12. On Generalized Conformal Maps

ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 4 Issue 4 (2012), Pages 99-108

ON GENERALIZED CONFORMAL MAPS

(COMMUNICATED BY KRISHNAN DUGGAL)

A. M. CHERIF, H. ELHENDI AND M. TERBECHE

Abstract. In this paper, we present some new properties of generalized con-formal maps as f-biharmonic between equidimensional Riemannian manifolds.

1. _Introduction

Consider a smooth mapϕ :M −→ N between Riemannian manifolds M = (Mm_{, g) and N = (N}n_{, h) and f : M ×N −→ (0,+∞) be a smooth positive} function, then thef-energy functional ofϕis defined by

Ef(ϕ) = 1 2 Z

M

f(x, ϕ(x))|dxϕ|2vg

(or over any compact subsetK⊂M).

A map is calledf-harmonic if it is a critical point of thef-energy functional. By [1] the mapϕisf-harmonic if and only if

τf(ϕ) = fϕτ(ϕ) +dϕ(gradMfϕ)−e(ϕ)(gradNf)◦ϕ= 0, (1) wherefϕ:Mm−→(0,+∞) be a smooth positive function defined by

fϕ(x) =f(x, ϕ(x)), ∀x∈M,

τ(ϕ) =traceg∇dϕis the tension field ofϕ, ande(ϕ) =1₂|dϕ|2is the energy density ofϕ.

τf(ϕ) is called the f-tension field of ϕ, and thef-bi-energy functional of ϕ is defined by

E2,f(ϕ) = 1 2

Z

M

|τf(ϕ)|2_vg.

0_{2000 Mathematics Subject Classification: 53A45, 53C20, 58E20.}

Keywords and phrases. Conformal maps; f-harmonic maps; f-biharmonic maps. c

2012 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted August 21, 2012. Published November 6, 2012.

A mapϕis calledf-biharmonic if it is a critical point of the f-energy functional. By [1] the mapϕisf-bi-harmonic if and only if

τ2,f(ϕ) = −fϕtracegRN(τf(ϕ), dϕ)dϕ−traceg∇ϕfϕ∇ϕτf(ϕ) (2) +e(ϕ)(∇N

τf(ϕ)grad

N_{f)◦ϕ−dϕ(grad}M_τ

f(ϕ)(f))

−τf(ϕ)(f)τ(ϕ)+<∇ϕ_{τf(ϕ), dϕ >(grad}N_{f)◦ϕ= 0,}

whereh, iis the inner product onT∗M⊗ϕ−1T N andRN is the curvature tensor ofN.

τ2,f(ϕ) is called thef-bi-tension field of ϕ.

Thef-biharmonic concept is a natural generalization of biharmonic maps ([3][4][7][8]).

2. _Conformalf_{-bi-harmonic maps.}

Theorem A. Let ϕ: (Mn_{, g)−→(N}n_{, h) be a conformal map with dilation λ.}

The f-bi-tension field of ϕis given by

τ2,f(ϕ) = F1dϕ(gradMlnλ) +F2dϕ(gradM(fN ◦ϕ))

+2(n−2)f_ϕ2dϕ(RicciM(gradMlnλ))−2fϕdϕ(RicciM(gradMfϕ)) +nfϕdϕ(RicciM(gradM(fN ◦ϕ))) + 2(n−2)f_ϕ2<∇dϕ,∇dlnλ >

−2fϕ<∇dϕ,∇dfϕ>+nfϕ<∇dϕ,∇d(fN ◦ϕ)> +(n−2)fϕ2dϕ(gradM∆(lnλ))−fϕdϕ(gradM∆(fϕ)) +n

2fϕdϕ(grad

M_{∆(fN◦ϕ)) + 4(n−2)fϕ∇}ϕ gradM_f

ϕdϕ(grad

M_lnλ)

−(n−2)2f_ϕ2∇ϕ_gradM_lnλdϕ(grad

M_lnλ)

