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Poincaré-Type Metrics and Their ∂‾ Estimates on Pseudoconvex Domains
Edgren, Neal G

2014, Doctor of Philosophy, Ohio State University, Mathematics.
The Bergman kernel and its associated metric are fundamental to the study of complex analysis. However, it is very difficult to compute these objects on domains whose boundaries are not described in terms of computationally simple functions, or on asymmetric domains. The Bergman metric on the unit ball in Cn is given by the complex Hessian of the function -(n+1)log(1 - |z|2). As r(z) = |z|2 - 1 is a defining function for the unit ball, the Bergman metric is thus described by a very simple potential function, and after some work, we can generalize this idea to bounded pseudoconvex domains with at least C2 boundary. We recall that on such domains Ω, defining functions can be modified to obtain a bounded plurisubharmonic exhaustion function r on Ω, and define a Poincaré-type metric to be the complex Hessian of -log(-r).

The main results we present on the geometry of Poincaré-type metrics are contained in Chapter 2. It turns out that they are all complete, much like the Bergman metric on domains with smooth boundary. It would be quite interesting to study the geodesics of these metrics, but we have not included that in this dissertation. On strongly pseudoconvex domains with C boundary, we show that these metrics are quasi-isometric under biholomorphisms. It is the author's hope that this fact will generalize to the weakly pseudoconvex case, although the problem is more difficult there. This is a partial generalization of the elementary fact that all biholomorphisms are isometries of the Bergman metric, regardless of conditions on the domain.

The remaining three chapters deal with ∂‾ theory and estimates related to Poincaré-type metrics. We first explore the Hodge theory of Poincaré-type metrics, concluding that they have trivial L2 Dolbeault cohomology except on (p,q)-forms where p + q = n. We then examine the twisted Cauchy-Riemann complex and prove an estimate on solutions to a ∂‾ problem appropriate for this setting. We finish the dissertation by discussing an application of this estimate to the old problem of constructing functions on the disk which do not extend past any point of its boundary.
Jeffery McNeal (Advisor)
Kenneth Koenig (Committee Member)
Fangyang Zheng (Committee Member)
73 p.

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Edgren, N. (2014). Poincaré-Type Metrics and Their ∂‾ Estimates on Pseudoconvex Domains. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Edgren, Neal. "Poincaré-Type Metrics and Their ∂‾ Estimates on Pseudoconvex Domains." Electronic Thesis or Dissertation. Ohio State University, 2014. OhioLINK Electronic Theses and Dissertations Center. 23 Nov 2017.

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Edgren, Neal "Poincaré-Type Metrics and Their ∂‾ Estimates on Pseudoconvex Domains." Electronic Thesis or Dissertation. Ohio State University, 2014. https://etd.ohiolink.edu/

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