## African Diaspora Journal of Mathematics

- Afr. Diaspora J. Math. (N.S.)
- Volume 20, Number 2 (2017), 45-68.

### About the Degenerate Spectrum of the Tension Field for Mappings into a Symmetric Riemannian Manifold

#### Abstract

Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds, where $(N,h)$ is symmetric, $v\in W^{1,2}((M,g),(N,h))$, and $\tau $ is the tension field for mappings from $(M,g)$ into $(N,h)$. We consider the nonlinear eigenvalue problem $\tau (u)-\lambda \exp _{u}^{-1}v=0$, for $u$ $\in W^{1,2}(M,N)$ such that $u_{\left\vert \partial M\right. }=v_{\left\vert \partial M\right.}$, and $\lambda \in \mathbb{R}$. We prove, under some assumptions, that the set of all $\lambda $, such that there exists a solution $(u,\lambda )$ of this problem and a non trivial Jacobi field $V$ along $u$, is contained in $\mathbb{R}_{+}$, is countable, and has no accumulation point in $\mathbb{R}$. This result generalizes a well known one about the spectrum of the Laplace-Beltrami operator $\Delta $ for functions from $(M,g)$ into $\mathbb{R}$.

#### Article information

**Source**

Afr. Diaspora J. Math. (N.S.), Volume 20, Number 2 (2017), 45-68.

**Dates**

First available in Project Euclid: 17 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.adjm/1494986435

**Mathematical Reviews number (MathSciNet)**

MR3652656

**Zentralblatt MATH identifier**

1378.58013

**Subjects**

Primary: 58C 58E 58J 49R50 35J 35D

**Keywords**

Riemannian manifold tension field Jacobi field eigenvalue eigenmapping Laplace-Beltrami operator bifurcation symmetry

#### Citation

Kourouma, Moussa. About the Degenerate Spectrum of the Tension Field for Mappings into a Symmetric Riemannian Manifold. Afr. Diaspora J. Math. (N.S.) 20 (2017), no. 2, 45--68. https://projecteuclid.org/euclid.adjm/1494986435