In this note, under certain regularity and transversality conditions we obtain a sharp lower bound of the first eigenvalue for certain self-adjoint elliptic operators on manifolds with long necks. This result is used in the gluing construction of 3-dimensional Seiberg-Witten moduli spaces along T^2. See [C1], [C2].
CHEN, Weimin (1997) "A lower bound of the first eigenvalue of certain self-adjoint elliptic operators on manifolds containing long necks," Turkish Journal of Mathematics: Vol. 21: No. 1, Article 9. Available at: https://journals.tubitak.gov.tr/math/vol21/iss1/9