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Journal of Inequalities and Applications
 
J Inequal Appl. 2017; 2017(1): 214.
Published online 2017 September 8. doi:  10.1186/s13660-017-1483-z
PMCID: PMC5591368

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

Abstract

Let Mm be a compact convex hypersurface in Rm+1. In this paper, we prove that if the principal curvatures λi of Mm satisfy 0 < λ1 ≤  ⋯  ≤ λm and 3λm<j=1m1λj, 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

Introduction

Let u:(Mmg) → (Nnh) be a smooth map between Riemannian manifolds (Mmg) and (Nnh). Recently, Kawai and Nakauchi [1] introduced a functional related to the pullback metric uh as follows:

Φ(u) = ¼∫Muh2 dvg
1

(see [24]), where uh is the symmetric 2-tensor defined by

(uh)(XY) = h(du(X), du(Y))

for any vector fields X, Y on M and  ∥ uh ∥  is given by

uh2=i,j=1m[h(du(ei),du(ej))]2,

with respect to a local orthonormal frame (e1, …, em) on (Mg). The map u is stationary for Φ if it is a critical point of Φ(u) with respect to any compact supported variation of u, and u is stable if the second variation for the functional Φ(u) is nonnegative. They showed the nonexistence of a nonconstant stable stationary map for Φ, either from Sm (m ≥ 5) to any manifold, or from any compact Riemannian manifold to Sn (n ≥ 5). In this paper, for a smooth function F:[0, ∞) → [0, ∞) such that F(0) = 0 and F(t) > 0 on t ∈ (0, ∞), 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 Sm(1) to any Riemannian manifold is constant if

Smuh2{F(uh24)uh2+(4m)F(uh24)}dvg<0,
2

or every F-stationary map from any compact Riemannian manifold Nn to Sm is constant if

Nnuh2{F(uh24)uh2+(4m)F(uh24)}dvg<0.
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.

Preliminaries

Let F:[0, ∞) → [0, ∞) be a C2-function such that F(0) = 0 and F(t) > 0 on t ∈ (0, ∞). For a smooth map u:(Mg) → (Nh) between compact Riemannian manifolds (Mg) and (Nh) 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 ΦF if

ddtΦF(ut)|t=0=0

for any compactly supported variation ut:M → N with u0u, where

ΦF(u)=MmF(uh24)dvg.

Let [nabla] and N[nabla] always denote the Levi-Civita connections of M and N, respectively. Let [nabla with tilde] be the induced connection on u−1TN defined by ˜XW=Ndu(X)W, where X is a tangent vector of M and W is a section of u−1TN. We choose a local orthonormal frame field {ei} on M. We define the F-tension field τΦF(u) of u by

τΦF(u)=δ(F(uh24)σu)=F(uh24)divg(σu)+σu(grad(F(uh24))),
4

where σu = ∑jh(du( ⋅ ), du(ej))du(ej), which was defined in [1].

We need the following second variation formula for F-stationary maps (cf. [5]). Let u:(Mg) → (Nh) be an F-stationary map. Let us,t:M → N (ε < st < ε) be a compactly supported two-parameter variation such that u0,0u, and set V=tus,t|s,t=0, W=sus,t|s,t=0. Then

2stΦF(us,t)|s,t=0=MF(uh24)˜V,σu˜W,σudvg+MF(uh24)i,j=1mh(˜eiV,˜ejW)h(du(ei),du(ej))dvg+MF(uh24)i,j=1mh(˜eiV,du(ej))h(˜eiW,du(ej))dvg+MF(uh24)i,j=1mh(˜eiV,du(ej))h(du(ei),˜ejW)dvg+MF(uh24)h(RN(V,du(ei))W,du(ej))h(du(ei),du(ej))dvg,

where 〈 ⋅ ,  ⋅ 〉 is the inner product on TM ⊗ u−1TN and RN is the curvature tensor of N.

