Classification of f-biharmonic submanifolds in Lorentz space forms

Li Du

doi:10.1515/math-2021-0084

Abstract

Abstract In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

Highlights

The theory of harmonic maps between manifolds is a very active and rich subject as an extension of important concepts of geodesics, minimal surfaces, and has been studied extensively by many mathematicians and physicists, we refer to [1,2,3,4,5] for details
On the one side is the analytic aspect from the point of view of partial differential equation (PDE): biharmonic maps are solutions of a fourth order strongly elliptic semilinear PDE, we refer to [7,8] for a review
The differential geometric aspect has driven attention to biharmonic submanifolds as those submanifolds whose isometric immersions are biharmonic maps. The study of such submanifolds was initiated by Jiang as applications of biharmonic maps, independently by Chen in his study on finite type submanifolds, and had become a very dynamic research subject in modern differential geometry

Summary

Introduction

The theory of harmonic maps between manifolds is a very active and rich subject as an extension of important concepts of geodesics, minimal surfaces, and has been studied extensively by many mathematicians and physicists, we refer to [1,2,3,4,5] for details. In [18], Ferrández and Lucas classified η-biharmonic hypersurfaces in high-dimensional Lorentz spaces n+1 under the assumption that the minimal polynomial of the shape operator is at most of degree two. We attempt to study f-biharmonic submanifolds (instead of hypersurfaces) in Lorentz space forms, where f is a function. When f is a function, we classify completely pseudo-umbilical submanifolds

Notions and formulas

The shape operator of submanifolds

The f-biharmonic submanifolds in pseudo-Riemannian manifolds

Main results and their proofs

Classification of f-biharmonic submanifolds