×

The Ahlfors Laplacian on a Riemannian manifold. (English) Zbl 0746.53012

Constantin Carathéodory: an international tribute. Vol. II, 1020-1048 (1991).
[For the entire collection see Zbl 0728.00004.]
On a Riemannian manifold \((M,g)\) the Ahlfors Laplacian \(L\) is defined as the second order differential operator \(S^*S\) where \(S^*\) is the formal adjoint of \(S:=L_ \cdot g-(2/n)(\text{div}\cdot)g\), i.e. \(SX=L_ Xg-(2/n)(\text{div} X)\cdot g\) for vector fields \(X\) on \(M\). Here \(L_ X\) is the standard Lie derivative. The authors then show that \(L\) is elliptic and that the kernel of \(L\) consists exactly of the conformal Killing vector fields on \(M\). As an application the problem of finding quasi-conformal deformations of a given transformation is studied.

MSC:

53A30 Conformal differential geometry (MSC2010)
58J05 Elliptic equations on manifolds, general theory
35J25 Boundary value problems for second-order elliptic equations

Biographic References:

Carathéodory, Constantin

Citations:

Zbl 0728.00004
PDFBibTeX XMLCite