Biharmonic and biconservative hypersurfaces in space forms

Abstract

We present some general properties of biharmonic and biconservative submanifolds and then survey recent results on such hypersurfaces in space forms. We also propose an alternative version for a well-known result of Nomizu and Smyth for hypersurfaces by replacing the CMC hypothesis with the more general condition of biconservativity.

Biharmonic hypersurfaces with constant scalar curvature in space forms

“…(c.f. [10]) A hypersurface φ : M n → N n+1 (c) is biconservative if the mean curvature H and the shape operator A on M n satisfy…”

“…In view of Lemma 3.3 and the Codazzi equation (2.14), we see at once that ω 3 22 = ω 2 33 = 0. In addition, it follows from (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) in [12] that ω 1 23 = ω 1 32 = 0. Therefore, we obtain that R(e 2 , e 3 )e 2 , e 3 = ω 1 22 ω 1 33 .…”

“…Many important research results have been carried out to investigate the existence and classification problems of biharmonic submanifolds in some model spaces. More details and an overview about the historic development concerning biharmonic submanifolds can be found in [10,24] and the references therein.…”

“…Since the class of biconservative hypersurfaces is a generalization of minimal (or CMC) hypersurfaces, it is very interesting to investigate whether Chern's conjecture holds for biconservative hypersurfaces instead of using minimality (or CMC) for hypersurfaces in space forms. In their recent survey [10], Fetcu and Oniciuc proposed a challenging problem on biconservative hypersurfaces as following:…”

Triharmonic CMC hypersurfaces in space forms with at most 3 distinct principal curvatures

“…In 1986, G. Jiang [17] studied the first and second variational formulae of E 2 . We refer the readers to the very recent book by Ou and Chen [33] and a survey article by Fetcu and Oniciuc [14] for the abundant progress on biharmonic maps. For k ≥ 3, the first and the second variational formulae of E k were obtained by S. Wang [34] in 1989 and by Maeta [20] in 2012 respectively.…”