Skip to content

Advertisement

  • Research
  • Open Access

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

Journal of Inequalities and Applications20172017:214

https://doi.org/10.1186/s13660-017-1483-z

  • Received: 25 December 2016
  • Accepted: 27 August 2017
  • Published:

Abstract

Let \(M^{m}\) be a compact convex hypersurface in \(R^{m+1}\). In this paper, we prove that if the principal curvatures \(\lambda_{i}\) of \(M^{m}\) satisfy \(0<\lambda_{1}\leq \cdots \leq \lambda_{m}\) and \(3\lambda_{m}<\sum_{j=1}^{m-1}\lambda_{j}\), then there exists no nonconstant stable F-stationary map between M and a compact Riemannian manifold when (6) or (7) holds.

Keywords

  • F-stationary map
  • compact convex hypersurfaces

MSC

  • 58E20
  • 53C21

1 Introduction

Let \(u:(M^{m},g)\rightarrow (N^{n},h)\) be a smooth map between Riemannian manifolds \((M^{m},g)\) and \((N^{n},h)\). Recently, Kawai and Nakauchi [1] introduced a functional related to the pullback metric \(u^{*}h\) as follows:
$$\begin{aligned} \Phi (u)=\frac{1}{4} \int_{M} \bigl\Vert u^{*}h \bigr\Vert ^{2}\,dv_{g}, \end{aligned}$$
(1)
(see [24]), where \(u^{*}h\) is the symmetric 2-tensor defined by
$$\begin{aligned} \bigl(u^{*}h \bigr) (X,Y)=h \bigl(du(X),du(Y) \bigr) \end{aligned}$$
for any vector fields X, Y on M and \(\Vert u^{*}h \Vert \) is given by
$$\begin{aligned} \bigl\Vert u^{*}h \bigr\Vert ^{2}=\sum _{i,j=1}^{m} \bigl[h \bigl(du(e_{i}),du(e_{j}) \bigr) \bigr]^{2}, \end{aligned}$$
with respect to a local orthonormal frame \((e_{1},\ldots ,e_{m})\) on \((M,g)\). The map u is stationary for Φ if it is a critical point of \(\Phi (u)\) with respect to any compact supported variation of u, and u is stable if the second variation for the functional \(\Phi (u)\) is nonnegative. They showed the nonexistence of a nonconstant stable stationary map for Φ, either from \(S^{m}\) (\(m\geq 5\)) to any manifold, or from any compact Riemannian manifold to \(S^{n}\) (\(n\geq 5\)). In this paper, for a smooth function \(F:[0,\infty )\rightarrow [0,\infty )\) such that \(F(0)=0\) and \(F'(t)>0\) on \(t\in (0,\infty )\), we are concerned with the instability of F-stationary maps which is the generalization of a stationary map for Φ introduced by Asserda in [4]. In [4], 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 [5] 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 \(S^{m}(1)\) to any Riemannian manifold is constant if
$$\begin{aligned} \int_{S^{m}} \bigl\Vert u^{*}h \bigr\Vert ^{2} \biggl\{ F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\Vert u^{*}h \bigr\Vert ^{2}+(4-m)F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggr\} \,dv_{g}< 0, \end{aligned}$$
(2)
or every F-stationary map from any compact Riemannian manifold \(N^{n}\) to \(S^{m}\) is constant if
$$\begin{aligned} \int_{N^{n}} \bigl\Vert u^{*}h \bigr\Vert ^{2} \biggl\{ F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\Vert u^{*}h \bigr\Vert ^{2}+(4-m)F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggr\} \,dv_{g}< 0. \end{aligned}$$
(3)
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.

2 Preliminaries

Let \(F:[0,\infty )\rightarrow [0,\infty )\) be a \(C^{2}\)-function such that \(F(0)=0\) and \(F'(t)>0\) on \(t\in (0,\infty )\). For a smooth map \(u:(M,g)\rightarrow (N,h)\) between compact Riemannian manifolds \((M,g)\) and \((N,h)\) with Riemannian metrics g and h, respectively, following Ara [6] for an F-harmonic map (also see [710]), Asserda in [4] gave the following definition.

Definition 2.1

We call u an F-stationary map for \(\Phi_{F}\) if
$$ \frac{d}{dt}\Phi_{F}(u_{t})|_{t=0}=0 $$
for any compactly supported variation \(u_{t}:M\rightarrow N\) with \(u_{0}=u\), where
$$ \Phi_{F}(u)= \int_{M^{m}}F \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\,dv_{g}. $$
Let and N always denote the Levi-Civita connections of M and N, respectively. Let ̃ be the induced connection on \(u^{-1}TN\) defined by \(\widetilde{\nabla } _{X}W=^{N}\nabla_{du(X)}W\), where X is a tangent vector of M and W is a section of \(u^{-1}TN\). We choose a local orthonormal frame field \(\{e_{i}\}\) on M. We define the F-tension field \(\tau_{\Phi_{F}}(u)\) of u by
$$\begin{aligned} \tau_{\Phi_{F}}(u) =&-\delta \biggl(F' \biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sigma_{u} \biggr) \\ =&F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\operatorname{div}_{g}( \sigma_{u})+\sigma_{u} \biggl(\operatorname{grad} \biggl(F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggr) \biggr), \end{aligned}$$
(4)
where \(\sigma_{u}=\sum_{j}h(du(\cdot ),du(e_{j}))du(e_{j})\), which was defined in [1].
We need the following second variation formula for F-stationary maps (cf. [5]). Let \(u:(M,g)\rightarrow (N,h)\) be an F-stationary map. Let \(u_{s,t}:M\rightarrow N\) (\(-\varepsilon < s,t< \varepsilon \)) be a compactly supported two-parameter variation such that \(u_{0,0}=u\), and set \(V=\frac{\partial }{\partial t}u_{s,t}|_{s,t=0}\), \(W=\frac{\partial }{ \partial s}u_{s,t}|_{s,t=0}\). Then
$$\begin{aligned} \frac{\partial^{2}}{\partial s\partial t}\Phi_{F}(u_{s,t})|_{s,t=0} &= \int_{M}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\langle \widetilde{\nabla }V,\sigma_{u}\rangle \langle \widetilde{\nabla }W,\sigma_{u}\rangle \,dv_{g} \\ &\quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j=1}^{m}h( \widetilde{\nabla }_{e_{i}}V,\widetilde{\nabla }_{e_{j}}W)h \bigl(du(e_{i}),du(e _{j}) \bigr)\,dv_{g} \\ &\quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j=1}^{m}h \bigl( \widetilde{\nabla }_{e_{i}}V,du(e_{j}) \bigr)h \bigl(\widetilde{\nabla }_{e_{i}}W,du(e _{j}) \bigr)\,dv_{g} \\ &\quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j=1}^{m}h \bigl( \widetilde{\nabla }_{e_{i}}V,du(e_{j}) \bigr)h \bigl(du(e_{i}), \widetilde{\nabla }_{e_{j}}W \bigr)\,dv_{g} \\ &\quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)h \bigl(R^{N} \bigl(V,du(e_{i}) \bigr)W,du(e_{j}) \bigr)h \bigl(du(e _{i}),du(e_{j}) \bigr) \,dv_{g}, \end{aligned}$$
where \(\langle \cdot, \cdot \rangle \) is the inner product on \(T^{*}M\otimes u^{-1}TN\) and \(R^{N}\) is the curvature tensor of N.
We put
$$\begin{aligned} I(V,W)=\frac{\partial^{2}}{\partial s\partial t}\Phi_{F}(u_{s,t})|_{s,t=0}. \end{aligned}$$
(5)

