On boundary regularity of analytic discs.

*(English)*Zbl 0985.32009The 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.

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.

Reviewer: V.Thilliez (Villaneuve d’Ascq)

##### MSC:

32H40 | Boundary regularity of mappings in several complex variables |

32F45 | Invariant metrics and pseudodistances in several complex variables |

32U05 | Plurisubharmonic functions and generalizations |