− ∇ϕ_gradM_fϕdϕ(grad

M_f ϕ) +n∇ϕ_gradMfϕdϕ(grad

M_{(fN ◦ϕ))−}n(n−2) 2 fϕ∇

ϕ

gradM(fN◦ϕ)dϕ(grad

M_lnλ)

−n(n−2)

2 ∇

ϕ

gradM_lnλdϕ(grad

M_(f

N◦ϕ))− n2

4 ∇ ϕ gradM_(f

N◦ϕ)dϕ(grad

M_(f N ◦ϕ))

+(n−2)dϕ(∇M

gradMlnλgrad

M_{(fN ◦ϕ)) + (n−2)dϕ(∇}M

gradM(fN◦ϕ)grad

M_lnλ)

−dϕ(∇M

gradM_fϕgrad

M_(f

N ◦ϕ))−dϕ(∇MgradM_(fN◦ϕ)grad

M_f ϕ) +n

2dϕ(grad

M_(|gradM_{(fN ◦ϕ)|}2_),

where

F1 = (n−2)|gradMfϕ|2+ (n−2)fϕ∆(fϕ)−(n−2)2g(gradMlnλ, gradM(fN◦ϕ))

+(n−2)g(gradMfϕ, gradM(fN ◦ϕ))−n(n−2)

2 |grad

M_(fN◦ϕ)|2_,

and

F2 = −(n−2)∆(lnλ) + ∆(fϕ)− n

Lemma A. Let ϕ: (Mn, g)−→(Nn, h)be a conformal map with dilationλ. The

f-tension field ofϕis given by

τf(ϕ) = (2−n)fϕdϕ(gradMlnλ) +dϕ(gradMfϕ)

−n

2dϕ grad M_(f

N◦ϕ)

,

where, at x∈M,fN :N−→(0,∞)defined byfN(y) =f(x, y) for ally∈N. Proof. Since τ(ϕ) = (2−n)dϕ(gradM_{lnλ), e(ϕ) =} n

2λ

2 _{and let {e}

i}ni=1 be an

orthonormal frame inM, then{fi}n

i=1, wherefi◦ϕ= 1

λdϕ(ei) for alli= 1, ..., n, be an orthonormal frame inN. Then

(gradNf)◦ϕ(x) = (gradNfN)ϕ(x)

= fi|ϕ(x)(fN)fi|ϕ(x)

= 1

λ2_(x)dϕ(ei)x(fN)dϕ(ei)x

= 1

λ2_(x)ei|x(fN ◦ϕ)dϕ(ei)x

= 1

λ2_(x)dϕ grad

M_{(fN ◦ϕ)} x,

for allx∈M.

Lemma B. Let γ:M −→Rbe a smooth function, then traceg(∇ϕ₎2_dϕ(gradM_{γ) = (2−n)∇}ϕ

gradM_γdϕ(grad

M_{lnλ) + 2dϕ(Ricci}M_gradM_γ)

−tracegRN(dϕ(gradMγ), dϕ)dϕ+ 2<∇dϕ,∇dγ > +dϕ(gradM∆(γ)),

where<∇dϕ,∇dγ >=∇dϕ(ei, ej)∇dγ(ei, ej) =∇dϕ(ei, ej)g(∇M eigrad

M_{γ, ej).}

Proof. Fix a point x0 ∈ M and let {ei}ni=1 be an orthonormal frame, such that

∇M

eiej= 0 atx0 for alli, j= 1, ..., n. Then calculating atx0, we have

traceg(∇ϕ₎2_dϕ(gradM_{γ) = ∇}ϕ ei∇

ϕ

eidϕ(grad

M_γ)

= ∇ϕei∇dϕ(ei, grad

M_{γ) +}

∇ϕeidϕ(∇

M eigrad

M_{γ) (3)}

= ∇ϕ

ei∇dϕ(ei, grad

M_{γ) +∇dϕ(ei,∇}M eigrad

M_γ)

