Biharmonic Immersion in Cartan Hadamard manifold

TL;DR

This paper proves that under certain curvature conditions, all proper biharmonic isometric immersions into a Cartan-Hadamard manifold are actually harmonic, extending understanding of biharmonic maps in curved spaces.

Contribution

It establishes a new rigidity result showing proper biharmonic immersions become harmonic under specific Ricci curvature bounds in Cartan-Hadamard manifolds.

Findings

Proper biharmonic immersions are harmonic under given curvature conditions.

The result applies to manifolds with Ricci curvature bounded below by a function G(r).

The theorem generalizes previous results by relaxing curvature constraints.

Abstract

If $(N^{m+p},h)$ is a Cartan-Hadamard manifold such that $Ric(h)\geq -G(r_{N}(x))$ where $G(0)\geq 1, G^{'}\geq 0$ and $G^{-1/2}\not\in L^{1}(+\infty)$ then every proper biharmonic isometric immersion $\phi : M^{m}\rightarrow(N^{m+p},h)$ is a harmonic map.