Existence Results for Super-Liouville Equations on the Sphere via Bifurcation Theory

Aleks Jevnikar,Andrea Malchiodiand Ruijun Wu

1Department of Mathematics,Computer Science and Physics,University of Udine,Via delle Scienze 206,33100 Udine,Italy;

2Scuola Normale Superiore,Piazza dei Cavalieri 7,56126 Pisa,Italy.

Abstract.We are concerned with super-Liouville equations on S2,which have variational structure with a strongly-indefinite functional.We prove the existence of nontrivial solutions by combining the use of Nehari manifolds,balancing conditions and bifurcation theory.

Key words:Super-Liouville equations,existence results,bifurcation theory,critical groups.

1 Introduction

In this paper we study the super-Liouville equations,arising from Liouville field theory in supergravity.Recall that the classical Liouville field theory describes the matterinduced gravity in dimension two:the super-Liouville field theory is a supersymmetric generalization of the classical one,by taking the spinorial super-partner into account,so that the bosonic and fermionic fields couple under the supersymmetry principle.Such models also play a role in superstring theory.For the physics of the Liouville field theory and super-Liouville field theory as well as their relations,one can refer to[9,39–42],and for the applications of Liouville field theory in other models of mathematical physics[44,46–48]and the references therein.It is almost impossible to have a complete references for the related theory.However,the existence theory of regular solutions of the super-Liouville equations on closed Riemann surfaces,especially on the sphere,is still far from satisfactory.

Liouville equations also have a relevant role in two-dimensional geometry.For example,on a Riemannian surface(M2,g),the Gaussian curvature K of a conformal metricBe2ug,with u∈C∞(M),is given by

Conversely,we have the prescribed curvature problem:which functionscan be the Gaussian curvatures of a Riemannian metric conformal to g?If M is a closed surface,the problem reduces to solving equation(1.1)in u for K˜g=assigned.This question has been widely studied in the last century,and the solvability of(1.1)depends on the geometry and the topology of the surface.For a surface with nonzero genus,this can be solved variationally,as long assatisfies some mild constraints,see[6,32,43].However when the genus is zero,namely M is a topological two-sphere,the problem has additional difficulties arising from the non-compactness of the automorphism group.Actually,since there is only one conformal structure on S2,we can take without loss of generality the standard round metric g=g0,which is the one induced by the embedding S2⊂R3with Gaussian curvature Kg0=1.Let x=(x1,x2,x3)be the standard coordinates of R3.It was shown in[32]that a necessary condition forto admit a solution u of(1.1)is that

where the volume form dvol and the gradient are taken with respect to g0.The above formula shows that,for example,affine functions cannot be prescribed conformally as Gaussian curvatures.

One of the first existence results for the problem on the sphere is due to Moser,see[38]:he proved that there exist solutions provided thatis an antipodally-symmetric function.Other important results were proven in[12,13],removing the symmetry condition and replacing it with an index-counting condition or some assumption of min-max type,see also[14].One fundamental tool in proving such results was an improved Moser-Trudinger inequality derived in[5]for functions satisfying a balancing condition,namely for which the conformal volume has zero center of mass in R3(where S2is embedded).This fact allowed to show that whenever solutions(or approximate solutions)of(1.1)blowup,they develop a single-bubbling behavior.With this information at hand,existence results were derived via asymptotic estimates and Morse-theoretical results.We should also mention that there are natural generalizations to higher dimensions,see e.g.[36]and references therein.

Recently Jost et al.in[26]considered a mathematical version of the super-Liouville equations on surfaces.Given a Riemann surface M with metric g,and S→M the spinor bundle with Dirac operator/D,they considered the Euler–Lagrange equations of the functional

In subsequent works,they performed blow-up analysis and studied the compactness of sequences of solutions under weak assumptions and in various settings;see e.g.[27–29]and the references therein.

In [24],we studied the existence issue from a variational viewpoint when M is a closed surface of genus γ>1,with the signs of some terms adapted to the background geometry.More precisely we consider a uniformized surface(M,g)with Kg=−1 and the following functional

where ρ>0 is a parameter.The pair(0,0)is clearly a trivial critical point of.Moreover,when ρ is not in the spectrum of the Dirac operator/D,we could find non-trivial solutions using min-max schemes.However,the method there does not directly apply to the sphere case,for two reasons.First,in the sphere case the trivial solution(0,0)is not isolated,but within a continuum of solutions connecting to it which are geometrically also trivial and induced by M¨obius maps.Second,there is neither local mountain-pass geometry nor local linking geometry in zero genus,preventing us from finding min-max critical points starting from(0,0).Thus,the problem in the sphere case is more challenging.

In this article,we use a Morse-theoretical approach combined with bifurcation theory to attack the problem.Taking the Gauss–Bonnet formula into account we consider the following functional

where g is a Riemannian metric on S2,dvolgis the induced volume form,and the last tail-term 4π=2πχ(S2)is simply needed to normalize the functional so that Jρ(0,0)=0.The Euler–Lagrange equations for Jρare the following

Let u∗be a solution of

whose existence follows from the uniformization theorem:then we have clearly a trivial solution(u∗,0)of(EL).However,in contrast to the higher genus case,here we have another explicit family of solutions with nonzero spinor component and constant function component,see below.Hence we are interested in finding non-trivial solutions with nonconstant components.

We remark that for the system(EL) to admit a solution with nonzero spinor component,it is necessary that ρ >1.Indeed,for every solution(u,ψ)(which is smooth by regularity theory,see[24,26])we can consider the metric guBe2ug on S2.The corresponding Dirac operator/Dguhas ρ as an eigenvalue since the second equation transforms into

for any metric g0on S2.In particular,we conclude that ρ >1 if ψ is not identically zero.

