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On boundary regularity of analytic discs. (English) Zbl 0985.32009
The authors study the boundary behavior of analytic discs near the zero set of a non-negative plurisubharmonic function on a complex manifold.
The main result can be stated as follows: let $$\Omega$$ be a complex manifold, $$\rho$$ a plurisubharmonic function in $$\Omega$$, and $$f:\Delta \to\Omega$$ a holomorphic map of the unit disc $$\Delta\subset\mathbb{C}$$ into $$\Omega$$ such that $$\rho\circ f\geq 0$$ and $$\rho\circ f(\zeta)\to 0$$ as $$\zeta$$ tends to an open $$\text{arc} \gamma \subset\partial \Delta$$. Assume that for some point $$a\in \partial\Delta$$, the cluster set $$C(f,a)$$ contains a point $$p$$ such that $$\rho$$ is strictly plurisubharmonic in a neighborhood of $$p$$. Then $$f$$ extends to a $${1\over 2}$$-Hölder map in a neighborhood of $$a$$ in $$\Delta \cup\gamma$$. Moreover, if $$\rho$$ takes only non-negative values and, for some $$\theta\in [{1\over 2},1]$$, the function $$\rho^\theta$$ is plurisubharmonic in a neighborhood of $$p$$, then $$f$$ is actually $${1\over 2\theta}$$-Hölder in a neighborhood of $$a$$ in $$\Delta\cup \gamma$$.
The authors describe a number of applications, especially to the study of the boundary behavior of analytic discs near totally real submanifolds of $$\mathbb{C}^n$$.
A key argument of the proof is an estimate of the Kobayashi metric which is interesting in itself, and has to be compared with former results by N. Sibony [Ann. Math. Stud. 100, 357-372 (1981; Zbl 0476.32033)] and F. Berteloot [Banach Cent. Publ. 31, 91-98 (1995; Zbl 0831.32012)], among others.

##### MSC:
 32H40 Boundary regularity of mappings in several complex variables 32F45 Invariant metrics and pseudodistances in several complex variables 32U05 Plurisubharmonic functions and generalizations
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