Polyharmonic hypersurfaces into pseudo-Riemannian space forms

TL;DR

The paper extends the theory of polyharmonic (r-harmonic) maps to hypersurfaces of pseudo-Riemannian space forms, deriving a sharp criterion for when a non-minimal CMC hypersurface with constant $\,\mathrm{Tr} olimits A^2\,$ is $r$-harmonic: either $\mathrm{Tr} olimits A^2=0$ or $\varepsilon(\mathrm{Tr} olimits A^2)^2 - m c \mathrm{Tr} olimits A^2 - (r-2) m^2 c \alpha^2 = 0$, with $\varepsilon = \langle \eta,\eta\rangle$ and $\alpha$ the mean curvature. The authors develop explicit constructions of new proper $r$-harmonic hypersurfaces, including diagonalizable cases like pseudo-spheres and generalized pseudo-Clifford tori, as well as non-diagonalizable, isoparametric Lorentz surfaces in 3D Lorentz space forms. They also provide a complete classification of proper $r$-harmonic isoparametric pseudo-Riemannian surfaces in 3-dimensional Lorentz space forms, and present concrete examples such as complex circles in ${\mathbb H}^3_1$ and $B$-scrolls. Their results highlight a richer set of $r$-harmonic phenomena in the pseudo-Riemannian setting compared to the Riemannian case, including rigidity and nonexistence in certain curvature regimes and new isoparametric configurations.

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.

Theorems & Definitions (42)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 2.1
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Remark 3.4
• Theorem 3.5
• Remark 3.6