2024/05/02 by Ognjen Milatovic, Milatovic, Ognjen
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2405.00926
openalex publication_date 2024/05/02 · openalex created_date 2024/05/05 · openalex updated_date 2026/07/28
Let M be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let E be a Hermitian vector bundle over M equipped with a metric covariant derivative ∇. We consider the operator HX,V=∇†∇+∇X+ V, where ∇† is the formal adjoint of ∇ with respect to the inner product in the space of square-integrable sections of E, X is a smooth (real) vector field on M, and V is a fiberwise self-adjoint, smooth section of the endomorphism bundle \textrmEnd E. We give a sufficient condition for the triviality of the L2-kernel of HX,V. As a corollary, putting X≡ 0 and working in the setting of a Clifford module equipped with a Clifford connection ∇, we obtain the triviality of the L2-kernel of D2, where D is the Dirac operator corresponding to ∇. In particular, when E=ΛℂkT^*M and D2 is the Hodge--deRham Laplacian on (complex-valued) k-forms, we recover some recent vanishing results for L2-harmonic (complex-valued) k-forms.