+dϕ(∇Mei∇

M eigrad

M_γ)

= ∇ϕ

ei∇dϕ(ei, grad

M_{γ)+<∇dϕ,∇dγ >}

+dϕ(traceg(∇M)2gradMγ) Let∇2_{dϕbe the third fundamental form defined by}

∇2dϕ(X, Y, Z) =∇ϕ_X∇dϕ(Y, Z)− ∇dϕ(∇ϕ_XY, Z)− ∇dϕ(Y,∇ϕ_XZ), (4) for allX, Y, Z∈Γ(T M). And we have

∇2dϕ(X, Y, Z) = ∇2dϕ(Z, Y, X) +dϕ(RM(Z, X)Y) (5)

By (4) and (5), the first term of (3) is

∇ϕ

ei∇dϕ(ei, grad

M_{γ) = ∇}2_{dϕ(ei, ei, grad}M_{γ) +∇dϕ(ei,∇}M eigrad

M_γ)

= ∇2dϕ(gradMγ, ei, ei) +dϕ(RM(gradMγ, ei)ei)

−RN(dϕ(gradMγ), dϕ(ei))dϕ(ei) +∇dϕ(ei,∇Meigrad

M γ)

= ∇ϕ_gradMγτ(ϕ) +dϕ(Ricci

M_gradM_γ)

−tracegRN(dϕ(gradMγ), dϕ)dϕ+<∇dϕ,∇dγ > .

From the equation

traceg(∇M₎2_gradM_γ=RicciM_(gradM_{γ) +grad}M_(∆γ),

the Lemma B follows.

Lemma C. Letγ:M −→Rbe a smooth function, then

<∇dϕ(gradMγ), dϕ >=nλ2g(gradMγ, gradMlnλ) +λ2∆(γ).

Proof. Let{ei}n

i=1 be an orthonormal frame, then

<∇dϕ(gradMγ), dϕ > = h(∇eidϕ(grad

M

γ), dϕ(ei))

= ei(h(dϕ(gradMγ), dϕ(ei)))

−h(dϕ(gradMγ),∇eidϕ(ei))

= ei(λ2g(gradMγ, ei))

−(2−n)h(dϕ(gradMγ), dϕ(gradMlnλ)) = g(gradMγ, gradMλ2)

+(n−2)λ2g(gradMγ, gradMlnλ) +λ2ei(g(gradMγ, ei))

= 2λ2g(gradMγ, gradMlnλ)

+(n−2)λ2g(gradMγ, gradMlnλ) +λ2∆(γ)

= nλ2g(gradMγ, gradMlnλ) +λ2∆(γ).

Let {ei}n

i=1 be an orthonormal frame, such that ∇

M

eiej = 0 at x0 ∈ M for all

i, j= 1, ..., n. Then calculating atx0, we have

−(2−n)traceg∇ϕ_fϕ∇ϕ_fϕdϕ(gradM_{lnλ) = (n−2)∇}ϕ eifϕ∇

ϕ

eifϕdϕ(grad

M_lnλ) = (n−2)∇ϕ_gradM_fϕfϕdϕ(gradMlnλ)

+(n−2)fϕ∇ϕ ei∇

ϕ

eifϕdϕ(grad

M_lnλ) = (n−2)|gradMfϕ|2dϕ(gradMlnλ)

+(n−2)fϕ∇ϕ_gradM_f

ϕdϕ(grad

M_lnλ)

+(n−2)fϕ∆(fϕ)dϕ(gradMlnλ) +2(n−2)fϕ∇ϕ_gradM_fϕdϕ(grad

M_lnλ)

+(n−2)f_ϕ2traceg(∇ϕ₎2_dϕ(gradM_lnλ),

and by the lemma B, we obtain

−(2−n)traceg∇ϕfϕ∇ϕfϕdϕ(gradMlnλ) = (n−2)|gradMfϕ|2dϕ(gradMlnλ) +3(n−2)fϕ∇ϕ_gradM_f

ϕdϕ(grad

M_lnλ) +(n−2)fϕ∆(fϕ)dϕ(gradMlnλ) +(n−2)f_ϕ2n(2−n)∇ϕ_gradM_lnλdϕ(grad

M_ln(6)λ)

+2dϕ(RicciMgradMlnλ)

−tracegRN(dϕ(gradMlnλ), dϕ)dϕ

+2<∇dϕ,∇dlnλ >+dϕ(gradM∆(lnλ))o.