We put

I(V,W)=2stΦF(us,t)|s,t=0.
5

An F-stationary map u is called stable if I(VV) ≥ 0 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 3.1

Let M ⊂ Rm+1 be a compact convex hypersurface. Assume that the principal curvatures λi of Mm satisfy 0 < λ1 ≤  ⋯  ≤ λm and 3λm<i=1m1λi. Then every nonconstant F-stationary map from M to any compact Riemannian manifold N is unstable if there exists a constant cF = inf{c ≥ 0|F(t)/tc is nonincreasing} such that

cF<14λm2min1im{λi(k=1mλk2λi2λm)}
6

or when F(t) = F(t) (for example, F(t) = exp(t))

uh2<1λm2min1im{λi(k=1mλk2λi2λm)}.
7

Proof

In order to prove the instability of u:Mm → N, we need to consider some special variational vector fields along u. To do this, we choose an orthonormal field {eiem+1}, i = 1, …, m, of Rm+1 such that {ei} are tangent to Mm ⊂ Rm+1, em+1 is normal to Mm and eiej|P = 0. Meanwhile, we take a fixed orthonormal basis EA, A = 1, …, m + 1, of Rm+1 and set

VA=i=1mvAiei,vAi=EA,ei,vAm+1=EA,em+1,
8

where 〈 ⋅ ,  ⋅ 〉 denotes the canonical Euclidean inner product. Then du(VA) ∈ Γ(u−1TN) and

AvAivAj=AEA,eiEA,ej=δij,
9

eiVA=vAm+1Bijej,
10

ei(eiVA)=vAkBikBijej+vAm+1(eihij)ej,
11

˜ei(du(eiVA))=vAkBikBijdu(ej)˜ei(du(eiVA))=+vAm+1(eiBij)du(ej)+vAm+1Bij˜eidu(ej),
12

where Bij denotes the components of the second fundamental form of Mm in Rm+1. Suppose that u:Mm → N is a nonconstant F-stationary map. Then the condition τF(u)=δ(F(uh24)σu)=0 implies that

AMmF(uh24)(du)(VA),σu(VA)dvg=AMmF(uh24)vAivAj(du)(ei),σu(ej)dvg=iMmF(uh24)(du)(ei),σu(ei)dvg=MmF(uh24)(du),σudvg=Mmδdu,δ(F(uh24)σu)dvg=0.
13

It follows from the Weitzenböck formula that

k=1mRN(du(X),du(ek))du(ek)+du(RicM(X))=du(X)+˜2du(X),
14

where X is any smooth vector field on Mm. With respect to the variational vector field du(VA) along u, it follows from (13) and (14) that

AI(du(VA),du(VA))=MF(uh24)A˜du(VA),σu2dvg+MF(uh24)i,j,Ah(˜eidu(VA),˜ejdu(VA))h(du(ei),du(ej))dvg+MF(uh24)i,j,Ah(˜eidu(VA),du(ej))h(˜eidu(VA),du(ej))dvg+MF(uh24)i,j,Ah(˜eidu(VA),du(ej))h(du(ei),˜ejdu(VA))dvgMF(uh24)Ah(du(RicMm(VA)),σu(VA))dvg+MF(uh24)Ah((˜2du)(VA),σu(VA))dvg.
15

For any fixed point P ∈ M, choose {ei} such that eiej|P = 0. We have

˜2du(VA)=˜ei˜ei(du(VA))2˜ei(du(eiVA))+du(eieiVA)
16

and

MF(uh24)A,ih(˜ei˜eidu(VA),σu(VA))dvg=MA,ih(˜eidu(VA),˜ei[F(uh24)σu(VA)])dvg=MA,ih(˜eidu(VA),˜ei[F(uh24)]σu(VA))dvgMA,ih(˜eidu(VA),F(uh24)˜eiσu(VA))dvg=MA,ih(˜eidu(VA),˜ei[F(uh24)]σu(VA))dvgMF(uh24)A,i,jh(˜eidu(VA),˜eidu(ej))h(du(VA),du(ej))dvgMF(uh24)A,i,jh(˜eidu(VA),du(ej))h(˜eidu(VA),du(ej))dvgMF(uh24)A,i,jh(˜eidu(VA),du(ej))h(du(VA),˜eidu(ej))dvg.
17

Substituting (16) and (17) into (15), we have

AI(du(VA),du(VA))=M{F(uh24)A˜du(VA),σu2h(˜eidu(VA),˜ei[F(uh24)]σu(VA))}dvg+MF(uh24)h(2˜ei(du(eiVA))+du(eieiVA)du(RicMm(VA)),σu(VA))dvg+MF(uh24)i,j,Ah(˜eidu(VA),˜ejdu(VA))h(du(ei),du(ej))dvg+MF(uh24)i,j,Ah(˜eidu(VA),du(ej))h(du(ei),˜ejdu(VA))dvgMF(uh24)A,i,jh(˜eidu(VA),˜eidu(ej))h(du(VA),du(ej))dvgMF(uh24)A,i,jh(˜eidu(VA),du(ej))h(du(VA),˜eidu(ej))dvg.
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 {eiem+1} such that Bijλiδij.

