f-Harmonic Morphisms Between Riemannian Manifolds?

Yelin OU

1 f-Harmonic Maps vs.F-Harmonic Maps

1.1 f-Harmonic maps

Letf:(M,g)→(0,∞)be a smooth function.Anf-harmonic map is a mapφ:(Mm,g)→(Nn,h)between Riemannian manifolds such thatφ|Ωis a critical point of thef-energy(see[10,18]),

for every compact domain Ω?M.The Euler-Lagrange equation gives thef-harmonic map equation(see[7,23])

whereτ(φ)=Trg?dφis the tension field ofφvanishing of which meansφis a harmonic map.

Example 1.1Let?,ψ,φ:R3→R2be defined as

Then,one can easily check that both?andψaref-harmonic map withf=?is a horizontally conformal submersion whilstψis not.Also,φis anf-harmonic map withf=,which is a submersion but not horizontally weakly conformal.

1.2 F-Harmonic map

LetF:[0,+∞)→[0,+∞)be aC2-function,strictly increasing on(0,+∞),and let?:(M,g)→(N,h)be a smooth map between Riemannian manifolds.Then?is said to be anF-harmonic map if?|Ωis a critical point of theF-energy functional

for every compact domain Ω?M.The equation ofF-harmonic maps is given by(see[2])

whereτ(?)denotes the tension field of?.

Harmonic maps,p-harmonic maps and exponential harmonic maps are examples ofF-harmonic maps withF(t)=t,F(t)=(p>4),andF(t)=et,respectively(see[2]).

In particular,p-harmonic map equation can be written as

1.3 Relationship between f-harmonic and F-harmonic maps

We can see from(1.1)that anf-harmonic map withf=const>0 is nothing but a harmonic map,so bothf-harmonic maps andF-harmonic maps are generalizations of harmonic maps.Though we were warned in[7]thatf-harmonic maps should not be confused withF-harmonic maps andp-harmonic maps,we observe that,apart from critical points,anyF-harmonic map is a specialf-harmonic map.More precisely we have the following corollary.

Corollary 1.1Any F-harmonic map ?:(M,g)→(N,h)without critical points,i.e.,|d?0for all x∈M,is an f-harmonic map with f=In particular,a pharmonic map without critical points is an f-harmonic map with f=

ProofSinceFis aC2-function and strictly increasing on(0,+∞)we have(t)>0 on(0,+∞).If theF-harmonic map?:(M,g)→(N,h)has no critical points,i.e.,for allx∈M,then the functionf:(M,g)→(0,+∞)withf=is smooth and we see from(1.1)–(1.2)that theF-harmonic map?is anf-harmonic map withf=The second statement follows from the fact that for ap-harmonic map,F(t)=and hence

Another relationship betweenf-harmonic maps and harmonic maps can be characterized as follows.

Corollary 1.2(see[18])A map φ:(Mm,g)→(Nn,h)with2is f-harmonic if and only if φ:→(Nn,h)is a harmonic map.

1.4 A physical motivation for the study of f-harmonic maps

In physics,the equation of motion of a continuous system of spins with inhomogeneous neighbor Heisenberg interaction(such a model is called the inhomogeneous Heisenberg ferromagnet)is given by

where Ω?Rmis a smooth domain in the Euclidean space,fis a real-valued function defined on Ω,u(x,t)∈S2,×denotes the cross products in R3and Δ is the Laplace operator on Rm.Physically,the functionfis called the coupling function,and is the continuum limit of the coupling constants between the neighboring spins.Sinceuis a map intoS2it is well-known that the tension field ofucan be written asτ(u)= Δu+|?u|2u,and one can easily check that the right-hand side of the inhomogeneous Heisenberg spin system(1.4)can be written asu×(fτ(u)+?f·?u).It follows thatuis a smooth stationary solution of(1.4)if and only iffτ(u)+?f·?u=0,i.e.,uis anf-harmonic map.So there exists a 1-1 correspondence between the set of the stationary solutions of the inhomogeneous Heisenberg spin system(1.4)on the domain Ω and the set off-harmonic maps from Ω into 2-sphere.The above inhomogeneous Heisenberg spin system(1.4)is also called inhomogeneous Landau-Lifshitz system(see[5–6,9,14,16–17]).

Using Corollary 1.2 we have the following example which provides many stationary solutions of the inhomogeneous Heisenberg spin system defined on R3.

Example 1.2u:→(Nn,h)is anf-harmonic map if and only if

is a harmonic map.In particular,there is a 1-1 correspondence between harmonic maps from 3-sphere

andf-harmonic maps withf=from Euclidean 3-space R3→(Nn,h).When(Nn,h)=S2,we have a 1-1 correspondence between the set of harmonic mapsS3→S2and the set of stationary solutions of the inhomogeneous Heisenberg spin system on R3.Similarly,there exists a 1-1 correspondence between harmonic maps from hyperbolic 3-space

andf-harmonic mapsfrom the unit disk in Euclidean 3-space.

