By Jean-Michel Bismut
This booklet provides the analytic foundations to the idea of the hypoelliptic Laplacian. The hypoelliptic Laplacian, a second-order operator performing on the cotangent package of a compact manifold, is meant to interpolate among the classical Laplacian and the geodesic circulate. Jean-Michel Bismut and Gilles Lebeau identify the fundamental useful analytic homes of this operator, that's additionally studied from the viewpoint of neighborhood index concept and analytic torsion. The ebook exhibits that the hypoelliptic Laplacian presents a geometrical model of the Fokker-Planck equations. The authors provide the right kind sensible analytic environment with a view to learn this operator and enhance a pseudodifferential calculus, which supplies estimates at the hypoelliptic Laplacian's resolvent. whilst the deformation parameter has a tendency to 0, the hypoelliptic Laplacian converges to the normal Hodge Laplacian of the bottom by means of a collapsing argument within which the fibers of the cotangent package deal cave in to some degree. For the neighborhood index conception, small time asymptotics for the supertrace of the linked warmth kernel are acquired. The Ray-Singer analytic torsion of the hypoelliptic Laplacian in addition to the linked Ray-Singer metrics at the determinant of the cohomology are studied in an equivariant atmosphere, leading to a key comparability formulation among the elliptic and hypoelliptic analytic torsions.
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The Hypoelliptic Laplacian and Ray-Singer Metrics
This publication provides the analytic foundations to the idea of the hypoelliptic Laplacian. The hypoelliptic Laplacian, a second-order operator performing on the cotangent package of a compact manifold, is meant to interpolate among the classical Laplacian and the geodesic circulation. Jean-Michel Bismut and Gilles Lebeau determine the elemental useful analytic houses of this operator, that is additionally studied from the point of view of neighborhood index concept and analytic torsion.
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Since A2φ,Hc −1 acts on S · (T ∗ X, π ∗ F )∗ , we get the identity in End (S · (T ∗ X, π ∗ F )∗ ), ∗ T ∗X 1 = dTHcX , dφ,Hc A2φ,Hc −1 . 16), we deduce that S · (T ∗ X, π ∗ F )∗ , dTHcX is exact. 13). 8) and the fact that A2φ,Hc is hΩ (T X,π F ) selfadjoint, if a ∈ S · (T ∗ X, π ∗ F )0 , a ∈ S · (T ∗ X, π ∗ F ), we get 0 a, A2n φ,Hc a hS · (T ∗ X,π∗ F ) = 0. 17) that S (T X, π F )0 and S (T X, π F )∗ are mutually hS (T X,π F ) · ∗ ∗ orthogonal. Since hS (T X,π F ) is nondegenerate, we get the second part of ∗ T ∗X our theorem.
First we use the canonical isomorphism Λ· (T X) Λn−· (T ∗ X) ⊗Λn (T X) . 6), we get the isomorphism Λ· (T ∗ T ∗ X) π ∗ Λ· (T ∗ X) ⊗Λn−· (T ∗ X) ⊗Λn (T X) . 7) will now be generated by e , . . , en . Set λ0 = ei iebi . 8) Then we conjugate the operator obtained by the first transformation by e−λ0 . Let T 0 , p be given by T 0 , p = T fαH , ei , p f α ei . 9) A final conjugation is done by conjugating the operator we obtained before M by exp T 0 , p . Starting from CM φ,H−ω H , Dφ,H−ω H , we obtain the operators M M Cφ,Hc −ωH , Dφ,Hc −ωH .
Let k : X → T ∗ X be the embedding of X as the zero section of T ∗ X. Let H · (T ∗ X, π ∗ F ) (resp. H c,· (T ∗ X, π ∗ F )) be the cohomology of T ∗ X (resp. with compact support) with coefficients in π ∗ F . Then, classically, H · (T ∗ X, π ∗ F ) = H · (X, F ) , H c,· (T ∗ X, π ∗ F ) = H ·−n (X, F ⊗ o(T X)) . 1) The first isomorphism comes from the maps k ∗ : Ω· (T ∗ X, π ∗ F ) → Ω· (X, F ) HODGE THEORY, THE HYPOELLIPTIC LAPLACIAN AND ITS HEAT KERNEL 45 and π ∗ : Ω· (X, F ) → Ω· (T ∗ X, π ∗ F ). The second is the Thom isomorphism.