A straightforward computation shows

Ah(˜eidu(VA),˜ei[F(uh24)]σu(VA))=AF(uh24)˜ei(uh24)h(vAm+1Bikdu(ek)+vAk˜eidu(ek),vAlσu(el))=F(uh24)˜ei(uh24)h(˜eidu(ek),σu(ek))=F(uh24)˜eidu,σu2
19

and

AF(uh24)˜du(VA),σu2=AF(uh24)vAm+1Bikdu(ek)+vAk˜eidu(ek),σu(ei)2=AF(uh24){BikBjlh(du(ek),σu(ei))h(du(el),σu(ej))+h(˜eidu(ek),σu(ei))h(˜ejdu(ek),σu(ej))}=AF(uh24){λiλjh(du(ei),σu(ei))h(du(ej),σu(ej))+˜eidu,σu2}.
20

Then it follows from (19) and (20) that

M{F(uh24)A˜du(VA),σu2h(˜eidu(VA),˜ei[F(uh24)]σu(VA))}dvg=MF(uh24)λiλjh(du(ei),σu(ei))h(du(ej),σu(ej))dvg.
21

From the Gauss equation it follows that

RicM(VA)=vAi(BkkBijBikBjk)ej.
22

Using (10), (11),(12) and (22), we have

MF(uh24)h(2˜ei(du(eiVA))+du(eieiVA)du(RicMm(VA)),σu(VA))dvg=MF(uh24){[h(2vAkBikBijdu(ej)vAm+1ei(Bij)du(ej)vAm+1Bij˜eidu(ej),vAlσu(el))]+h(vAkBikBijdu(ej)+vAm+1(eiBij)du(ej),vAlσu(el))+h(vAkBikBijdu(ej)vAiBkkBijdu(ej),vAlσu(el))}dvg=MF(uh24){h(2vAkBikBijdu(ej)vAiBkkBijdu(ej),vAlσu(el))}dvg=MF(uh24)i{[2λi(kλk)]λih(du(ei),σu(ei))}dvg.
23

A straightforward computation shows

MF(uh24)i,j,Ah(˜eidu(VA),˜ejdu(VA))h(du(ei),du(ej))dvg=MF(uh24)h(vAm+1Bikdu(ek)+vAk˜eidu(ek),vAm+1Bjldu(el)+vAl˜ejdu(el))h(du(ei),du(ej))dvg=MF(uh24){BikBjlh(du(ek),du(el))h(du(ei),du(ej))+h(˜eidu(ek),˜ejdu(ek))h(du(ei),du(ej))}dvg=MF(uh24){λiλjh(du(ei),du(ej))h(du(ei),du(ej))+h(˜eidu(ek),˜ejdu(ek))h(du(ei),du(ej))}dvg
24

and

MF(uh24)i,j,Ah(˜eidu(VA),du(ej))h(du(ei),˜ejdu(VA))dvg=MF(uh24){h(vAm+1Bikdu(ek)+vAk˜eidu(ek),du(ej))×h(du(ei),vAm+1Bjkdu(ek)+vAk˜ejdu(ek))}dvg=MF(uh24){λiλjh(du(ei),du(ej))h(du(ei),du(ej))+h(˜eidu(ek),du(ej))h(˜ejdu(ek),du(ei))}dvg
25

and

MF(uh24)A,i,jh(˜eidu(VA),˜eidu(ej))h(du(VA),du(ej))dvg=MF(uh24){h(vAm+1Bikdu(ek)+vAk˜eidu(ek),˜eidu(ej))×h(vAldu(el),du(ej))}dvg=MF(uh24)h(˜eidu(ek),˜eidu(ej))h(du(ek),du(ej))dvg
26

and

MF(uh24)A,i,jh(˜eidu(VA),du(ej))h(du(VA),˜eidu(ej))dvg=MF(uh24){h(vAm+1Bikdu(ek)+vAk˜eidu(ek),du(ej))×h(vAldu(el),˜eidu(ej))}dvg=MF(uh24)h(˜eidu(ek),du(ej))h(du(ek),˜eidu(ej))dvg.
27

