Harmonic functions on manifolds

William Minicozzi, Johns Hopkins University


The classical Liouville theorem states that positive, and thus bounded, harmonic functions in Euclidean space must be constant. It then follows that a harmonic function which grows polynomially on $\RR^n$ must be a harmonic polynomial. S.T. Yau generalized the Liouville theorem to manifolds with nonnegative Ricci curvature and conjectured furthermore that the spaces of harmonic functions of polynomial growth on these manifolds are finite dimensional.

In this talk, we will describe joint work with Tobias Colding including the proof of this conjecture, sharp polynomial bounds on the dimension, and some generalizations. In addition, the relationship with certain eigenvalue problems on compact manifolds and applications to minimal submanifolds will be discussed.