Liouville-Riemann-Roch theorems on abelian coverings

Abstract

Chapter 1. Introduction ix Chapter 2. Preliminaries 2.1. Periodic elliptic operators on abelian coverings 2.2. Floquet transform 2.3. Bloch and Fermi varieties 2.4. Floquet-Bloch functions and solutions 2.5. Liouville theorem on abelian coverings 2.6. Some properties of spaces V p N (A) 2.7. Explicit formulas for dimensions of spaces V ∞ N (A) 2.8. The Nadirashvili-Gromov-Shubin version of the Riemann-Roch theorem for elliptic operators on noncompact manifolds Chapter 3. The main results 3.1. Non-empty Fermi surface 3.2. Empty Fermi surface Chapter 4. Proofs of the main results 4.1. Proof of Theorem 3.6 4.2. Proof of Theorem 3.10 4.3. Proof of Theorem 3.13 4.4. Proof of Theorem 3.23 Chapter 5. Specific examples of Liouville-Riemann-Roch theorems 5.1. Self-adjoint operators 5.2. Non-self-adjoint second order elliptic operators Chapter 6. Auxiliary statements and proofs of technical lemmas 6.1. Properties of Floquet functions on abelian coverings 6.2. Basic properties of the family {A(k)} k∈C d 6.3. Properties of Floquet transforms on abelian coverings 6.4. A Schauder type estimate 6.5. A variant of Dedekind's lemma 6.6. Proofs of some other technical statements Chapter 7. Final Remarks and Acknowledgments 7.1. Remarks and conclusions 7.2. Acknowledgements Bibliography v