An F-stationary map u is called stable if \(I(V,V)\geq 0\) for any compactly supported vector field V along u.

3 F-stationary maps from compact convex hypersurfaces

In this section, we obtain the following result.

Theorem 3.1

Let \(M\subset R^{m+1}\) be a compact convex hypersurface. Assume that the principal curvatures \(\lambda_{i}\) of \(M^{m}\) satisfy \(0<\lambda_{1} \leq \cdots \leq \lambda_{m}\) and \(3\lambda_{m}< \sum_{i=1}^{m-1} \lambda_{i}\). Then every nonconstant F-stationary map from M to any compact Riemannian manifold N is unstable if there exists a constant \(c_{F}=\operatorname{inf}\{c\geq 0| F'(t)/t^{c}\ \textit{is nonincreasing}\}\) such that
$$\begin{aligned} c_{F}< \frac{1}{4\lambda^{2}_{m}} \min_{1\leq i\leq m} \Biggl\{ \lambda_{i} \Biggl( \sum_{k=1}^{m} \lambda_{k}-2\lambda_{i}-2\lambda_{m} \Biggr) \Biggr\} \end{aligned}$$
(6)
or when \(F''(t)=F'(t)\) (for example, \(F(t)=\exp (t)\))
$$\begin{aligned} \bigl\Vert u^{*}h \bigr\Vert ^{2}< \frac{1}{\lambda_{m}^{2}} \min_{1\leq i\leq m} \Biggl\{ \lambda _{i} \Biggl(\sum_{k=1}^{m}\lambda_{k}-2 \lambda_{i}-2\lambda_{m} \Biggr) \Biggr\} . \end{aligned}$$
(7)

