Polyharmonic hypersurfaces into pseudo-Riemannian space forms

TL;DR

This paper investigates polyharmonic hypersurfaces in pseudo-Riemannian space forms, deriving conditions for their existence and classifying proper r-harmonic surfaces, including new examples and uniqueness results under specific curvature assumptions.

Contribution

It provides new conditions for r-harmonicity of hypersurfaces, constructs novel examples, and classifies proper r-harmonic isoparametric surfaces in Lorentz space forms.

Findings

Existence of new families of proper r-harmonic hypersurfaces with diagonalizable shape operator.

Results indicating these examples may be unique under certain principal curvature conditions.

Complete classification of proper r-harmonic isoparametric surfaces in 3D Lorentz space forms.

Abstract

In this paper we shall assume that the ambient manifold is a pseudo-Riemannian space form $N^{m+1}_t(c)$ of dimension $m+1$ and index $t$ ($m\geq2$ and $1 \leq t\leq m$). We shall study hypersurfaces $M^{m}_{t'}$ which are polyharmonic of order $r$ (briefly, $r$-harmonic), where $r\geq 3$ and either $t'=t$ or $t'=t-1$. Let $A$ denote the shape operator of $M^{m}_{t'}$. Under the assumptions that $M^{m}_{t'}$ is CMC and $Tr A^2$ is a constant, we shall obtain the general condition which determines that $M^{m}_{t'}$ is $r$-harmonic. As a first application, we shall deduce the existence of several new families of proper $r$-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 $r$-harmonic hypersurfaces ($r \geq 3$). Finally, we shall obtain the complete classification of proper $r$-harmonic isoparametric pseudo-Riemannian surfaces into a $3$-dimensional Lorentz space form.