From (18), (21), (23), (24), (25), (26), (27) and ˜eidu(ej)=˜ejdu(ei), we obtain

AI(du(VA),du(VA))=MF(uh24)λiλjh(du(ei),σu(ei))h(du(ej),σu(ej))dvg+MF(uh24)i{[2λi(kλk)]λih(du(ei),σu(ei))}dvg+2MF(uh24)λiλjh(du(ei),du(ej))h(du(ei),du(ej))dvgMF(uh24)λiλjh(du(ei),σu(ei))h(du(ej),σu(ej))dvg+MF(uh24)i{[2λi(kλk)]λih(du(ei),σu(ei))}dvg+2MF(uh24)λiλmh(du(ei),σu(ei))dvg=MF(uh24)λiλjh(du(ei),σu(ei))h(du(ej),σu(ej))dvg+MF(uh24)i{[2λi+2λm(kλk)]λih(du(ei),σu(ei))}dvg.
28

If F(t) = F(t), then (28) leads to the following inequality:

AI(du(VA),du(VA))MF(uh24)λm2uh4dvg+MF(uh24)max1im{[2λi+2λm(kλk)]λi}uh2dvg=MF(uh24)uh2{λm2uh2+max1im{[2λi+2λm(kλk)]λi}}dvg.
29

If there exists a constant cF such that F(t)tcF is nonincreasing, it follows that F(t)t ≤ cFF(t) on t ∈ (0, ∞), thus (28) implies

AI(du(VA),du(VA))M4cFF(uh24)λm2uh2dvg+MF(uh24)max1im{[2λi+2λm(kλk)]λi}uh2dvg=MF(uh24)uh2{4cFλm2+max1im{[2λi+2λm(kλk)]λi}}dvg.
30

If u is nonconstant and (6) or (7) holds, we have

AI(du(VA),du(VA))<0
31

and u is unstable.

Corollary 3.2

Let u:Sm → N be a nonconstant F-stationary map and m > 4. If cF<m41 or uh2 < m − 4, then u is unstable.

F-stationary maps into compact convex hypersurfaces

In this section, we obtain the following result.

Theorem 4.1

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

Proof

In order to prove the instability of u:Nn → Mm, we need to consider some special variational vector fields along u. To do this, we choose an orthonormal field {ϵαϵm+1}, α = 1, …, m, of Rm+1 such that {ϵα} are tangent to Mm ⊂ Rm+1, ϵm+1 is normal to Mm, ϵαMmϵβ|P=0 and Bαβλαδαβ, where Bαβ denotes the components of the second fundamental form of Mm in Rm+1. Meanwhile, take a fixed orthonormal basis EA, A = 1, …, m + 1, of Rm+1 and set

VA=α=1mvAαϵα,vAα=EA,ϵα,vAm+1=EA,ϵm+1,
32

where 〈 ⋅ ,  ⋅ 〉 denotes the canonical Euclidean inner product. We shall consider the second variation

AI(VA,VA)=NF(uh24)˜VA,σu˜VA,σudvg+NF(uh24)i,j=1mh(˜eiVA,˜ejVA)h(du(ei),du(ej))dvg+NF(uh24)i,j=1mh(˜eiVA,du(ej))h(˜eiVA,du(ej))dvg+NF(uh24)i,j=1mh(˜eiVA,du(ej))h(du(ei),˜ejVA)dvg+NF(uh24)ih(RMm(VA,du(ei))VA,σu(ei))dvg,
33

where {e1, …, en} is the local orthonormal frame of Nn.

Firstly, we compute the first term of (33)

ANF(uh24)˜VA,σu˜VA,σudvg=NF(uh24)[ih(˜eiVA,σu(ei))]2dvg=NF(uh24)[ih(Mmdu(ei)VA,σu(ei))]2dvg=NF(uh24)[iuiαh(MmϵαVA,σu(ei))]2dvg=NF(uh24)[ivAm+1uiαBαβh(ϵβ,σu(ei))]2dvg=NF(uh24)[ivAm+1uiαλαh(ϵα,σu(ei))]2dvg=NF(uh24)λαλβh(uiαϵα,σu(ei))h(ujβϵβ,σu(ej))dvg.
34

The second term of (33)