1.5 A little more about f-harmonic maps

Corollary 1.3If φ:(Mm,g)→(Nn,h)is an f1-harmonic map and also an f2-harmonic map,thengrad()∈kerdφ.

ProofThis follows from

and hence

Proposition 1.1A conformal immersion φ:(Mm,g)→(Nn,h)withis f-harmonic if and only if it is m-harmonic and f=In particular,an isometric immersion is f-harmonic if and only if f=const and hence it is harmonic.

ProofIt is not difficult to check(see[24])that for a conformal immersionφ:(Mm,g)→(Nn,h)withthe tension field is given by

so we can compute thef-tension field to have

whereηis the mean curvature vector of the submanifoldφ(M)?N.Noting thatηis normal part whilst dφ(grad lnf)is the tangential part ofτf(φ),we conclude thatτf(φ)=0 if and only if

It follows thatη=0 and grad=0 sinceφis an immersion.From these we see thatφis a minimal conformal immersion which means it is anm-harmonic map(see[24])and thatf=Thus,we obtain the first statement.The second statement follows from the first one withλ=1.

2 f-Harmonic Morphisms

A horizontally weakly conformal map is a map?:(M,g)→(N,h)between Riemannian manifolds such that for eachx∈Mat whichthe restrictionis conformal and surjective,where the horizontal subspaceis the orthogonal complement ofVx=kerd?xinTxM.It is not difficult to see that there exists a numberλ(x)∈(0,∞)such thath(d?(X),d?(Y))=(x)g(X,Y)for anyX,Y∈Hx.At the pointx∈Mwhere d?x=0 one can letλ(x)=0 and obtain a continuous functionλ:M→R which is called the dilation ofa horizontally weakly conformal map?.A non-constant horizontally weakly conformal map?is called horizontally homothetic if the gradient of(x)is vertical meaning thatX()≡0 for any horizontal vector fieldXonM.Recall that aC2map?:(M,g)→(N,h)is ap-harmonic morphism(p>1)if it preserves the solutions ofp-Laplace equation in the sense that for anyp-harmonic functionf:U→R,defined on an open subsetUofNwith(U)non-empty,f??:(U)→R is ap-harmonic function.Ap-harmonic morphism can be characterized as a horizontally weakly conformalp-harmonic map(see[3,11,15,19,21]).

Definition 2.1Let f:(M,g)→(0,∞)be a smooth function.A-function u:U→Rdefined on an open subset U of M is called f-harmonic if

A continuous map φ:(,g)→(,h)is called an f-harmonic morphism if for every harmonic function u defined on an open subset V of N such that(V)is non-empty,the composition u?φ is f-harmonic onV).

Theorem 2.1Let φ:(,g)→(,h)be a smooth map.Then,the following are equivalent:

(1)φ is an f-harmonic morphism;

(2)φ is a horizontally weakly conformal f-harmonic map;

(3)There exists a smooth function on M such that

for any -function u defined on(an open subset of)N.

ProofWe will need the following lemma to prove the theorem.

Lemma 2.1(see[15])For any point q∈(,h)and any constants==0,there exists a harmonic function u on a neighborhood of q such that

Letφ:(,g)→(,h)be a map and letp∈M.Suppose that

is the local expression ofφwith respect to the local coordinates{}in the neighborhood(V)ofpand{}in a neighborhoodVofq=φ(p)∈N.Letu:V→R be defined on an open subsetVofN.Then,a straightforward computation gives

By Lemma 2.1,we can choose a local harmonic functionuonV?Nsuch that(q)==0,=1,2,···,n,=1(αβ),and all other=0,and substitute it into

(2.2)to have

Note that the choice of such functions implies

Another choice of harmonic functionuwith=1,=?1(α1)and all other,=0 for(2.2)gives

Note also that for these choices of harmonic functionsuwe have

It follows from(2.3)–(2.6)that thef-harmonic morphismφis a horizontally weakly conformal map

Substituting horizontal conformality equation(2.7)into(2.2),we have

for any functionudefined(locally)onN.By special choice of harmonic functionuwe conclude that thef-harmonic morphism is anf-harmonic map.Thus,we obtain the implication “(1)?(2)”.Note that the only assumption we used to obtain(2.8)is the horizontal conformality(2.7).Therefore,it follows from(2.8)that“(2)?(3)”.Finally,“(3)?(1)” is clearly true.Thus,we complete the proof of Theorem 2.1.

Similar to harmonic morphisms we have the following regularity result.

Corollary 2.1For m≥3,an f-harmonic morphism φ:is smooth.

