A Liouville-Riemann-Roch theorem on abelian coverings
When
Speaker
Abstract
A generalization by Nadirashvili, and then Gromov and Shubin of the classical Riemann-Roch theorem describes the index of an elliptic operator on a compact manifold with a divisor of prescribed zeros and allowed singularities. On the other hand, Liouville type theorems count the number of solutions of a given polynomial growth of the LaplaceBeltrami (or more general elliptic) equation on a non-compact manifold. The solution of a 1975 Yau’s conjecture by Colding and Minicozzi implies in particular, that such dimensions are finite for Laplace-Beltrami equation on a nilpotent co-compact covering. In the case of an abelian covering, much more complete Liouville theorems (including exact formulas for dimensions) have been obtained by Kuchment and Pinchover. One wonders whether such results have a combined generalization that would allow for a divisor that ”includes the infinity.” Surprisingly, combining the two types of results turns out being non-trivial. The talk will present such a result obtained in a joint work with Peter Kuchment.