Polyharmonic hypersurfaces into space forms

TL;DR

This paper investigates polyharmonic hypersurfaces in space forms, proving minimality under certain conditions in non-positive curvature spaces and characterizing and constructing new examples in spheres, linking them to Eells-Sampson energy.

Contribution

It provides new geometric characterizations and existence results for proper r-harmonic hypersurfaces in space forms, especially spheres, and connects them to Eells-Sampson r-energy functional.

Findings

Minimality of r-harmonic hypersurfaces in non-positive curvature spaces under constant mean curvature and shape operator norm.

Characterization and bounds for r-harmonic hypersurfaces with constant mean curvature in spheres.

Existence of new proper r-harmonic isoparametric hypersurfaces in spheres.

Abstract

In this paper we shall assume that the ambient manifold is a space form $N^{m+1}(c)$ and we shall consider polyharmonic hypersurfaces of order $r$ (briefly, $r$-harmonic), where $r\geq 3$ is an integer. For this class of hypersurfaces we shall prove that, if $c \leq 0$, then any $r$-harmonic hypersurface must be minimal provided that the mean curvature function and the squared norm of the shape operator are constant. When the ambient space is $\mathbb{S}^{m+1}$, we shall obtain the geometric condition which characterizes the $r$-harmonic hypersurfaces with constant mean curvature and constant squared norm of the shape operator, and we shall establish the bounds for these two constants. In particular, we shall prove the existence of several new examples of proper $r$-harmonic isoparametric hypersurfaces in $\mathbb{S}^{m+1}$ for suitable values of $m$ and $r$. Finally, we shall show that all these $r$-harmonic hypersurfaces are also $ES-r$-harmonic, i.e., critical points of the Eells-Sampson $r$-energy functional.