Moebius geometry of submanifolds in ?ⁿ

Abstract:

In this paper we define a Moebius invariant metric and a Moebius invariant second fundamental form for submanifolds in ?ⁿ and show that in case of a hypersurface with n≥ 4 they determine the hypersurface up to Moebius transformations. Using these Moebius invariants we calculate the first variation of the moebius volume functional. We show that any minimal surface in ?ⁿ is also Moebius minimal and that the image in ?ⁿ of any minimal surface in ℝⁿ unter the inverse of a stereographic projection is also Moebius minimal. Finally we use the relations between Moebius invariants to classify all surfaces in ?³ with vanishing Moebius form.