Branding V, Montaldo S, Oniciuc C, Ratto A
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy.
Ann Mat Pura Appl. 2023;202(2):877-899. doi: 10.1007/s10231-022-01263-1. Epub 2022 Sep 16.
In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form of dimension and index ( and ). We shall study hypersurfaces which are polyharmonic of order (briefly, -harmonic), where and either or . Let denote the shape operator of . Under the assumptions that is CMC and is a constant, we shall obtain the general condition which determines that is -harmonic. As a first application, we shall deduce the existence of several new families of proper -harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper -harmonic hypersurfaces ( ). Finally, we shall obtain the complete classification of proper -harmonic isoparametric pseudo-Riemannian surfaces into a three-dimensional Lorentz space form.
在本文中，我们假设环境流形是一个维度为(n)且指标为(q)（(n\geq3)且(1\leq q\leq n - 1)）的伪黎曼空间形式。我们将研究(m)阶多重调和（简称为(m -)调和）的超曲面，其中(m\geq1)且要么(m\geq2)要么(m = 1)。设(S)表示(M)的形状算子。在(M)是常平均曲率（CMC）且(H)是常数的假设下，我们将得到确定(M)是(m -)调和的一般条件。作为第一个应用，我们将推导出几个具有可对角化形状算子的新的恰当(m -)调和超曲面族的存在性，并且我们还将在主曲率满足某些假设的情况下得到一些结果，表明我们的例子是唯一可能的。接下来，我们专注于研究形状算子不可对角化的等参超曲面，并且在这种情况下我们也将证明一些恰当(m -)调和超曲面（(m\geq2)）的新例子的存在性。最后，我们将得到恰当(m -)调和等参伪黎曼曲面到三维洛伦兹空间形式的完全分类。
