Holomorphic extensions of maps from the boundary of Kähler manifolds
The classical theorem on global holomorphic extensions (of Bochner) asserts that if \(f\) is a smooth function on \(\partial \Omega \), \(\Omega \) be a bounded domain in \({\mathbb C}^n\) with smooth connected boundary, which satisfies the tangential Cauchy-Riemann equation \(\overline{\partial}_bf=0\) on \(\partial \Omega\), then \(f\) can be extended from \(\partial \Omega \) to \(\overline{\Omega}\) so that \(f\) is holomorphic in \( \Omega \). If \(M\) is a complex manifold, then even the local extension problem for a hypersurface is not necessarily solvable in general. Let \(D\) be a domain in a Kaehler manifold \(M\) with smooth compact boundary \(\partial \Omega \). Then \(D\) is called hyperconvex if \(D\) has a smooth defining function with nonnegative trace of the Levi-form on \(\partial D\). According to Siu, a Kaehler manifold \(N\) has strongly seminegative curvature if \(R\) denotes the Riemannian curvature tensor, then \(\sum_{\alpha,\beta,\gamma,\delta}R_{\alpha \overline{\beta}\gamma \overline{\delta}}(A^\alpha \overline{B^\beta}-C^\alpha \overline{D^\beta})(\overline{A^\delta}B^\gamma -\overline{C^\delta}D^\gamma)\) is nonnegative for arbitrary complex numbers \(A^\alpha,B^\alpha,C^\alpha \) and \(D^\alpha \) when \(A^\alpha \overline{B^\beta}-C^\alpha \overline{D^\beta}\neq 0\) for at least one pair of indices \((\alpha,\beta)\). Theorem 1.1. Let \(M(\dim_{{\mathbb C}}M=m>1)\) be a Kaehler manifold and \(D\) a hyperconvex domain in \(M\) with compact closure and non-empty smooth boundary \(\partial D\). Suppose there is a plurisubharmonic function \(\eta \in C^2(\overline{D})\) so that the complex Hessian \(H(\eta)\) has at least two positive eigenvalues at some point in \(D\). Suppose \(N\) is a complete Kaehler manifold with strongly seminegative curvature. Let \(f:\overline{D}\rightarrow N\) be a smooth map satisfying the tangential Cauchy-Riemann equation \(\overline{\partial}_bf=0\) on \(\partial D\). Then there exists a unique holomorphic extension of \(f\). Corollary 1.2. Let \(M(\dim_{{\mathbb C}}M=m>1)\) be a Kaehler manifold and \(D\) a hyperconvex domain in \(M\) with compact closure and non-empty smooth boundary \(\partial D\). Let \(N\) be a complete Kaehler manifold with strongly seminegative curvature. Suppose there are two holomorphic functions \(\phi_1,\phi_2\in C^\infty (\overline{D})\) such that \(\partial \phi_1,\partial \phi_2\) are linearly independent at some point in \(D\). If \(f:\overline{D}\rightarrow N\) is a smooth map satisfying the tangential Cauchy-Riemann equation \(\overline{\partial}_bf=0\) on \(\partial D\), then there exists a unique holomorphic extension of \(f\).
scientific article; zbMATH DE number 1140491
scientific article