Nonexistence of stable F-stationary maps of a functional related to pullback metrics

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M^{m}$\end{document}Mm be a compact convex hypersurface in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R^{m+1}$\end{document}Rm+1. In this paper, we prove that if the principal curvatures \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda_{i}$\end{document}λi of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M^{m}$\end{document}Mm satisfy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\lambda_{1}\leq \cdots \leq \lambda_{m}$\end{document}0<λ1≤⋯≤λm and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$3\lambda_{m}<\sum_{j=1}^{m-1}\lambda_{j}$\end{document}3λm<∑j=1m−1λj, then there exists no nonconstant stable F-stationary map between M and a compact Riemannian manifold when (6) or (7) holds.


Introduction
Let u : (M m , g) → (N n , h) be a smooth map between Riemannian manifolds (M m , g) and (N n , h). Recently, Kawai and Nakauchi [] introduced a functional related to the pullback metric u * h as follows: (see [-]), where u * h is the symmetric -tensor defined by for any vector fields X, Y on M and u * h is given by with respect to a local orthonormal frame (e  , . . . , e m ) on (M, g). The map u is stationary for if it is a critical point of (u) with respect to any compact supported variation of u, and u is stable if the second variation for the functional (u) is nonnegative. They showed the nonexistence of a nonconstant stable stationary map for , either from S m (m ≥ ) to any manifold, or from any compact Riemannian manifold to S n (n ≥ ). In this paper, for a smooth function F : [, ∞) → [, ∞) such that F() =  and F (t) >  on t ∈ (, ∞), we are concerned with the instability of F-stationary maps which is the generalization of a stationary map for introduced by Asserda in []. In [], they obtained some monotonicity formulas for F-stationary maps via the coarea formula and the comparison theorem. Also, by using monotonicity formulas, they got some Liouville type results for these maps. The authors in [] obtained the first and second variation formula for F-stationary maps. By using the second variation formula, they proved that every stable F-stationary map from S m () to any Riemannian manifold is constant if or every F-stationary map from any compact Riemannian manifold N n to S m is constant if In this paper, we obtain the results on the instability of F-stationary maps which are from or into the compact convex hypersurfaces in the Euclidean space.

Preliminaries
Let ∇ and N ∇ always denote the Levi-Civita connections of M and N , respectively. Let ∇ be the induced connection on u - TN defined by ∇ X W = N ∇ du(X) W , where X is a tangent vector of M and W is a section of u - TN. We choose a local orthonormal frame field {e i } on M. We define the F-tension field τ F (u) of u by where σ u = j h(du(·), du(e j ))du(e j ), which was defined in []. We need the following second variation formula for F-stationary maps (cf. []). Let u : (M, g) → (N, h) be an F-stationary map. Let u s,t : M → N (-ε < s, t < ε) be a compactly supported two-parameter variation such that u , = u, and set V = ∂ ∂t u s,t | s,t= , W = ∂ ∂s u s,t | s,t= . Then where ·, · is the inner product on T * M ⊗ u - TN and R N is the curvature tensor of N . We put An F-stationary map u is called stable if I(V , V ) ≥  for any compactly supported vector field V along u.

F-stationary maps from compact convex hypersurfaces
In this section, we obtain the following result.
Theorem . Let M ⊂ R m+ be a compact convex hypersurface. Assume that the principal curvatures λ i of M m satisfy  < λ  ≤ · · · ≤ λ m and λ m < m- i= λ i . Then every nonconstant F-stationary map from M to any compact Riemannian manifold N is unstable if there exists a constant c F = inf{c ≥ |F (t)/t c is nonincreasing} such that Proof In order to prove the instability of u : M m → N , we need to consider some special variational vector fields along u. To do this, we choose an orthonormal field {e i , e m+ }, i = , . . . , m, of R m+ such that {e i } are tangent to M m ⊂ R m+ , e m+ is normal to M m and ∇ e i e j | P = . Meanwhile, we take a fixed orthonormal basis E A , A = , . . . , m + , of R m+ and set where ·, · denotes the canonical Euclidean inner product. Then du(V A ) ∈ (u - TN) and where B ij denotes the components of the second fundamental form of It follows from the Weitzenböck formula that where X is any smooth vector field on M m . With respect to the variational vector field du(V A ) along u, it follows from () and () that For any fixed point P ∈ M, choose {e i } such that ∇ e i e j | P = . We have

Substituting () and () into (), we have
In the following, we shall estimate each term in (). Because trace is independent of the choice of orthonormal basis, we can take pointwisely Then it follows from () and () that and and

Using (), (),() and (), we have
and If F (t) = F (t), then () leads to the following inequality: If u is nonconstant and () or () holds, we have and u is unstable.
Corollary . Let u : S m → N be a nonconstant F-stationary map and m > . If c F < m  - or u * h  < m -, then u is unstable.

F-stationary maps into compact convex hypersurfaces
In this section, we obtain the following result.
where ·, · denotes the canonical Euclidean inner product. We shall consider the second where {e  , . . . , e n } is the local orthonormal frame of N n .
Firstly, we compute the first term of () The second term of () The third term of () The fourth term of () The fifth term of ()  If F (t) = F (t), then () leads to the following inequality: If there exists a constant c F such that F (t) t c F is nonincreasing, it follows that F (t)t ≤ c F F (t) on t ∈ (, ∞), thus () implies