Of the same method, we obtain

−traceg∇ϕfϕ∇ϕdϕ(gradMfϕ) = −∇ ϕ

gradM_fϕdϕ(grad

M_f ϕ)

−fϕn(2−n)∇ϕ_gradM_f

ϕdϕ(grad

M_lnλ)

+2dϕ(RicciMgradMfϕ) (7)

−tracegRN(dϕ(gradMfϕ), dϕ)dϕ +2<∇dϕ,∇dfϕ>+dϕ(gradM∆(fϕ))o,

n

2traceg∇ ϕ_f

ϕ∇ϕdϕ(gradM(fN◦ϕ)) = n 2∇

ϕ

gradM_fϕdϕ(grad

M_(f N ◦ϕ))

+n 2fϕ

n

(2−n)∇ϕ_gradM_(f

N◦ϕ)dϕ(grad

M_lnλ)

+2dϕ(RicciMgradM(fN ◦ϕ)) (8)

−tracegRN(dϕ(gradM(fN ◦ϕ)), dϕ)dϕ

by Lemma A we have

e(ϕ)∇N

τf(ϕ)grad

N_{f =} n 2λ

2n_(2−n)∇ϕ gradM_lnλ

1

λ2dϕ(grad

M_(f N ◦ϕ))

+∇ϕ_gradM_fϕ

1

λ2dϕ(grad

M_(f N◦ϕ))

−n

2∇ ϕ

gradM_(fN◦ϕ)

1

λ2dϕ(grad

M_(f N ◦ϕ))

o

= n

2λ

2n−2(2−n)

λ2 |grad

M_lnλ|2_dϕ(gradM_{(fN ◦ϕ)) (9)}

+2−n λ2 ∇

ϕ

gradM_lnλdϕ(grad

M_(fN◦ϕ))

− 2

λ2(grad

M_{fϕ)(lnλ)dϕ(grad}M_(fN◦ϕ))

+ 1 λ2∇

ϕ gradM_f

ϕdϕ(grad

M_(fN◦ϕ))

+n λ2(grad

M_(f

N ◦ϕ))(lnλ)dϕ(gradM(fN ◦ϕ))

− n

2λ2∇

ϕ gradM_(f

N◦ϕ)dϕ(grad

M_{(fN ◦ϕ))}o_.

The functionτf(ϕ)(f) is given by

τf(ϕ)(f) = τf(ϕ)(fN)

= (2−n)(gradMlnλ)(fN ◦ϕ) + (gradMfϕ)(fN◦ϕ)

−n

2|grad

M_{(fN ◦ϕ)|}2_,

then

−dϕ(gradMτf(ϕ)(f)) = (n−2)dϕ(gradM((gradMlnλ)(fN ◦ϕ)))

−dϕ(gradM((gradMfϕ)(fN ◦ϕ))) +n

2dϕ(grad

M_(|gradM_(f

N ◦ϕ)|2)) = (n−2)dϕ(∇M

gradM_lnλgradM(fN ◦ϕ)) (10)

+(n−2)dϕ(∇M_gradM_(f

N◦ϕ)grad

M lnλ)

−dϕ(∇M gradM_f

ϕgrad

M_(fN◦ϕ))

−dϕ(∇M

gradM_(fN◦ϕ)grad

M_f ϕ) +n

2dϕ(grad

M_(|gradM_{(fN ◦ϕ)|}2_)),

−τf(ϕ)(f)τ(ϕ) = (n−2) n

(2−n)(gradMlnλ)(fN ◦ϕ) + (gradMfϕ)(fN◦ϕ)

