A Hodge theory for some non-compact manifolds.
Let \(\Omega^q(M)\) be the set of smooth differential forms on a differentiable manifold \(M\) with \(q\leq\dim(M)\). The theorem of de Rham states that if \(H^q(M)\equiv \{\omega\in \Omega^q(M);\;d\,\omega=0\}/\{d\,\omega;\;\omega \in \Omega^{q-1}(M)\}\), then \(H^q(M)\) can be identified with the \(q\)\,th Čech cohomology space of \(M\) over \(\mathbb R\). The de Rham equivalence class of a closed form \(\omega\in \Omega^q(M)\) is \([\omega]=\omega+d\,\Omega^{q-1}\). Let \(\lambda_M\) denote the associated Riemannian measure for a compact Riemannian manifold \(M\). One interpretation of Hodge's refinement to de Rham's theorem asserts the existence of a unique \(\eta\in [\omega]\) for each closed \(\omega\in\Omega^q(M)\) with minimal \(L^2(\lambda_M; \Lambda^q(TM^*))\)-norm. The proof of Hodge's theory leads, via elementary variational calculus, first to the introduction of the formal adjoint \(\delta=d^{*\lambda_M}\) of \(d\) in \(L^2(\lambda_M; \Lambda^q(TM^*))\), then to the Hodge operator \(H = d\,\delta+\delta\, d\), and finally to showing that each \(\omega\) admits a Hodge decomposition \(\omega=\eta+d\,\xi_1+\delta\,\xi_2\), for some forms \(\xi_1\), \(\xi_2\), and a unique \(H\)-harmonic form \(\eta\). One way to prove Hodge's decomposition theorem is to study the Cauchy initial value problem for the parabolic equation \((*)\,\partial_t\omega_t+H\omega_t=0\) on \((0,\infty)\times M\) with \(\omega_0=\omega\). One can check the semigroup of operators \(Q^0_t = e^{-tH}\) which solve \((*)\) (in the sense that \(\omega_t=Q^0_t\omega\)) is just as bounded as the scalar heat flow semigroup of operators \(P^0_t = e^{t\Delta}\). In particular, each \(Q^0_t\) determines a unique extension as a self-adjoint contraction \(\overline{Q^0_t}\) on \(L^2(\lambda_M; \Lambda^q(TM^*))\). The semigroup \(\overline{Q^0_t}\) is compact. If \(-\overline H\) is the generator of the semigroup \(\{\overline{Q^0_t}:\;t>0\}\), then \(\ker (\overline H)=\ker (I-\overline{Q^0_1})\). Because \(\overline{Q^0_t}\) is compact \(\dim \ker (\overline H)\) is finite, and if \(\eta\) is the orthogonal projection of \(\omega\) onto \(\ker (\overline H)\), then \(\omega-\eta=\xi-Q^0_1\xi\) for some form \(\xi\), and \(\xi-Q^0_1\xi=d\,\xi_1+\delta\,\xi_2\), where \(\xi_1\) and \(\xi_2\) are special forms. This shows the Hodge decomposition of \(\omega\). In this paper, the authors consider the same problem when they drop the assumption that \(M\) is compact. They replace the Riemannian measure \(\lambda_M\) by the measure \(m^U (dx) = e^{-U(x)}\lambda_M(dx)\), where \(U: M\to [0,\infty)\) is chosen to grow so fast at infinity that \(m^U\) is finite. Also, the original Hodge operator \(H\) is replaced by the weighted Hodge operator \(H^U\equiv\delta^Ud+d\,\delta^U\), where \(\delta^U\) is the formal adjoint of \(d\) computed relative to \(m^U\). The authors develop the Hodge decomposition theorem corresponding to the weighted Hodge operator \(H^U\) and they apply it to obtain a Hodge theory for the de Rham cohomology of a complete, connected, finite dimensional Riemannian manifold \(M\).
scientific article; zbMATH DE number 1782639