Riemannian geometry of the space of volume preserving immersions
The authors consider the manifold \(\text{Imm}(M,N)\) of immersions from a compact manifold \(M\) to a Riemannian manifold \(N\) of bounded geometry, and its subset \(\text{Imm}_{\mu}(M,N)\) of those immersions with the property that the volume-form of the pull-back metric is equal to \(\mu\). The interest in the space \(\text{Imm}_{\mu}(M,N)\) can be motivated from applications in the study of biological membranes, where the volume density of the surface remains constant during certain biological deformations.NEWLINENEWLINEThe authors first prove that the set of non-minimal elements of \(\text{Imm}_{\mu}(M,N)\) form a splitting submanifold. On this submanifold they consider the Levi-Civita connection for various natural Sobolev metrics and write down the geodesic equation. In many cases they can show the local well-posedness of the equation.NEWLINENEWLINEThis question is a natural generalization of the corresponding well-posedness question for the group of volume-preserving diffeomorphisms, which was solved in connection with studies of the motion of an incompressible fluid in a fundamental paper by \textit{D. G. Ebin} and \textit{J. Marsden} [Ann. Math. (2) 92, 102--163 (1970; Zbl 0211.57401)]. The methods developed by Ebin and Marsden have inspired the methods employed in the present paper.
scientific article; zbMATH DE number 6655141
scientific article