Defining Functions and Cores of Unbounded Domains
Harz
Tobias
Harz, Tobias
aut
2015-07-14
2015-08-07
en
We show that every strictly pseudoconvex domain Ω with smooth boundary in a complex manifold <i><font size="+1">M</font></i> admits a global defining function, i.e., a smooth plurisubharmonic function <i>φ</i>: <i>U</i> → ℝ defined on an open neighbourhood <i>U</i> ⊂ <i><font size="+1">M</font></i> of <span style="text-decoration: overline";>Ω</span> such that Ω = {<i>φ</i> < 0}, <i>dφ</i> ≠ 0 on <i>b</i>Ω and <i>φ</i> is strictly plurisubharmonic near <i>b</i>Ω. We then introduce the notion of the core <i>c</i>Ω of an arbitrary domain Ω ⊂ <i><font size="+1">M</font></i>} as the set of all points where every smooth and bounded from above plurisubharmonic function on Ω fails to be strictly plurisubharmonic. If Ω is not relatively compact in <i><font size="+1">M</font></i>, then in general <i>c</i>(Ω) is nonempty, even in the case when <i><font size="+1">M</font></i> is Stein. It is shown that every strictly pseudoconvex domain Ω ⊂ <i><font size="+1">M</font></i> with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of <i>c</i>(Ω). We then investigate properties of the core. Among other results, we prove 1-pseudoconcavity of the core, we show that in general the core does not possess an analytic structure, and we investigate Liouville type properties of the core.
urn:nbn:de:hbz:468-20150807-101856-5
2015-08-04T13:22:39.312Z
2015-09-01T13:09:26.085Z
published
Diss
fbc/mathematik/diss2015/harz