Proof

In order to prove the instability of \(u:M^{m}\rightarrow N\), we need to consider some special variational vector fields along u. To do this, we choose an orthonormal field \(\{e_{i},e_{m+1}\}\), \(i=1,\ldots ,m\), of \(R^{m+1}\) such that \(\{e_{i}\}\) are tangent to \(M^{m}\subset R^{m+1}\), \(e_{m+1}\) is normal to \(M^{m}\) and \(\nabla_{e_{i}}e_{j}|_{P}=0\). Meanwhile, we take a fixed orthonormal basis \(E_{A}\), \(A=1,\ldots ,m+1\), of \(R^{m+1}\) and set
$$\begin{aligned} V_{A}=\sum_{i=1}^{m}v_{A}^{i}e_{i}, \quad v_{A}^{i}=\langle E_{A},e_{i} \rangle, v_{A}^{m+1}=\langle E_{A},e_{m+1} \rangle, \end{aligned}$$
(8)
where \(\langle \cdot, \cdot \rangle \) denotes the canonical Euclidean inner product. Then \(du(V_{A})\in \Gamma (u^{-1}TN)\) and
$$\begin{aligned}& \sum_{A}v_{A}^{i}v_{A}^{j} = \sum_{A}\langle E_{A},e_{i} \rangle \langle E_{A},e_{j}\rangle= \delta_{ij}, \end{aligned}$$
(9)
$$\begin{aligned}& \nabla_{e_{i}}V_{A}=v_{A}^{m+1}B_{ij}e_{j}, \end{aligned}$$
(10)
$$\begin{aligned}& \nabla_{e_{i}}(\nabla_{e_{i}}V_{A}) = -v_{A}^{k}B_{ik}B_{ij}e_{j}+v _{A}^{m+1}(\nabla_{e_{i}}h_{ij})e_{j}, \end{aligned}$$
(11)
$$\begin{aligned}& \widetilde{\nabla }_{e_{i}} \bigl(du(\nabla_{e_{i}}V_{A}) \bigr) = -v_{A}^{k}B _{ik}B_{ij} du(e_{j}) \\& \hphantom{\widetilde{\nabla }_{e_{i}} (du(\nabla_{e_{i}}V_{A}) )= }{} +v_{A}^{m+1}(\nabla_{e_{i}}B_{ij}) du(e_{j})+v_{A}^{m+1}B_{ij} \widetilde{ \nabla }_{e_{i}}du(e_{j}), \end{aligned}$$
(12)
where \(B_{ij}\) denotes the components of the second fundamental form of \(M^{m}\) in \(R^{m+1}\). Suppose that \(u:M^{m}\rightarrow N\) is a nonconstant F-stationary map. Then the condition \(\tau_{F}(u)=- \delta (F'(\frac{\Vert u^{*}h \Vert ^{2}}{4})\sigma_{u})=0\) implies that
$$\begin{aligned}& \sum_{A} \int_{M^{m}}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\langle (\triangle du) (V _{A}),\sigma_{u}(V_{A}) \bigr\rangle \, dv_{g} \\& \quad = \sum_{A} \int_{M^{m}}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)v_{A}^{i}v^{j}_{A} \bigl\langle ( \triangle du) (e_{i}),\sigma_{u}(e_{j}) \bigr\rangle \, dv_{g} \\& \quad = \sum_{i} \int_{M^{m}}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\langle (\triangle du) (e _{i}),\sigma_{u}(e_{i}) \bigr\rangle \, dv_{g} \\& \quad = \int_{M^{m}} F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\langle ( \triangle du),\sigma _{u} \bigr\rangle \,dv_{g} \\& \quad = \int_{M^{m}} \biggl\langle \delta du,\delta \biggl(F' \biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sigma _{u} \biggr) \biggr\rangle \,dv_{g} =0. \end{aligned}$$
(13)
It follows from the Weitzenböck formula that
$$\begin{aligned} -\sum_{k=1}^{m}R^{N} \bigl(du(X), du(e_{k}) \bigr)du(e_{k})+du \bigl( \operatorname{Ric}^{M}(X) \bigr)=\triangle du(X)+\widetilde{\nabla }^{2}du(X), \end{aligned}$$
(14)
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 (13) and (14) that
$$\begin{aligned}& \sum_{A}I \bigl(du(V_{A}),du(V_{A}) \bigr) \\& \quad = \int_{M}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum_{A} \bigl\langle \widetilde{\nabla } du(V_{A}),\sigma_{u}\bigr\rangle ^{2} \,dv_{g} \\& \quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j,A}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),\widetilde{\nabla }_{e_{j}}du(V _{A}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\& \quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j,A}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),du(e_{j}) \bigr)h \bigl(\widetilde{\nabla } _{e_{i}}du(V_{A}),du(e_{j}) \bigr) \,dv_{g} \\& \quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j,A}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),du(e_{j}) \bigr)h \bigl(du(e_{i}), \widetilde{\nabla }_{e_{j}}du(V_{A}) \bigr)\,dv_{g} \\& \quad \quad {}- \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{A}h \bigl(du \bigl(\operatorname{Ric}^{M^{m}}(V_{A}) \bigr), \sigma_{u}(V_{A}) \bigr)\,dv_{g} \\& \quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{A}h \bigl( \bigl(\widetilde{\nabla }^{2} du \bigr) (V_{A}),\sigma_{u}(V_{A}) \bigr) \,dv_{g}. \end{aligned}$$
(15)
For any fixed point \(P\in M\), choose \(\{e_{i}\}\) such that \(\nabla _{e_{i}}e_{j}|_{P}=0\). We have
$$\begin{aligned} \widetilde{\nabla }^{2}du(V_{A})= \widetilde{\nabla }_{e_{i}} \widetilde{\nabla }_{e_{i}} \bigl(du(V_{A}) \bigr)-2\widetilde{\nabla }_{e_{i}} \bigl(du( \nabla_{e_{i}}V_{A}) \bigr) +du(\nabla_{e_{i}} \nabla_{e_{i}}V_{A}) \end{aligned}$$
(16)
and
$$\begin{aligned} & \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{A,i}h \bigl(\widetilde{\nabla } _{e_{i}} \widetilde{\nabla }_{e_{i}}du(V_{A}),\sigma_{u}(V_{A}) \bigr)\,dv_{g} \\ &\quad =- \int_{M}\sum_{A,i}h \biggl( \widetilde{\nabla }_{e_{i}}du(V_{A}), \widetilde{\nabla }_{e_{i}} \biggl[F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \sigma_{u}(V _{A}) \biggr] \biggr)\,dv_{g} \\ &\quad =- \int_{M}\sum_{A,i}h \biggl( \widetilde{\nabla }_{e_{i}}du(V_{A}), \widetilde{\nabla }_{e_{i}} \biggl[F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggr] \sigma_{u}(V _{A}) \biggr)\,dv_{g} \\ &\quad \quad {}- \int_{M}\sum_{A,i}h \biggl( \widetilde{\nabla }_{e_{i}}du(V_{A}),F' \biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\widetilde{\nabla }_{e_{i}}\sigma_{u}(V_{A}) \biggr)\,dv_{g} \\ &\quad =- \int_{M}\sum_{A,i}h \biggl( \widetilde{\nabla }_{e_{i}}du(V_{A}), \widetilde{\nabla }_{e_{i}} \biggl[F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggr] \sigma_{u}(V _{A}) \biggr)\,dv_{g} \\ &\quad \quad {}- \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{A,i,j}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),\widetilde{\nabla }_{e_{i}}du(e _{j}) \bigr)h \bigl(du(V_{A}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad \quad {}- \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{A,i,j}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),du(e_{j}) \bigr)h \bigl(\widetilde{\nabla } _{e_{i}}du(V_{A}),du(e_{j}) \bigr) \,dv_{g} \\ &\quad \quad {}- \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{A,i,j}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),du(e_{j}) \bigr)h \bigl(du(V_{A}), \widetilde{\nabla }_{e_{i}}du(e_{j}) \bigr) \,dv_{g}. \end{aligned}$$
(17)
Substituting (16) and (17) into (15), we have
