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    • PI3K Akt mTOR Compound Library receptor The scattering opera

    2024-03-07

    The scattering operators associated to are defined as follows: for any and , , , there exists a unique solution to the following equation such that Then the scattering operator is defined by Here is an elliptic pseudodifferential operator of order , which is conformally covariant on the boundary. Moreover, can be extended meromorphically to , where K is the same as above. The poles at , or are of first order. For simplicity we define the renormalised scattering operators by If g is approximate Einstein, i.e. then for are GJMS operators. In particular, is the Yamabe operator. See [12] for more details. If the conformal infinity is of positive Yamabe type, the scatting operators are studied by Guillarmou and Qing [16]. They showed that
    A new interpretation of the renormalised scattering operator is given by Case–Chang in [2] as a generalised Dirichlet-to-Neumann map on naturally associated smooth metric measure spaces. The authors exhibited some PI3K Akt mTOR Compound Library receptor identities for on the boundary in terms of energies in the compact space . This connects the positivity of renormalised scattering operators to the positivity of curvature terms for the compactified metric . In particular, while a Poincaré–Einstein metric has positive conformal infinity, Qing [31] showed that there exists a suitable compactification such that has positive scalar metric, which implies that has positive spectrum for from Case–Chang's energy identity. In this paper, we consider a complex manifold X of complex dimension , with strictly pseudoconvex boundary . The complex structure on X naturally induces a CR-structure on M, where , and is defined by . Let ρ be a smooth boundary defining function such that on . Assume the function is plurisubharmonic. We consider the Kähler metric g induced by Kähler form The metric g is asymptotically complex hyperbolic in the sense that the holomorphic sectional curvature has limit −4 when approaching to the boundary. More explicitly, g takes the form near M, where and have Taylor series in ρ at . In particular, gives the contact form on M and induces a pseudo-Hermitian metric on H. Moreover, the conformal class of the boundary pseudo-Hermitian structure is independent of choice of boundary defining function. A standard example is the complex hyperbolic space : it is the ball equipped with Kähler metric induced from the boundary defining function . The spectrum and resolvent of the Laplacian operator were studied by Epstein–Melrose–Mendoza [6] and Vash–Wunsch [33]. Actually in both papers, the authors studied more general ACH manifolds. Similar like the real case, the spectrum of consists of two disjoint parts: the absolute continuous spectrum and the pure point spectrum . More explicitly, The resolvent is a bounded operator on for , and has finite meromorphic extension to . The smoothness of ρ implies that the metric has even asymptotic expansion in the sense of [6]. If g is Kähler–Einstein, or equivalently if ρ is a solution to the complex Monge–Ampére equation, then generally speaking ρ is not smooth up to boundary and it has logarithmic terms in the Taylor expansion at boundary with respect to smooth coordinates. See [23] for more details. The analysis of spectrum and resolvent is still valid except that the mapping property of changes a bit. The scattering operators associated to is defined in a similar way as the real case. For any and , , and , consider the equation There exists a unique solution u such that Then the scattering operator is defined by which is a pseudo-differential operator of Heisenberg class of order , conformally covariant and having meromorphic extension to . See [6], [17], [29] for more details on the scattering theory and [1], [5], [10], [30], [32] for the Heisenberg calculus. For simplicity, we also define the renormalised scattering operators as If g is approximate Einstein (see Definition 1), then for are CR-GJMS operators of order 2k. In particular, is the CR-Yamabe operator. See [18]. This gives a different approach to construct the CR-invariant powers of sub-Laplacian studied by Grover–Graham [13] as well as the Q-curvature by Fefferman–Hirachi [8].