Bounds for the first Dirichlet eigenvalue of domains in Kaehler manifolds
The authors study the first eigenvalue \(\mu_ 1(M)\) of the Dirichlet problem \[ \Delta f=\mu f\text{ on }M\text{ and }f=0\text{ on }\partial M, \] where \(M\) is a connected compact Kähler manifold of real dimension \(2n\) with smooth boundary \(\partial M\). They find some geometric conditions for \(M\) and \(\partial M\) to get a lower bound \(\mu_ 1(B^ \mathbb{C} _ r)\) for \(\mu_ 1(M)\). Here \(B^ \mathbb{C} _ r\) (resp. \(S^ \mathbb{C} _ r\)) denotes the geodesic ball (resp. sphere) of radius \(r\) in \(\mathbb{C} P^ n(\lambda)\). If \(\partial M\) is connected they prove that the equality \(\mu_ 1(M)=\mu_ 1(B^ \mathbb{C} _ r)\) is equivalent to some equality for \(r\). Under the same conditions it is proved there is a holomorphic isometry \(i: M\rightarrow B^ \mathbb{C} _ r\), such that \(i(\partial M)=S^ \mathbb{C} _ r=\partial B^ \mathbb{C} _ r\).
scientific article; zbMATH DE number 12474