Biharmonic homogeneous hypersurfaces in compact symmetric spaces. (English) Zbl 1336.58009
Biharmonic submanifolds generalize minimal ones in the sense that they solve a fourth-order partial differential equation that is satisfied by all minimal submanifolds. A proper biharmonic submanifold is one that is not minimal. The existence of proper biharmonic submanifolds is a difficult question in general, and can be answered best in the presence of special geometric structures. In this paper, the authors are looking for biharmonic hypersurfaces in irreducible symmetric spaces of compact type, which is a very natural question. Assuming some structure called “commutative Hermann actions of cohomogeneity one” (requiring another structure, a compact symmetric triad), they study hypersurfaces given by orbits of the Hermann actions. The structures mentioned have been classified, and they appear in 18 (families of) examples. The authors group the examples into three classes. In three cases, there is a unique regular orbit which is proper biharmonic. In seven more cases, there are exactly two regular orbits which are proper biharmonic. In the remaining eight cases, all regular biharmonic orbits are already minimal. With this classification, the authors have found all biharmonic hypersurfaces in compact symmetric spaces which are orbits of commutative Hermann actions.
Reviewer: Andreas Gastel (Essen)
Keywords:
symmetric space; symmetric triad; Hermann action; harmonic map; biharmonic map; minimal submanifoldsReferences:
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