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Mathematical Physics and Harmonic Analysis Seminar

Date: March 9, 2018

Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: Peter Kuchment, Texas A&M


Title: On Liouville-Riemann-Roch theorems on co-compact abelian coverings

Abstract: A generalization by Gromov and Shubin [2-3] 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 Laplace-Beltrami (or more general elliptic) equation on a non-compact manifold. The solution of a 1975 Yau's conjecture [6] by Colding and Minicozzi [1] 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 [4-5]. 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 rather non-trivial. The talk will present such a result obtained recently in a joint work with Minh Kha (former A&M PhD student, currently postdoc at U. Arizona).
[1] Colding, T. H., Minicozzi, W. P.: Harmonic functions on manifolds, Ann. of Math. 146 (1997), 725–747.
[2] M. Gromov and M. A. Shubin, The Riemann-Roch theorem for elliptic operators, I. M. Gel'fand Seminar, 1993, pp. 211--241.
[3] --" -- , The Riemann-Roch theorem for elliptic operators and solvability of elliptic equations with additional conditions on compact subsets, Invent. Math. 117 (1994), no. 1, 165--180.
[4] P. Kuchment and Y. Pinchover, Integral representations and Liouville theorems for solutions of periodic elliptic equations, J. Funct. Anal. 181 (2001), no. 2, 402--446.
[5] --"-- , Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds, Trans. Amer. Math. Soc. 359 (2007), no. 12, 5777--5815.
[6] Yau, S. T.: Harmonic functions on complete Riemannian manifolds, Commun. Pure Appl. Math. 28 (1975), 201–228.