$$\begin{aligned} &\sum_{A}I \bigl(du(V_{A}),du(V_{A}) \bigr) \\ &\quad = \int_{M} \biggl\{ F'' \biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum_{A} \bigl\langle \widetilde{\nabla }du(V_{A}),\sigma _{u} \bigr\rangle ^{2} \\ &\quad \quad {}-h \biggl(\widetilde{\nabla }_{e_{i}} du(V_{A}),\widetilde{ \nabla }_{e_{i}} \biggl[F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggr]\sigma_{u}(V_{A}) \biggr) \biggr\} \,dv_{g} \\ &\quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)h \bigl(-2\widetilde{\nabla }_{e_{i}} \bigl(du( \nabla_{e_{i}}V_{A}) \bigr) \\ &\quad \quad {}+du(\nabla_{e_{i}}\nabla_{e_{i}}V_{A})-du \bigl(\operatorname{Ric}^{M^{m}}(V_{A}) \bigr),\sigma _{u}(V_{A}) \bigr)\,dv_{g} \\ &\quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j,A}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),\widetilde{\nabla }_{e_{j}}du(V _{A}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j,A}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),du(e_{j}) \bigr)h \bigl(du(e_{i}), \widetilde{\nabla }_{e_{j}}du(V_{A}) \bigr)\,dv_{g} \\ &\quad \quad {}- \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{A,i,j}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),\widetilde{\nabla }_{e_{i}}du(e _{j}) \bigr)h \bigl(du(V_{A}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad \quad {}- \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{A,i,j}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),du(e_{j}) \bigr)h \bigl(du(V_{A}), \widetilde{\nabla }_{e_{i}}du(e_{j}) \bigr) \,dv_{g}. \end{aligned}$$
(18)
In the following, we shall estimate each term in (18). Because trace is independent of the choice of orthonormal basis, we can take pointwisely \(\{e_{i},e_{m+1}\}\) such that \(B_{ij}=\lambda_{i}\delta _{ij}\).
A straightforward computation shows
$$\begin{aligned} &\sum_{A}h \biggl(\widetilde{ \nabla }_{e_{i}}du(V_{A}),\widetilde{\nabla } _{e_{i}} \biggl[F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggr] \sigma_{u}(V_{A}) \biggr) \\ &\quad =\sum_{A}F'' \biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\widetilde{\nabla }_{e_{i}} \biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)h \bigl(v_{A}^{m+1}B_{ik} du(e_{k}) +v_{A}^{k} \widetilde{\nabla }_{e_{i}} du(e_{k}),v_{A}^{l} \sigma_{u}(e_{l}) \bigr) \\ &\quad =F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \widetilde{\nabla }_{e_{i}} \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)h \bigl( \widetilde{\nabla }_{e_{i}}du(e_{k}),\sigma_{u}(e _{k}) \bigr) \\ &\quad =F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \langle \widetilde{ \nabla }_{e_{i}}du,\sigma _{u} \rangle^{2} \end{aligned}$$
(19)
and
$$\begin{aligned} &\sum_{A}F'' \biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\langle \widetilde{\nabla } du(V_{A}),\sigma_{u} \bigr\rangle ^{2} \\ &\quad =\sum_{A}F''\biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\langle v_{A}^{m+1}B_{ik} du(e_{k})+v _{A}^{k}\widetilde{\nabla}_{e_{i}}du(e_{k}),\sigma_{u}(e_{i} )\bigr\rangle ^{2} \\ &\quad =\sum_{A}F'' \biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\{ B_{ik}B_{jl}h \bigl(du(e_{k}), \sigma_{u}(e_{i}) \bigr)h \bigl(du(e_{l}),\sigma_{u}(e_{j}) \bigr) \\ &\quad \quad {}+h \bigl(\widetilde{\nabla }_{e_{i}}du(e_{k}), \sigma_{u}(e_{i}) \bigr)h \bigl( \widetilde{\nabla }_{e_{j}}du(e_{k}),\sigma_{u}(e_{j}) \bigr) \bigr\} \\ &\quad =\sum_{A}F'' \biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\{ \lambda_{i} \lambda_{j} h \bigl(du(e _{i}),\sigma_{u}(e_{i}) \bigr)h \bigl(du(e_{j}),\sigma_{u}(e_{j}) \bigr) +\langle \widetilde{\nabla }_{e_{i}}du,\sigma_{u} \rangle^{2} \bigr\} . \end{aligned}$$
(20)
Then it follows from (19) and (20) that
$$\begin{aligned} & \int_{M} \biggl\{ F'' \biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum_{A} \bigl\langle \widetilde{\nabla }du(V _{A}),\sigma_{u} \bigr\rangle ^{2} \\ &\quad \quad {}-h \biggl(\widetilde{\nabla }_{e_{i}} du(V_{A}),\widetilde{\nabla }_{e_{i}} \biggl[F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggr]\sigma_{u}(V_{A}) \biggr) \biggr\} \,dv_{g} \\ &= \int_{M}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\lambda_{i}\lambda_{j} h \bigl(du(e _{i}), \sigma_{u}(e_{i}) \bigr)h \bigl(du(e_{j}), \sigma_{u}(e_{j}) \bigr) \,dv_{g}. \end{aligned}$$
(21)
From the Gauss equation it follows that
$$\begin{aligned} \operatorname{Ric}^{M}(V_{A})=v_{A}^{i}(B_{kk}B_{ij}-B_{ik}B_{jk})e_{j}. \end{aligned}$$
(22)
Using (10), (11),(12) and (22), we have
$$\begin{aligned} & \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)h \bigl(-2\widetilde{\nabla }_{e_{i}} \bigl(du( \nabla_{e_{i}}V_{A}) \bigr) \\ &\quad \quad {}+du(\nabla_{e_{i}}\nabla_{e_{i}}V_{A})-du \bigl( \operatorname{Ric}^{M^{m}}(V_{A}) \bigr),\sigma _{u}(V_{A}) \bigr)\,dv_{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\{ \bigl[h \bigl(2v_{A}^{k}B_{ik}B_{ij} du(e _{j})-v_{A}^{m+1}\nabla_{e_{i}}(B_{ij}) du(e_{j}) \\ &\quad \quad {}-v_{A}^{m+1}B_{ij}\widetilde{\nabla }_{e_{i}}du(e_{j}),v_{A}^{l} \sigma_{u}(e_{l}) \bigr) \bigr] \\ &\quad \quad {}+h \bigl(-v_{A}^{k}B_{ik}B_{ij} du(e_{j})+v_{A}^{m+1}(\nabla_{e_{i}}B_{ij}) du(e _{j}),v_{A}^{l}\sigma_{u}(e_{l}) \bigr) \\ &\quad \quad {}+h \bigl(v_{A}^{k}B_{ik}B_{ij} du(e_{j})-v^{i}_{A}B_{kk}B_{ij} du(e_{j}),v _{A}^{l}\sigma_{u}(e_{l}) \bigr) \bigr\} \,dv_{g} \\ &\quad =\int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\{ h\bigl(2v_{A}^{k}B_{ik}B_{ij}du(e_{j})-v^{i}_{A}B_{kk}B_{ij}du(e_{j}),v_{A}^{l}\sigma_{u}(e_{l})\bigr)\bigr\} \,dv _{g} \\ &\quad =\int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum_{i} \biggl\{ \biggl[2\lambda_{i}-\biggl( \sum_{k}\lambda_{k} \biggr) \biggr] \lambda_{i}h \bigl(du(e_{i}),\sigma_{u}(e_{i}) \bigr) \biggr\} \,dv_{g}. \end{aligned}$$
(23)
A straightforward computation shows
$$\begin{aligned} & \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j,A}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),\widetilde{\nabla }_{e_{j}}du(V _{A}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)h \bigl(v_{A}^{m+1}B_{ik}du(e_{k})+v _{A}^{k}\widetilde{\nabla }_{e_{i}} du(e_{k}), \\ &\quad \quad v_{A}^{m+1}B_{jl}du(e_{l})+v_{A}^{l} \widetilde{\nabla }_{e_{j}}du(e _{l}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\{ B_{ik}B_{jl}h \bigl(du(e_{k}),du(e _{l}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr) \\ &\quad \quad {}+h \bigl(\widetilde{\nabla }_{e_{i}}du(e_{k}), \widetilde{ \nabla }_{e_{j}}du(e _{k}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr) \bigr\} \,dv_{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\{ \lambda_{i}\lambda_{j}h \bigl(du(e _{i}),du(e_{j}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr) \\ &\quad \quad {}+h \bigl(\widetilde{\nabla }_{e_{i}}du(e_{k}), \widetilde{ \nabla }_{e_{j}}du(e _{k}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr) \bigr\} \,dv_{g} \end{aligned}$$