ProofIn fact,by Corollary 1.1,ifm2 andφ:→(Nn,h)is anf-harmonic morphism,thenφ:→is a harmonic map and hence a harmonic morphism,which is known to be smooth(see[4]).

It is well-known that the composition of harmonic morphisms is again a harmonic morphism.The composition law forf-harmonic morphisms,however,will need to be modified accordingly.In fact,by the definitions of harmonic morphisms andf-harmonic morphisms we have the following result.

Corollary 2.2Let φ:be an f-harmonic morphism with dilation λ1and ψ:a harmonic morphism with dilation λ2.Then the composition ψ? φ:is an f-harmonic morphism with dilation

More generally,we can prove thatf-harmonic morphisms pull back harmonic maps tofharmonic maps.

Proposition 2.1Let φ:be an f-harmonic morphism with dilation λ and ψ:a harmonic map.Then the composition ψ? φ:is an f-harmonic map.

ProofIt is well-known(see[4,Proposition 3.3.12])that the tension field of the composition map is given by

from which we have thef-tension of the compositionψ?φgiven by

Sinceφis anf-harmonic morphism and hence a horizontally weakly conformalf-harmonic map with dilationλ,we can choose local orthonormal frames{e1,···,en,en+1,···,em}aroundp∈Mandaroundφ(p)∈Nso that

Using these local frames we compute

Substituting this into(2.9)we have

from which the proposition follows.

Theorem 2.2Let φ:→be a horizontally weakly conformal map withThen,any two of the following conditions imply the other one:

(1)φ is an f-harmonic map and hence an f-harmonic morphism.

(2)grad(is vertical.

(3)φ has minimal fibers.

ProofIt can be check(see[4])that the tension field of a horizontally weakly conformal mapφ:→is given by

whereλis the dilation of the horizontally weakly conformal mapφandμis the mean curvature vector field of the fibers.It follows that thef-tension field ofφcan be written as

or,equivalently,

From this we obtain the theorem.

An immediate consequence is the following result.

Corollary 2.3(a)A horizontally homothetic map(in particular,a Riemannian submersion)φ:(Mm,g)→(Nn,h)is an f-harmonic morphism if and only if?(m?n)μ+gradlnf is vertical.

(b)A weakly conformal map φ:(Mm,g)→(Nm,h)with conformal factor λof same dimension spaces is f-harmonic and hence an f-harmonic morphism if and only if f=for some constant C>0.

(c)A horizontally weakly conformal map φ:(Mm,g)→(N2,h)is an f-harmonic map and hence an f-harmonic morphism if and only if?(m?2)μ+gradlnf is vertical.

Using the characterizations off-harmonic morphisms andp-harmonic morphisms and Corollary 1.1 we have the following corollary which provides many examples off-harmonic morphisms.

Corollary 2.4A map φ:(Mm,g)→(Nn,h)between Riemannian manifolds is a pharmonic morphism without critical points if and only if it is an f-harmonic morphism with f=|dφ|p?2.

Example 2.1The Mbius transformationφ:Rm{0}→Rm{0}defined by

is anf-harmonic morphism withf(x)=In fact,it is well-known that the Mbius transformation is a conformal map between the same dimensional spaces with the dilationλ=It follows from[20]thatφis anm-harmonic morphism,and hence by Corollary 2.4,the inversion is anf-harmonic morphism with

The next example is anf-harmonic morphism that does not come from ap-harmonic morphism.

Example 2.2The map from Euclidean 3-space into the hyperbolic planeφ:R×R×≡R×{0}×withφ(x,y,z)=(x,0,is anf-harmonic morphism withSimilarly,we know from[12]that the mapφ:H3≡R×R×R+,→H2≡R×{0}×withφ(x,y,z)=is a harmonic morphism.It follows from Example 1.2 that the map from Euclidean space into the hyperbolic planeφ:(R×R×→H2≡withφ(x,y,z)=is anf-harmonic map withf=.Since this map is also horizontally weakly conformal it is anf-harmonic morphism by Theorem 2.1.

Example 2.3Any harmonic morphismφ:(Mm,g)→(Nn,h)is anf-harmonic morphism for a positive functionfonMwith vertical gradient,i.e.,dφ(gradf)=0.In particular,the radial projectionφ:{0}→Sm,φ(x)=is anf-harmonic morphism forf=α(|x|),whereα:(0,∞)→(0,∞)is any smooth function.In fact,we know from[4]that the radial projection is a harmonic morphisms and on the other hand,one can check that the functionf=α(|x|)is positive and has vertical gradient.

Using the property off-harmonic morphisms and Sacks-Uhlenbeck’s well-known result on the existence of harmonic 2-spheres we have the following proposition which gives many examples off-harmonic maps from Euclidean 3-space into a manifold whose universal covering space is not contractible.

