A Hodge theory for some non-compact manifolds.
type: Item
instance of: scholarly article
title: A Hodge theory for some non-compact manifolds.
zbMATH Open document ID: 1073.58001
author: Zulfikar M. Ahmed
author: Daniel W. Stroock
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: 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 \(\ker (\overline H)=\ker (I-\overline{Q^0_1})\). B…
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 58A14
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 1782639
zbMATH Keywords: De Rham theorem
zbMATH Keywords: Hodge theory
zbMATH Keywords: diffusion theory
zbMATH Keywords: Hodge decomposition theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214342150
OpenAlex ID: W1490999340
Wikidata QID: Q115194385
DOI: 10.4310/JDG/1214342150
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description: scientific article; zbMATH DE number 1782639