(24)
and
$$\begin{aligned} & \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j,A}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),du(e_{j}) \bigr)h \bigl(du(e_{i}), \widetilde{\nabla }_{e_{j}}du(V_{A}) \bigr)\,dv_{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\{ h \bigl(v_{A}^{m+1}B_{ik}du(e_{k})+v _{A}^{k}\widetilde{\nabla }_{e_{i}} du(e_{k}),du(e_{j}) \bigr) \\ &\quad \quad {}\times h \bigl(du(e_{i}),v_{A}^{m+1}B_{jk} du(e_{k})+v_{A}^{k}\widetilde{\nabla } _{e_{j}}du(e_{k}) \bigr) \bigr\} \,dv_{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\{ \lambda_{i}\lambda_{j}h \bigl(du(e _{i}),du(e_{j}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr) \\ &\quad \quad {}+h \bigl(\widetilde{\nabla }_{e_{i}}du(e_{k}),du(e_{j}) \bigr)h \bigl( \widetilde{\nabla }_{e_{j}}du(e_{k}),du(e_{i}) \bigr) \bigr\} \,dv_{g} \end{aligned}$$
(25)
and
$$\begin{aligned} & \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{A,i,j}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),\widetilde{\nabla }_{e_{i}}du(e _{j}) \bigr)h \bigl(du(V_{A}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\{ h \bigl(v_{A}^{m+1}B_{ik}du(e_{k})+v _{A}^{k}\widetilde{\nabla }_{e_{i}} du(e_{k}), \widetilde{\nabla }_{e _{i}}du(e_{j}) \bigr) \\ &\quad \quad {}\times h \bigl(v_{A}^{l} du(e_{l}),du(e_{j}) \bigr) \bigr\} \,dv_{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)h \bigl( \widetilde{\nabla }_{e_{i}}du(e _{k}), \widetilde{ \nabla }_{e_{i}}du(e_{j}) \bigr)h \bigl(du(e_{k}),du(e_{j}) \bigr)\,dv _{g} \end{aligned}$$
(26)
and
$$\begin{aligned} & \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{A,i,j}h \bigl( \widetilde{\nabla }_{e_{i}} du(V_{A}),du(e_{j}) \bigr)h \bigl(du(V_{A}), \widetilde{\nabla }_{e_{i}}du(e_{j}) \bigr)\,dv_{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\{ h \bigl(v_{A}^{m+1}B_{ik}du(e_{k})+v _{A}^{k}\widetilde{\nabla }_{e_{i}} du(e_{k}),du(e_{j}) \bigr) \\ &\quad \quad {}\times h \bigl(v_{A}^{l} du(e_{l}), \widetilde{\nabla }_{e_{i}}du(e_{j}) \bigr) \bigr\} \,dv _{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)h \bigl( \widetilde{\nabla }_{e_{i}}du(e _{k}),du(e_{j}) \bigr)h \bigl(du(e_{k}),\widetilde{\nabla }_{e_{i}} du(e_{j}) \bigr)\,dv _{g}. \end{aligned}$$
(27)
From (18), (21), (23), (24), (25), (26), (27) and \(\widetilde{\nabla }_{e_{i}}du(e_{j})=\widetilde{\nabla }_{e_{j}}du(e _{i})\), we obtain
$$\begin{aligned} &\sum_{A}I \bigl(du(V_{A}),du(V_{A}) \bigr) \\ &\quad = \int_{M}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\lambda_{i}\lambda_{j} h \bigl(du(e _{i}), \sigma_{u}(e_{i}) \bigr)h \bigl(du(e_{j}), \sigma_{u}(e_{j}) \bigr) \,dv_{g} \\ &\quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i} \biggl\{ \biggl[2\lambda_{i}- \biggl( \sum_{k}\lambda_{k} \biggr) \biggr] \lambda_{i}h \bigl(du(e_{i}),\sigma_{u}(e_{i}) \bigr) \biggr\} \,dv_{g} \\ &\quad \quad {}+2 \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{i}\lambda_{j}h \bigl(du(e _{i}),du(e_{j}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad \leq \int_{M}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\lambda_{i}\lambda_{j} h \bigl(du(e _{i}), \sigma_{u}(e_{i}) \bigr)h \bigl(du(e_{j}), \sigma_{u}(e_{j}) \bigr) \,dv_{g} \\ &\quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i} \biggl\{ \biggl[2\lambda_{i}- \biggl( \sum_{k}\lambda_{k} \biggr) \biggr] \lambda_{i}h \bigl(du(e_{i}),\sigma_{u}(e_{i}) \bigr) \biggr\} \,dv_{g} \\ &\quad \quad {}+2 \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{i}\lambda_{m}h \bigl(du(e _{i}), \sigma_{u}(e_{i}) \bigr)\,dv_{g} \\ &\quad = \int_{M}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\lambda_{i}\lambda_{j} h \bigl(du(e _{i}), \sigma_{u}(e_{i}) \bigr)h \bigl(du(e_{j}), \sigma_{u}(e_{j}) \bigr) \,dv_{g} \\ &\quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i} \biggl\{ \biggl[2\lambda_{i}+2 \lambda_{m}- \biggl(\sum_{k} \lambda_{k} \biggr) \biggr]\lambda_{i}h \bigl(du(e_{i}), \sigma_{u}(e _{i}) \bigr) \biggr\} \,dv_{g}. \end{aligned}$$
(28)
If \(F''(t)=F'(t)\), then (28) leads to the following inequality:
$$\begin{aligned} &\sum_{A}I \bigl(du(V_{A}),du(V_{A}) \bigr) \\ &\quad \leq \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{m}^{2} \bigl\Vert u^{*}h \bigr\Vert ^{4}\,dv _{g} \\ &\quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\max _{1\leq i\leq m} \biggl\{ \biggl[2\lambda _{i}+2 \lambda_{m}- \biggl(\sum_{k} \lambda_{k} \biggr) \biggr]\lambda_{i} \biggr\} \bigl\Vert u^{*}h \bigr\Vert ^{2}\,dv _{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\Vert u^{*}h \bigr\Vert ^{2} \biggl\{ \lambda_{m} ^{2} \bigl\Vert u^{*}h \bigr\Vert ^{2} \\ &\quad \quad {}+\max_{1\leq i\leq m} \biggl\{ \biggl[2 \lambda_{i}+2 \lambda_{m}- \biggl( \sum _{k} \lambda_{k} \biggr) \biggr] \lambda_{i} \biggr\} \biggr\} \,dv_{g}. \end{aligned}$$
(29)
If there exists a constant \(c_{F}\) such that \(\frac{F'(t)}{t^{c_{F}}}\) is nonincreasing, it follows that \(F''(t)t\leq c_{F}F'(t)\) on \(t\in (0,\infty )\), thus (28) implies
$$\begin{aligned} &\sum_{A}I \bigl(du(V_{A}),du(V_{A}) \bigr) \\ &\quad \leq \int_{M}4c_{F}F' \biggl( \frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\lambda_{m}^{2} \bigl\Vert u^{*}h \bigr\Vert ^{2}\,dv_{g} \\ &\quad \quad {}+ \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\max _{1\leq i\leq m} \biggl\{ \biggl[2\lambda _{i}+2 \lambda_{m}- \biggl(\sum_{k} \lambda_{k} \biggr) \biggr]\lambda_{i} \biggr\} \bigl\Vert u^{*}h \bigr\Vert ^{2}\,dv _{g} \\ &\quad = \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\Vert u^{*}h \bigr\Vert ^{2} \biggl\{ 4c_{F} \lambda _{m}^{2} \\ &\quad \quad {}+\max_{1\leq i\leq m} \biggl\{ \biggl[2 \lambda_{i}+2\lambda_{m}- \biggl(\sum _{k} \lambda_{k} \biggr) \biggr] \lambda_{i} \biggr\} \biggr\} \,dv_{g}. \end{aligned}$$
(30)
If u is nonconstant and (6) or (7) holds, we have
$$\begin{aligned} \sum_{A}I \bigl(du(V_{A}),du(V_{A}) \bigr)< 0 \end{aligned}$$
(31)
and u is unstable. □