ANF(uh24)i,j=1mh(˜eiVA,˜ejVA)h(du(ei),du(ej))dvg=NF(uh24)h(Mmdu(ei)VA,Mmdu(ej)VA)h(du(ei),du(ej))dvg=NF(uh24)uiαujβh(MmϵαVA,MmϵβVA)h(du(ei),du(ej))dvg=NF(uh24)uiαujβBαγBβδh(ϵγ,ϵδ)h(du(ei),du(ej))dvg=NF(uh24)λαλβh(uiαϵα,ujβϵβ)h(du(ei),du(ej))dvg=NF(uh24)λα2h(uiαϵα,du(ej))h(du(ei),du(ej))dvg.
35

The third term of (33)

ANF(uh24)i,j=1mh(˜eiVA,du(ej))h(˜eiVA,du(ej))dvg=NF(uh24)λαλβh(uiαϵα,du(ej))h(uiβϵβ,du(ej))dvg.
36

The fourth term of (33)

ANF(uh24)i,j=1mh(˜eiVA,du(ej))h(du(ei),˜ejVA)dvg=NF(uh24)λαλβh(uiαϵα,du(ej))h(du(ei),ujβϵβ)dvg.
37

The fifth term of (33)

ANF(uh24)ih(RMm(VA,du(ei))VA,σu(ei))dvg=NF(uh24)vAαvAβh(RMm(ϵα,du(ei))ϵβ,σu(ei))dvg=NF(uh24)uiγujδh(RMm(ϵα,ϵγ)ϵα,ϵδ)h(du(ei),du(ej))dvg=NF(uh24)uiγujδ[BαδBγαBααBγδ]h(du(ei),du(ej))dvg=NF(uh24)uiαujα[λα2(βλβ)λα]h(du(ei),du(ej))dvg=NF(uh24)[λα2(βλβ)λα]h(uiαϵα,ujγϵγ)h(du(ei),du(ej))dvg=NF(uh24)[λα2(βλβ)λα]h(uiαϵα,du(ej))h(du(ei),du(ej))dvg.
38

From (33)-(38), we have

AI(VA,VA)=NF(uh24)λαλβh(uiαϵα,σu(ei))h(ujαϵα,σu(ej))dvg+NF(uh24)[2λα2(βλβ)λα]h(uiαϵα,du(ej))h(du(ei),du(ej))dvg+NF(uh24)λαλβh(uiαϵα,du(ej))h(uiβϵβ,du(ej))dvg+NF(uh24)λαλβh(uiαϵα,du(ej))h(du(ei),ujβϵβ)dvgNF(uh24)λαλβh(uiαϵα,σu(ei))h(ujαϵα,σu(ej))dvg+NF(uh24)[2λα2(βλβ)λα]h(uiαϵα,du(ej))h(du(ei),du(ej))dvg+NF(uh24)2λαλmh(uiαϵα,du(ej))h(du(ei),du(ej))dvgNF(uh24)λαλβh(uiαϵα,σu(ei))h(ujαϵα,σu(ej))dvg+NF(uh24)[2λα2+2λαλm(βλβ)λα]h(uiαϵα,σu(ei))dvg.
39

If F(t) = F(t), then (39) leads to the following inequality:

AI(VA,VA)NF(uh24)uh2{uh2λm2+max1αm[2λα2+2λαλm(βλβ)λα]}dvg.
40

If there exists a constant cF such that F(t)tcF is nonincreasing, it follows that F(t)t ≤ cFF(t) on t ∈ (0, ∞), thus (39) implies

AI(VA,VA)MF(uh24)uh2{4cFλm2+max1αm{[2λα+2λm(βλβ)]λα}}dvg.
41

Now, if u:N → Mm is a nonconstant F-stationary map and (6) or (7) holds, then, from (41) or (40), we know that AI(VAVA) < 0 and u is unstable.

Corollary 4.2

Let u:N → Sm be a nonconstant F-stationary map with m > 4, where N is any compact Riemannian manifold. If cF<m41 or uh2 < m − 4, then u is unstable.

Conclusions

In this paper, we investigate F-stationary maps between the compact convex hypersurface Mm and any compact Riemannian manifold N. Assume that the principal curvatures λi of Mm satisfy 0 < λ1 ≤  ⋯  ≤ λm and 3λm<i=1m1λi, then every nonconstant F-stationary map from Mm to N or from N to Mm 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 Sm as a special case of compact convex hypersurfaces and obtain similar inferences.

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).

Footnotes

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Contributor Information

Jing Li, moc.361@999321gnijil.

Fang Liu, nc.ude.tsujn@87gnafuil.

Peibiao Zhao, nc.ude.tsujn@oahzbp.

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