Proposition 2.2For any Riemannian manifold whose universal covering space is not contractible,there exists an f-harmonic map φ:→(Nn,h)from Euclidean3-space with f(x)=

ProofLet ddenote the Euclidean metric on R3.It is well-known that we can use the inverse of the stereographic projection to identifywith

the Euclidean 3-sphere minus the north pole.In fact,the identification is given by the isometry

with

One can check that under this identification,the Hopf fiberation

can be written as

where

It is well-known(see[4])that the Hopf fiberationφis a harmonic morphism with dilationλ=2.So,by Corollary 1.1,φ:is anf-harmonic map withIt is easy to see that this map is also horizontally conformal submersion and hence,by Theorem 2.1,it is anf-harmonic morphism.On the other hand,by a well-known result of Sacks-Uhlenbeck’s,we know that there exists a harmonic mapρ:S2→(Nn,h)from 2-sphere into a manifold whose covering space is not contractible.It follows from Proposition 2.1 that the compositionρ?φ:→(Nn,h)is anf-harmonic map withf=

Remark 2.1We notice that the authors in[8]and[14]used the heat flow method to study the existence off-harmonic maps from closed unit diskD2→S2sending boundary to a single point.Thef-harmonic morphismφ:→S2in Proposition 2.2 clearly restrict to anf-harmonic mapφ:→S2from 3-dimensional open disk intoS2.It would be interesting to know if there exists anyf-harmonic map from higher dimensional closed disk into two-sphere.Though we know thatφ:(Mm,g)→(Nn,h)beingf-harmonic implies

being harmonic we need to be careful trying to use results from harmonic maps theory since a conformal change of metric may change the curvature and the completeness of the original manifold(Mm,g).

As we remark in Example 2.3 that any harmonic morphism is anf-harmonic morphism providedfis positive with vertical gradient,however,such a function need not always exist as the following proposition shows.

Proposition 2.3A Riemannian submersion φ:(Mm,g)→(Nn,h)from non-negatively curved compact manifold with minimal fibers is an f-harmonic morphism if and only if f=C>0.In particular,there exists no nonconstant positive function onso that the Hopf fi beration φ:(Nn,h)is an f-harmonic morphism.

ProofBy Corollary 2.3,a Riemannian submersionφ:(,g)→(,h)with minimal fibers is anf-harmonic morphism if and only if gradlnfis vertical,i.e.,dφ(gradlnf)=0.This,together with the following lemma will complete the proof of the proposition.

Lemma 2.1Let φ:(Mm,g)→(Nn,h)be any Riemannian submersion of a compact positively curved manifold M.Then,there exists no(nonconstant)function f:M→Rsuch thatdφ(gradlnf)=0.

ProofSuppose thatf:(Mm,g)→R has vertical gradient.Consider

whereε>0 is a sufficiently small constant.

Ifεis small enough,thenis positively curved.One can check that

is a horizontally homothetic submersion with dilationsincefhas vertical gradient.By the main theorem in[22]we conclude that the mapφdefined in(2.10)is a Riemannian submersion,which implies that the dilation and hence the functionfhas to be a constant.

Remark 2.2It would be very interesting to know if there exists anyf-harmonic morphism(orf-harmonic map)φ:with non-constantf.Note that for the case ofn=2,the problem of classifying allf-harmonic morphismsφ:(whereg0denotes the standard Euclidean metric on the 3-sphere)amounts to classifying all harmonic morphismsφ:→(N2,h)from conformally flat 3-spheres.A partial result on the latter problem was given in[13]in which the author proved that a submersive harmonic morphismφ:→with non-vanishing horizontal curvature is the Hopf fiberation up to an isometry ofThis implies that there exists no submersivef-harmonic morphismφ:with non-constantfand the horizontal curvature

Proposition 2.4For m>n≥2,a polynomial map(i.e.,a map whose component functions are polynomials)φ:Rm→Rnis an f-harmonic morphism if and only if φ is a harmonic morphism and f has vertical gradient.

ProofLetφ:Rm→Rnbe a polynomial map(i.e.,a map whose component functions are polynomials).Ifφis anf-harmonic morphism,then,by Theorem 2.1,it is a horizontally weakly conformalf-harmonic map.It was proved in[1]that any horizontally weakly conformal polynomial map between Euclidean spaces has to be harmonic.This implies thatφis also a harmonic morphism,and in this case we have dφ(gradf)=0 from(1.1).

Example 2.4φ:R3R×C→C withφ(t,z)=p(z),wherep(z)is any polynomial function inz,is anf-harmonic morphism withf(t,z)=α(t)for any positive smooth functionα.

AcknowledgementThe author is very grateful to Fred Wilhelm for some useful conversations during the preparation of the paper,especially,the author would like to thank him for offering the proof of the lemma in the proof of Proposition 2.3.

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