Corollary 3.2

Let \(u:S^{m}\rightarrow N\) be a nonconstant F-stationary map and \(m>4\). If \(c_{F}<\frac{m}{4}-1\) or \(\Vert u^{*}h \Vert ^{2}< m-4\), then u is unstable.

4 F-stationary maps into compact convex hypersurfaces

In this section, we obtain the following result.

Theorem 4.1

With the same assumption on \(M^{m}\) as in Theorem 3.1, every nonconstant F-stationary map from any compact Riemannian manifold N to \(M^{m}\) is unstable if (6) or (7) holds.

Proof

In order to prove the instability of \(u:N^{n}\rightarrow M^{m}\), we need to consider some special variational vector fields along u. To do this, we choose an orthonormal field \(\{\epsilon_{\alpha },\epsilon _{m+1}\}\), \(\alpha =1,\ldots ,m\), of \(R^{m+1}\) such that \(\{ \epsilon_{\alpha }\}\) are tangent to \(M^{m}\subset R^{m+1}\), \(\epsilon_{m+1}\) is normal to \(M^{m}\), \(^{M^{m}} \nabla_{\epsilon_{\alpha }}\epsilon_{\beta }|_{P}=0\) and \(B_{\alpha \beta }=\lambda_{\alpha }\delta_{\alpha \beta }\), where \(B_{\alpha \beta }\) denotes the components of the second fundamental form of \(M^{m}\) in \(R^{m+1}\). Meanwhile, take a fixed orthonormal basis \(E_{A}\), \(A=1,\ldots ,m+1\), of \(R^{m+1}\) and set
$$\begin{aligned} V_{A}=\sum_{\alpha =1}^{m}v_{A}^{\alpha } \epsilon_{\alpha },\quad v_{A}^{\alpha }=\langle E_{A},\epsilon_{\alpha }\rangle, v_{A}^{m+1}= \langle E_{A},\epsilon_{m+1}\rangle, \end{aligned}$$
(32)
where \(\langle \cdot, \cdot \rangle \) denotes the canonical Euclidean inner product. We shall consider the second variation
$$\begin{aligned} \sum_{A}I(V_{A},V_{A}) =& \int_{N}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \langle \widetilde{\nabla }V_{A},\sigma_{u}\rangle \langle \widetilde{\nabla }V_{A},\sigma _{u}\rangle \,dv_{g} \\ & {}+ \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j=1}^{m}h( \widetilde{\nabla }_{e_{i}}V_{A},\widetilde{\nabla }_{e_{j}}V_{A})h \bigl(du(e _{i}),du(e_{j}) \bigr)\,dv_{g} \\ & {}+ \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j=1}^{m}h \bigl( \widetilde{\nabla }_{e_{i}}V_{A},du(e_{j}) \bigr)h \bigl(\widetilde{ \nabla }_{e _{i}}V_{A},du(e_{j}) \bigr) \,dv_{g} \\ & {}+ \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j=1}^{m}h \bigl( \widetilde{\nabla }_{e_{i}}V_{A},du(e_{j}) \bigr)h \bigl(du(e_{i}), \widetilde{\nabla }_{e_{j}}V_{A} \bigr)\,dv_{g} \\ &{}+ \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i}h \bigl(R^{M^{m}} \bigl(V_{A},du(e _{i}) \bigr)V_{A},\sigma_{u}(e_{i}) \bigr)\,dv_{g}, \end{aligned}$$
(33)
where \(\{e_{1},\ldots ,e_{n}\}\) is the local orthonormal frame of \(N^{n}\).
Firstly, we compute the first term of (33)
$$\begin{aligned} &\sum_{A} \int_{N}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \langle \widetilde{\nabla }V _{A},\sigma_{u}\rangle \langle \widetilde{ \nabla }V_{A},\sigma_{u}\rangle\, dv_{g} \\ &\quad = \int_{N}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggl[\sum_{i}h \bigl( \widetilde{\nabla }_{e_{i}}V_{A},\sigma_{u}(e_{i}) \bigr) \biggr]^{2}\,dv_{g} \\ &\quad = \int_{N}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggl[\sum_{i}h \bigl({}^{M^{m}} \nabla_{du(e_{i})}V_{A},\sigma_{u}(e_{i}) \bigr) \biggr]^{2}\,dv_{g} \\ &\quad = \int_{N}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggl[\sum_{i}u_{i}^{\alpha }h \bigl({}^{M ^{m}}\nabla_{\epsilon_{\alpha }}V_{A}, \sigma_{u}(e_{i}) \bigr) \biggr]^{2} \,dv_{g} \\ &\quad = \int_{N}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggl[\sum_{i}v_{A}^{m+1}u_{i}^{ \alpha }B_{\alpha \beta } h \bigl(\epsilon_{\beta },\sigma_{u}(e_{i}) \bigr) \biggr]^{2}\,dv _{g} \\ &\quad = \int_{N}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggl[\sum_{i}v_{A}^{m+1}u_{i}^{ \alpha } \lambda_{\alpha }h \bigl(\epsilon_{\alpha },\sigma_{u}(e_{i}) \bigr) \biggr]^{2}\,dv _{g} \\ &\quad = \int_{N}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{\alpha }\lambda_{ \beta }h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },\sigma_{u}(e_{i}) \bigr)h \bigl(u_{j} ^{\beta }\epsilon_{\beta }, \sigma_{u}(e_{j}) \bigr)\,dv_{g}. \end{aligned}$$
(34)
The second term of (33)
$$\begin{aligned} &\sum_{A} \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j=1}^{m}h( \widetilde{\nabla }_{e_{i}}V_{A},\widetilde{\nabla }_{e_{j}}V_{A})h \bigl(du(e _{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad =\int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert {}^{2}}{4} \biggr)h \bigl({}^{M^{m}}\nabla_{du(e_{i})}V _{A},{}^{M^{m}} \nabla_{du(e_{j})}V_{A} \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)u_{i}^{\alpha }u_{j}^{\beta }h \bigl({}^{M ^{m}}\nabla_{\epsilon_{\alpha }}V_{A},{}^{M^{m}} \nabla_{\epsilon_{ \beta }}V_{A} \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)u_{i}^{\alpha }u_{j}^{\beta }B _{\alpha \gamma }B_{\beta \delta }h(\epsilon_{\gamma },\epsilon_{ \delta })h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{\alpha }\lambda_{ \beta }h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },u_{j}^{\beta }\epsilon_{ \beta } \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{\alpha }^{2} h \bigl(u_{i} ^{\alpha } \epsilon_{\alpha },du(e_{j}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g}. \end{aligned}$$
(35)
The third term of (33)
$$\begin{aligned} &\sum_{A} \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j=1}^{m}h \bigl( \widetilde{\nabla }_{e_{i}}V_{A},du(e_{j}) \bigr)h \bigl(\widetilde{ \nabla }_{e _{i}}V_{A},du(e_{j}) \bigr) \,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{\alpha }\lambda_{ \beta }h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },du(e_{j}) \bigr)h \bigl(u_{i}^{\beta } \epsilon_{\beta },du(e_{j}) \bigr)\,dv_{g}. \end{aligned}$$
