(Pluri)potential compactifications. (English) Zbl 1473.32012
The author constructs biholomorphically invariant compactifications for some kind of complex manifolds including bounded domains in \(\mathbb{C}^n\) by using pluricomplex Green functions, which is called the pluripotential compactification.
In order to do so, the author first shows the existence of a norming volume form V on a connected complex manifold \(M\), and then observes that all negative plurisubharmonic functions on \(M\) are in \(L^1(M,V)\) and shows that the set \(PSH^-_1(M)\) of such functions with the norm not exceeding \(1\) is compact.
Assuming \(M\) is a locally uniformly pluri-Greenian complex manifold, constructing a mapping \(\Phi_V: M\to \operatorname{PSH}^-_1(M)\), \(\Phi_V(w) =\frac{g_M(\cdot,w)}{\|g_M(\cdot,w)\|_V}\), where \(g_M(\cdot,w)\) is the pluricomplex Green function on \(M\), the author gets an imbedding of \(M\) into a compact set in \(L^1(M,V)\). The pluripotential compactification of \(M\) is then defined as the closure of \(\Phi_V(M)\) in \(L^1(M,V)\). This approach also gives the Martin compactification in the real case.
In the last section, the author computes some pluripotential compactifications for the ball, the polydisk and smooth strongly convex domains in \(\mathbb{C}^n\).
In order to do so, the author first shows the existence of a norming volume form V on a connected complex manifold \(M\), and then observes that all negative plurisubharmonic functions on \(M\) are in \(L^1(M,V)\) and shows that the set \(PSH^-_1(M)\) of such functions with the norm not exceeding \(1\) is compact.
Assuming \(M\) is a locally uniformly pluri-Greenian complex manifold, constructing a mapping \(\Phi_V: M\to \operatorname{PSH}^-_1(M)\), \(\Phi_V(w) =\frac{g_M(\cdot,w)}{\|g_M(\cdot,w)\|_V}\), where \(g_M(\cdot,w)\) is the pluricomplex Green function on \(M\), the author gets an imbedding of \(M\) into a compact set in \(L^1(M,V)\). The pluripotential compactification of \(M\) is then defined as the closure of \(\Phi_V(M)\) in \(L^1(M,V)\). This approach also gives the Martin compactification in the real case.
In the last section, the author computes some pluripotential compactifications for the ball, the polydisk and smooth strongly convex domains in \(\mathbb{C}^n\).
Reviewer: Xiangyu Zhou (Beijing)
MSC:
| 32U05 | Plurisubharmonic functions and generalizations |
| 31C35 | Martin boundary theory |
| 32U15 | General pluripotential theory |
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