Morse-Novikov cohomology of Riemannian manifolds with boundary

Abstract

We consider θ to be a real closed one form belongïng to the first de Rham cohomology group HdR1 (M). When the exterior derivative is perturbed with θ, the Morse-Novikov coboundary differential operator dθ is defined as dθ ω = dω + θ ∧ ω for each ω ∈ Ωk (M), and the resulting is the Morse-Novikov cohomology group Hθk (M). The principal purpose of this paper is to reveal more geometric and topological insight of Hθk (M) in the non-vanishing boundary case. To this end, we define a certain boundary value problem and showing the elipticity of it implies new varying decompositions of the space of differential forms Ωk (M) in terms of the operators dθ and its adjoint δθ . Consequently, we define two different candidates of Hθk (M) and show the classes have unique representative (finite dimensional) θ-harmonic k-fields with certain prescribed boundary conditions, so this proves the finiteness of the dimension of Morse-Novikov cohomology when the boundary is present. This construction means that the elipticity implicates to give the relevant version of the Hodge-Morrey-Friedrichs decomposition theorem in terms of the operators dθ , δθ and the corresponding Morse-Novikov-Laplacian Δθ . Eventually, these results contribute effectively to imply new verified properties of Morse-Novikov cohomology group of M with boundary and generalize (in a different argument) some results.