(36)
The fourth term of (33)
$$\begin{aligned} &\sum_{A} \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i,j=1}^{m}h \bigl( \widetilde{\nabla }_{e_{i}}V_{A},du(e_{j}) \bigr)h \bigl(du(e_{i}), \widetilde{\nabla }_{e_{j}}V_{A} \bigr)\,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{\alpha }\lambda_{ \beta }h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },du(e_{j}) \bigr)h \bigl(du(e_{i}),u _{j}^{\beta }\epsilon_{\beta } \bigr)\,dv_{g}. \end{aligned}$$
(37)
The fifth term of (33)
$$\begin{aligned} &\sum_{A} \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)\sum _{i}h \bigl(R^{M^{m}} \bigl(V _{A},du(e_{i}) \bigr)V_{A},\sigma_{u}(e_{i}) \bigr)\,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)v_{A}^{\alpha }v_{A}^{\beta }h \bigl(R ^{M^{m}} \bigl(\epsilon_{\alpha },du(e_{i}) \bigr) \epsilon_{\beta },\sigma_{u}(e _{i}) \bigr) \,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)u_{i}^{\gamma }u_{j}^{\delta }h \bigl(R ^{M^{m}}(\epsilon_{\alpha },\epsilon_{\gamma }) \epsilon_{\alpha }, \epsilon_{\delta } \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)u_{i}^{\gamma }u_{j}^{\delta }[B _{\alpha \delta }B_{\gamma \alpha }-B_{\alpha \alpha }B_{\gamma \delta }]h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)u_{i}^{\alpha }u_{j}^{\alpha } \biggl[ \lambda_{\alpha }^{2}- \biggl(\sum_{\beta } \lambda_{\beta } \biggr) \lambda_{\alpha } \biggr]h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggl[ \lambda_{\alpha }^{2}- \biggl( \sum _{\beta } \lambda_{\beta } \biggr) \lambda_{\alpha } \biggr]h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },u_{j}^{\gamma } \epsilon_{\gamma } \bigr)h \bigl(du(e_{i}),du(e _{j}) \bigr)\,dv_{g} \\ &\quad = \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggl[ \lambda_{\alpha }^{2}- \biggl( \sum _{\beta } \lambda_{\beta } \biggr) \lambda_{\alpha } \biggr]h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },du(e_{j}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr) \,dv_{g}. \end{aligned}$$
(38)
From (33)-(38), we have
$$\begin{aligned} &\sum_{A}I(V_{A},V_{A}) \\ &\quad = \int_{N}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{\alpha }\lambda_{\beta }h \bigl(u_{i}^{\alpha } \epsilon_{\alpha }, \sigma_{u}(e_{i}) \bigr)h \bigl(u_{j}^{\alpha }\epsilon_{\alpha },\sigma_{u}(e _{j}) \bigr)\,dv_{g} \\ &\quad \quad {}+ \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggl[2 \lambda_{\alpha }^{2}- \biggl( \sum _{\beta } \lambda_{\beta } \biggr) \lambda_{\alpha } \biggr]h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },du(e_{j}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad \quad {}+ \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{\alpha }\lambda_{ \beta }h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },du(e_{j}) \bigr)h \bigl(u_{i}^{\beta } \epsilon_{\beta },du(e_{j}) \bigr)\,dv_{g} \\ &\quad \quad {}+ \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{\alpha }\lambda_{ \beta }h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },du(e_{j}) \bigr)h \bigl(du(e_{i}),u _{j}^{\beta }\epsilon_{\beta } \bigr)\,dv_{g} \\ &\quad \leq \int_{N}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{\alpha } \lambda_{\beta }h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },\sigma_{u}(e_{i}) \bigr)h \bigl(u _{j}^{\alpha }\epsilon_{\alpha },\sigma_{u}(e_{j}) \bigr)\,dv_{g} \\ &\quad \quad {}+ \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggl[2 \lambda_{\alpha }^{2}- \biggl( \sum _{\beta } \lambda_{\beta } \biggr) \lambda_{\alpha } \biggr]h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },du(e_{j}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv_{g} \\ &\quad \quad {}+ \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr)2 \lambda_{\alpha }\lambda_{m} h \bigl(u _{i}^{\alpha } \epsilon_{\alpha },du(e_{j}) \bigr)h \bigl(du(e_{i}),du(e_{j}) \bigr)\,dv _{g} \\ &\quad \leq \int_{N}F'' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \lambda_{\alpha } \lambda_{\beta }h \bigl(u_{i}^{\alpha } \epsilon_{\alpha },\sigma_{u}(e_{i}) \bigr)h \bigl(u _{j}^{\alpha }\epsilon_{\alpha },\sigma_{u}(e_{j}) \bigr)\,dv_{g} \\ &\quad \quad {}+ \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \biggl[2 \lambda_{\alpha }^{2}+2 \lambda_{\alpha } \lambda_{m}- \biggl(\sum_{\beta } \lambda_{\beta } \biggr) \lambda_{\alpha } \biggr]h \bigl(u_{i}^{\alpha } \epsilon_{\alpha }, \sigma_{u}(e _{i}) \bigr) \,dv_{g}. \end{aligned}$$
(39)
If \(F''(t)=F'(t)\), then (39) leads to the following inequality:
$$\begin{aligned} \sum_{A}I(V_{A},V_{A})& \leq \int_{N}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\Vert u^{*}h \bigr\Vert ^{2} \biggl\{ \bigl\Vert u^{*}h \bigr\Vert ^{2} \lambda_{m}^{2} \\ & \quad {}+ \max_{1\leq \alpha \leq m} \biggl[2\lambda_{\alpha }^{2}+2 \lambda_{\alpha }\lambda_{m}- \biggl(\sum _{\beta }\lambda_{\beta } \biggr) \lambda_{\alpha } \biggr] \biggr\} \,dv_{g} . \end{aligned}$$
(40)
If there exists a constant \(c_{F}\) such that \(\frac{F'(t)}{t^{c_{F}}}\) is nonincreasing, it follows that \(F''(t)t\leq c_{F}F'(t)\) on \(t\in (0,\infty )\), thus (39) implies
$$\begin{aligned} \sum_{A}I(V_{A},V_{A})&\leq \int_{M}F' \biggl(\frac{\Vert u^{*}h \Vert ^{2}}{4} \biggr) \bigl\Vert u^{*}h \bigr\Vert ^{2} \biggl\{ 4c_{F} \lambda_{m}^{2} \\ & \quad {}+\max_{1\leq \alpha \leq m} \biggl\{ \biggl[2 \lambda_{\alpha }+2 \lambda_{m}- \biggl(\sum_{\beta } \lambda_{\beta } \biggr) \biggr] \lambda_{\alpha } \biggr\} \biggr\} \,dv _{g} . \end{aligned}$$
(41)
Now, if \(u:N\rightarrow M^{m}\) is a nonconstant F-stationary map and (6) or (7) holds, then, from (41) or (40), we know that \(\sum_{A}I(V_{A},V_{A})<0\) and u is unstable. □

