, Nodal sets for solutions of elliptic equations, J. Diff. Geom. (1989).

Date and time info
Thursday 13.15 - 14.45

Keywords
Nodal set theory, harmonic functions, polynomial growth harmonic functions, heat kernel, wave kernel

Language
English