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Let Ω c C" be a strictly pseudoconvex domain, γ an admissible weight, and K γ (z, C) the reproducing (or y-Bergman) kernel for L 2 //(Ω,y), the space of square integrable functions, with respect to the measure γdμ, which are holomorphic in Ω (dμ is the Lebesgue measure in R 2n ), cf. e.g. Z. . Consider the complex tensor field: and the corresponding real tangent (0,2)-tensor field g γ given by:where χ(Ω) is the C°°(Ω)-module of all real tangent vector fields on Ω. Under suitable conditions (cf. section 2) g γ is a Kahlerian metric on Ω, hence ω γ = -idδ logK γ (z,z) is a symplectic structure (the Kahler 2-form of g γ ). One of the problems we take up in the present paper may be stated as follows. Let F : Ω -• Ω be a symplectomorphism of (Ω,ω y ) in itself, smooth up to the boundary. Does F : 3Ω -> 3Ω preserve the contact structure of the boundary?Our interest may be motivated as follows. If F : Ω -> Ω is a biholomorphism then, by a celebrated result of C. Fefferman (cf. Theorem 1 in [4], p. 2) F is smooth up to the boundary, hence F : 3Ω -> 3Ω is a CR diffeomorphism, and in particular a contact transformation. Also biholomorphisms are known to be isometries of the Bergman metric g\ (cf. e.g. [7], p. 370) hence symplectomorphisms of (Ω,ωi). On the other hand, one may weaken the assumption on F by requesting only that F be a C 00 diffeomorphism and F*a>\ =ω\.Then, by a result of A. Koranyi and H. M. Reimann [11], if F is smooth up to the boundary then F : 3Ω -• dΩ is a contact transformation.The main ingredient in the proof of A. Koranyi and H. M. Reimann's result is the fact that, when γ = 1, a certain negative power of the Bergman kernel (p(z) = K\(z,z)~ι^n +i>} ) is a defining function of Ω (allowing one to relate the symplectic structure of Ω to the contact structure of its boundary). In turn, this is a consequence of C. Fefferman's asymptotic expansion of K\(z, ζ) (cf. Theorem 2 in
Let Ω c C" be a strictly pseudoconvex domain, γ an admissible weight, and K γ (z, C) the reproducing (or y-Bergman) kernel for L 2 //(Ω,y), the space of square integrable functions, with respect to the measure γdμ, which are holomorphic in Ω (dμ is the Lebesgue measure in R 2n ), cf. e.g. Z. . Consider the complex tensor field: and the corresponding real tangent (0,2)-tensor field g γ given by:where χ(Ω) is the C°°(Ω)-module of all real tangent vector fields on Ω. Under suitable conditions (cf. section 2) g γ is a Kahlerian metric on Ω, hence ω γ = -idδ logK γ (z,z) is a symplectic structure (the Kahler 2-form of g γ ). One of the problems we take up in the present paper may be stated as follows. Let F : Ω -• Ω be a symplectomorphism of (Ω,ω y ) in itself, smooth up to the boundary. Does F : 3Ω -> 3Ω preserve the contact structure of the boundary?Our interest may be motivated as follows. If F : Ω -> Ω is a biholomorphism then, by a celebrated result of C. Fefferman (cf. Theorem 1 in [4], p. 2) F is smooth up to the boundary, hence F : 3Ω -> 3Ω is a CR diffeomorphism, and in particular a contact transformation. Also biholomorphisms are known to be isometries of the Bergman metric g\ (cf. e.g. [7], p. 370) hence symplectomorphisms of (Ω,ωi). On the other hand, one may weaken the assumption on F by requesting only that F be a C 00 diffeomorphism and F*a>\ =ω\.Then, by a result of A. Koranyi and H. M. Reimann [11], if F is smooth up to the boundary then F : 3Ω -• dΩ is a contact transformation.The main ingredient in the proof of A. Koranyi and H. M. Reimann's result is the fact that, when γ = 1, a certain negative power of the Bergman kernel (p(z) = K\(z,z)~ι^n +i>} ) is a defining function of Ω (allowing one to relate the symplectic structure of Ω to the contact structure of its boundary). In turn, this is a consequence of C. Fefferman's asymptotic expansion of K\(z, ζ) (cf. Theorem 2 in
<p>In this study, we characterize $ LP $-Kenmotsu manifolds admitting $ * $-Ricci–Yamabe solitons ($ * $-RYSs) and gradient $ * $-Ricci–Yamabe solitons (gradient $ * $-RYSs). It is shown that an $ LP $-Kenmotsu manifold of dimension $ n $ admitting a $ * $-Ricci–Yamabe soliton obeys Poisson's equation. We also determine the necessary and sufficient conditions under which the Laplace equation is satisfied by $ LP $-Kenmotsu manifolds. Finally, by using a non-trivial example of an $ LP $-Kenmotsu manifold, we verify some results of our paper.</p>
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