Corollary 4.2

Let \(u:N\rightarrow S^{m}\) be a nonconstant F-stationary map with \(m>4\), where N is any compact Riemannian manifold. If \(c_{F}< \frac{m}{4}-1\) or \(\Vert u^{*}h \Vert ^{2}< m-4\), then u is unstable.

5 Conclusions

In this paper, we investigate F-stationary maps between the compact convex hypersurface \(M^{m}\) and any compact Riemannian manifold N. Assume that the principal curvatures \(\lambda_{i}\) of \(M^{m}\) satisfy \(0<\lambda_{1}\leq \cdots \leq \lambda_{m}\) and \(3\lambda_{m}< \sum_{i=1}^{m-1}\lambda_{i}\), then every nonconstant F-stationary map from \(M^{m}\) to N or from N to \(M^{m}\) is unstable if (6) or (7) holds. We mainly use the second variation formula for F-stationary maps (cf. [5]) to get the instability. In particular, we consider \(S^{m}\) as a special case of compact convex hypersurfaces and obtain similar inferences.

Declarations

Acknowledgements

The first author wishes to thank Professor Yingbo Han for his guidance. This research was supported by the NNSF of China (No. 11371194; No. 11501292), by a Grant-in-Aid for Science Research from Nanjing University of Science and Technology (No. 30920140132035) and by the NUST Research Funding (No. CXZZ11-0258; No. AD20370).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094, P.R. China

References

  1. Kawai, S, Nakauchi, N: Some result for stationary maps of a functional related to pullback metrics. Nonlinear Anal. 74, 2284-2295 (2011) MathSciNetView ArticleMATHGoogle Scholar
  2. Nakauchi, N: A variational problem related to conformal maps. Osaka J. Math. 48, 719-741 (2011) MathSciNetMATHGoogle Scholar
  3. Nakauchi, N, Takenaka, Y: A variational problem for pullback metrics. Ric. Mat. 60, 219-235 (2011) MathSciNetView ArticleMATHGoogle Scholar
  4. Asserda, S: Liouville-type results for stationary maps of a class of functional related to pullback metrics. Nonlinear Anal. 75, 3480-3492 (2012) MathSciNetView ArticleMATHGoogle Scholar
  5. Han, YB, Feng, SX: Monotonicity formulas and the stability of F-stationary maps with potential. Houst. J. Math. 40, 681-713 (2014) MathSciNetMATHGoogle Scholar
  6. Ara, M: Geometry of F-harmonic maps. Kodai Math. J. 22, 243-263 (1999) MathSciNetView ArticleMATHGoogle Scholar
  7. Dong, YX, Wei, SS: On vanishing theorems for vector bundle valued p-forms and their applications. Commun. Math. Phys. 304, 329-368 (2011) MathSciNetView ArticleMATHGoogle Scholar
  8. Dong, YX, Lin, HZ, Yang, GL: Liouville theorems for F-harmonic maps and their applications. Results Math. 69, 105-127 (2016) MathSciNetView ArticleMATHGoogle Scholar
  9. Kassi, M: A Liouville theorems for F-harmonic maps with finite F-energy. Electron. J. Differ. Equ. 2006, 15 (2006) MathSciNetMATHGoogle Scholar
  10. Liu, JC: Liouville theorems of stable F-harmonic maps for compact convex hypersurfaces. Hiroshima Math. J. 36, 221-234 (2006) MathSciNetMATHGoogle Scholar

Copyright

© The Author(s) 2017

Advertisement