−n

2|grad M_(f

N ◦ϕ)|2 o

Then by lemma C, we have

<∇τf(ϕ), dϕ > (gradNf)◦ϕ = n(2−n)nλ2|gradMlnλ|2_{+ (2−n)λ}2_∆(lnλ)

+nλ2g(gradMfϕ, gradMlnλ) +λ2∆(fϕ)

−n

2

2 λ

2_g(gradM_(f

N ◦ϕ), gradMlnλ) (12)

−n

2λ

2_∆(f

N◦ϕ) o1

λ2dϕ(grad

M_(f

N ◦ϕ)).

The TheoremA, follows from (2), (6), (7), (8), (9), (10), (11) and (12).

Particular Cases

(1) Iff = 1, we obtain

τ2,f(ϕ) = τ2(ϕ)

= 2(n−2)dϕ(RicciM(gradMlnλ)) + 2(n−2)<∇dϕ,∇dlnλ > +(n−2)dϕ(gradM∆(lnλ))−(n−2)∇ϕ_gradM_lnλdϕ(grad

M_lnλ).

Is the natural bi-tension field ofϕ.

(2) If f1 :M −→(0,∞), be a smooth positif function. If f(x, y) =f1(x) for

all (x, y)∈M×N, then thef-bi-tension field ofϕis given by

τ2,f(ϕ) = τ2,f1(ϕ)

= (n−2)|gradMf1|2dϕ(gradMlnλ) + (n−2)f1∆(f1)dϕ(gradMlnλ)

+2(n−2)f₁2dϕ(RicciM(gradMlnλ))−2f1dϕ(RicciM(gradMf1))

+2(n−2)f₁2<∇dϕ,∇dlnλ >−2f1<∇dϕ,∇df1>

+(n−2)f₁2dϕ(gradM∆(lnλ))−f1dϕ(gradM∆(f1))

+4(n−2)f1∇

ϕ gradM_f

1dϕ(grad

M

lnλ)−(n−2)2f12∇

ϕ

gradM_lnλdϕ(grad

M lnλ)

−∇ϕ_gradM_f1dϕ(grad

M_f

1),

Theorem B. Let ϕ : (Mn, g) −→ (Nn, h) be a conformal map with dilation λ. Thenϕisf-bi-harmonic if and only if

0 = F3gradMlnλ+F4gradM(fN◦ϕ) +F5gradMfϕ

+2(n−2)fϕ2RicciM(gradMlnλ)−2fϕRicciM(gradMfϕ) +nfϕRicciM(gradM(fN ◦ϕ))−4fϕ∇M

gradM_lnλgradMfϕ

+(n−2)fϕ2grad M

∆(lnλ)−fϕgradM∆(fϕ) +n 2fϕgrad

M

∆(fN◦ϕ)

+4(n−2)fϕ∇M

gradMfϕgrad

M_lnλ−n(n−2) 2 fϕ∇

M

gradM(fN◦ϕ)grad

M_lnλ

+(n−2)∇M gradM_(f

N◦ϕ)grad

M_{lnλ− ∇}M gradM_(f

N◦ϕ)grad

M_fϕ

−(n−2)(n−6)

2 f

2

ϕgrad

M_(|gradM_lnλ|2₎

+2nfϕ−

(n−2)2

2

∇M

gradM_lnλgrad

M_(f N ◦ϕ)

−1

2grad

M_(|gradM_f

ϕ|2) + (n−1)∇MgradM_fϕgrad

M_(f N ◦ϕ)

−n(n−2)

4 grad

M_(|gradM_{(fN ◦ϕ)|}2_),

where

F3 = F1−2(n−2)fϕ2∆(lnλ) + 2fϕ∆(fϕ)−nfϕ∆(fN ◦ϕ)

