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Positivity Conditions in Several Complex Variables

Mernik, Luka
http://orcid.org/0000-0002-7607-1839
http://rave.ohiolink.edu/etdc/view?acc_num=osu1586522855649564

Abstract Details

2020, Doctor of Philosophy, Ohio State University, Mathematics.
My research is concerned with positivity conditions that arise naturally when studying important questions, such as regularity of the dbar-Neumann and the Bergman operators, in Several Complex Variables. The positivity conditions that appear in my work are convexity, pseudoconvexity, plurisubharmonicity, convexifiability, admitting a holomorphic support function, D'Angelo (finite) type, and Diederich-Fornaess index. These positivity conditions are intertwined to greatly controls the holomorphic function theory on the domain. More specifically, my research involved understanding the difference between D'Angelo singular and regular 1-type and under which conditions the two notions coincide. The second set of positivity conditions appearing in my work are convexifiability and admitting a holomorphic support functions. At the beggining my attention was focused on domains in C^2 of low D'Angelo 1-type, as these domains are more tractable and easier to study. Later, the generalizations to higher D'Angelo 1-type as well as domains in C^n were obtained. Finally, I studied the necessary and sufficient conditions for pseudoconvex domains to admit a local plurisubharmonic defining function on the boundary. For some domains in C^2 I showed that the necessary conditions are also sufficient and give an algorithm for producing a local plurisubharmonic defining function provided one exists.
Jeffery McNeal, Dr. (Advisor)
92 p.
Mathematics
DAngelo 1-type; holomorphic support surfaces; plurisubharmonic defining functions

Recommended Citations

Citations

  • Mernik, L. (2020). Positivity Conditions in Several Complex Variables [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1586522855649564

    APA Style (7th edition)

  • Mernik, Luka. Positivity Conditions in Several Complex Variables. 2020. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1586522855649564.

    MLA Style (8th edition)

  • Mernik, Luka. "Positivity Conditions in Several Complex Variables." Doctoral dissertation, Ohio State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1586522855649564

    Chicago Manual of Style (17th edition)

osu1586522855649564
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