Vanishing theorems on complete manifolds with weighted Poincaré inequality and applications (Q2894133)

In this paper, the authors prove two vanishing theorems for harmonic maps and \(L^2\) harmonic one-forms on complete noncompact Riemannian manifolds under certain geometric assumptions.NEWLINENEWLINELet \(M\) be an \(n\)-dimensional complete noncompact Riemannian manifold. The authors show that if the Ricci curvature of \(M\) has the lower bound \({\text{Ric}}_M(x) \geq -(n-1)\tau(x)\), where \(\tau(x)\) satisfies the Poincaré inequality \(\delta \int_M \tau \varphi^2 \leq \int_M |\nabla \varphi|^2\) for \(\varphi \in C_0^\infty(M)\) and \(\delta > (n-2)^2/n\), then there are no nontrivial \(L^2\) harmonic one-forms on \(M\). On the other hand, the authors prove that if \(M\) is as above with \(\delta >n-2\) and \(N\) is a manifold of nonpositive sectional curvature, then any harmonic map \(f: M \to N\) is constant, provided that its energy density satisfies \(\int_{B(R)} |df|^{2\beta} = {\text o}(R)\). There two results are generalizations of known results.