Tohoku Mathematical Journal
2023
December
SECOND SERIES VOL. 75, NO. 4

Tohoku Math. J.
75 (2023), 509-526

Title WEIGHTED $L^2$ HARMONIC 1-FORMS AND THE TOPOLOGY AT INFINITY OF COMPLETE NONCOMPACT WEIGHTED MANIFOLDS

Author Keomkyo Seo and Gabjin Yun

(Received September 8, 2021, revised May 13, 2022)
Abstract. In this paper, we prove that a complete noncompact weighted manifold supporting the weighted Sobolev inequality has at least linear weighted volume growth. We also obtain vanishing results and finiteness theorems for the weighted $L^2_f$ $f$-harmonic 1-forms on a complete noncompact weighted manifold supporting the weighted Sobolev inequality.

Mathematics Subject Classification. Primary 53C21; Secondary 53C20, 58J05.
Key words and phrases. Weighted manifolds, weighted harmonic forms, topology at infinity, weighted Sobolev inequality.

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