−(n−2)2f_ϕ2|gradMlnλ|2_+|gradM_fϕ|2

−ng(gradMfϕ, gradM(fN◦ϕ)) +n

2

4 |grad

M_{(fN ◦ϕ)|}2_,

F4 = F2+ng(gradMfϕ, gradMlnλ)−

n(n−2)

2 (fϕ+ 1)|grad M_lnλ|2

−n

2

2 g(grad

M_{(fN ◦ϕ), grad}M_lnλ),

and

F5 = 4(n−2)fϕ|gradMlnλ|2−2g(gradMfϕ, gradMlnλ)

+ng(gradMlnλ, gradM(fN ◦ϕ)).

Proof. The Theorem B follows from Theorem A and the following formulae h(<∇dϕ,∇dlnγ >, dϕ(X)) = g(2λ2∇gradM_lnλgradMlnγ (13)

−λ2∆(lnγ)gradMlnλ, X),

h(∇ϕ_Xdϕ(gradMlnγ), dϕ(Y)) = g(λ2X(lnλ)gradMlnγ−λ2X(lnγ)gradMlnλ + λ2dlnλ(gradMlnγ)X+λ2∇M

Xgrad

M_{lnγ, Y),} (14)

Corollary A. Let ϕ : (Mn, g) −→ (Nn, h) be a conformal map with constant dilation λ. Then ϕ is f-bi-harmonic if and only if the function f satisfies the equation

0 = ∆(fϕ)−n

2∆(fN ◦ϕ)

gradM(fN◦ϕ)−2fϕRicciM(gradMfϕ)

+nfϕRicciM(gradM(fN ◦ϕ))−fϕgradM∆(fϕ) +n 2fϕgrad

M_{∆(fN ◦ϕ)}

−∇M gradM_(f

N◦ϕ)grad

M_fϕ−1 2grad

M_(|gradM_fϕ|2₎

+(n−1)∇M

gradM_fϕgrad

M_{(fN ◦ϕ)−}n(n−2) 4 grad

M_(|gradM_{(fN ◦ϕ)|}2_).

Corollary B. Let ϕ : (Mn_{, g) −→ (N}n_{, h) be a conformal map with constant}

dilation λ. Iffϕ=fN◦ϕ=γ, thenϕ isf-bi-harmonic if and only if the function

γ satisfies the equation

0 = 2−n

2 ∆(γ)grad

M_{γ−(2−n)γ Ricci}M_(gradM_γ)

−2−n

2 γ grad

M_∆(γ)−(2−n)2

4 +

1 2

gradM(|gradMγ|2_).

Example A. Letϕ:R2−→(N2, h)be a conformal map with constant dilationλ.

If fϕ=fN ◦ϕ=γ, thenϕisf-bi-harmonic if and only if

( _∂γ ∂x

∂2γ ∂x2 +

∂γ ∂y

∂2γ ∂x∂y = 0, ∂γ

∂y ∂2_γ

∂y2 +

∂γ ∂x

∂2_γ

∂x∂y = 0.

Example B. Let ϕ: (M2_{, g)−→(N}2_{, h)be a conformal map with dilation λ. If}

fϕ =fN ◦ϕ= lnλ, then ϕisf-bi-harmonic if and only if the function λsatisfies the equation

gradM(|gradMlnλ|2) = 0.

Moreover, if M is related, then ϕ is f-bi-harmonic if and only if the function |gradM_{lnλ| is constant inM.}

Acknowledgements. The authors are highly thankful to the referee for his valuable suggestions towards the improvement of the paper.

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Ahmed Cherif Mohamed

Laboratory of Geometry, Analysis, Contrle and Applications. Saida University,BP 138, 200000, Algeria

E-mail address:[email protected]

Elhendi Hichem

Laboratory of Geometry, Analysis, Controle and Applications. Saida University,BP 138, 200000, Algeria.

E-mail address:[email protected]

Terbeche Mekki