Without loss of generality,we may consider the standard round sphere(S2,g0)with Kg0≡1.This is due to the conformal covariance of the system(EL),see Section3.Then the trivial solutions are simplyand its Mo¨bius transformations,see again Section 3.On the round sphere we know that the eigenspinors corresponding to the eigenvalue λ1=1∈Spec(/Dg0)=Z{0}has constant length,i.e.if/Dϕ1=ϕ1,then the function|ϕ|:S2→R is constant.Such spinors constitute a vector space of real dimension 4.This allows us to construct another family of solutions,namely choosing u to be the constant function such that ρeu=λ1and then choosing ψ∈Eigen(/Dg0,λ1)of a length such that the first equation of(EL)holds.Therefore,for any ρ ≥ 1,let ϕ1∈Eigen(/Dg0,1)be an eigenspinor of unit length:then the pair

is a solution of(EL).Note that these solutions converge to the trivial solution θ=(0,0)as ρ→1,which highlights a bifurcation phenomenon at the first eigenvalue ρ=λ1.We will see that this is actually a more general phenomenon.For later convenience we call a solution(u,ψ)non-trivial if the function component u is not constant and the pair(u,ψ)is not in the conformal orbit of constant functions.Note that u=const.implies that|ψ|=const.,which is only the case if ψ is a Killing spinor and ρeu=1.Also,the eigenspinors for λk>1 do not have constant length,see[10,Section 2.2]and[22,Section 4.2].

Theorem 1.1.Let ρ=λk∈Spec(/D)with λk>1.Then,ρ is a bifurcation point for(EL)on S2,i.e.there exists a sequence ρl→ρ=λksuch that(EL)admits a non-trivial solution on S2for ρ=ρl.

The metric g in the above statement is suppressed:once we proved it for the round metric g0,then it also holds for any other(smooth)metric g by conformal and diffeomorphism transformations since,as we recalled,the sphere admits only one conformal class of metrics.

Note that there exists a 3-dimensional family of quaternionic structures on the spinor bundle S,which are fibrewise automorphisms preserving the connection,metric and Clifford multiplication.Thus,once we get a solution with nonzero spinor component,we automatically get a three-dimensional family of solutions for free.

There also exists the real volume element ω =e1·e2,where(e1,e2)denotes a local oriented orthonormal frame of S2and the dot is the Clifford multiplication in the Clifford bundle Cl(S2).It is readily checked that ω is globally well-defined.The endomorphism γ(ω)≡γ(e1)γ(e2)∈End(S)is an almost-complex structure,parallel with respect to the spin connection, but anti-commutative with the Dirac operator:/D(γ(ω)ψ)=−γ(ω)/Dψ.Therefore,if(u,ψ)is a solution to(EL),then the pair(u,γ(ω)ψ)solves the system

That is,we can allow a change of sign in front of the Dirac part in the functional Jρ(u,ψ),without affecting the result.

The main observation is that the second equation in(EL)has the form of a weighted eigenvalue equation.This suggests to employ a bifurcation argument to search for nontrivial solutions.Recall that a theorem by Krasnosel’skii states that for a pure(nonlinear)eigenvalue problem,any eigenvalue is a bifurcation point for the eigenvalue equation,see e.g.[3,Chapter 5,Appendix]and[33]with the references therein.Here we are adopting a Morse-theoretical approach in the spirit of[37],see also[2,Section 12],which differently from e.g.[15]exploits the variational structure of the problem instead of information on the multiplicity of eigenvalues,lacking here.However,note that here the presence of the Dirac operator makes the functional strongly indefinite and the Morse-theoretical groups are generally not well-defined,meanwhile the critical points are not isolated because of the symmetries of the functional.To overcome these difficulties we introduce some natural constraints,based on spectral decomposition and balancing conditions,to remove most of the negative directions which decreases the functional and also kill the redundancy of the conformal orbits.We also refer to[8,16,45]for related approaches to strongly indefinite problems in other contexts.Restricted to this Nehari type manifold,the origin is now an isolated critical point,and though the functional is still indefinite,we are able to count the index of the origin within the Nehari manifold and hence get the well-defined local critical groups.In doing so we reduce ourselves to a more classical setting and the problem is tractable:see also[17]for related issues treated via spectral flows.

The paper is organized as follows.First we recall some preliminary facts about the Dirac operator and set up the variational framework.Then we introduce a class of Nehari manifolds and show that they are natural constraints.After showing the validity of the Palais-Smale condition,we analyze the local behavior of the functional around the origin and define the critical groups there.In the end we use a parametrized flow to show the bifurcation result,hence obtaining the existence of non-trivial solutions.

2 Preliminaries

Recall that S2admits a unique conformal structure up to diffeomorphism and consider the Riemannian metric g0induced from the embedding S2⊂R3.The spectrum of the Laplace operator−∆S2=−∆g0is explicitly known:the eigenvalues are given by µk=k(k+1),for k=0,1,2,...,the multiplicity of µ0=0 is 1(eigenfunctions given by constants),that of µ1=2 is 3(with eigenfunctions given by affine functions on R3restricting to S2;a basis is given by the coordinate functions{x1,x2,x3}),with multiplicities of µk(k≥2)given by the binomial coefficients

and with eigenfunctions given by homogeneous harmonic polynomials on R3restricted to S2,see e.g.[1,Chapter 4].

The two-sphere admits a non-compact group of conformal automorphisms,which constitutes the M¨obius group Aut(C∪{∞})=PSL(2;C).In terms of the Riemann sphere C∪{∞},these are the fractional linear transformations,which are nothing but compositions of translations,rotations,dilations and inversions.Note that with zero spinor components,the functional

is invariant under the M¨obius group action.Indeed,each element ϕ∈PSL(2;C)is a conformal diffeomorphism with ϕ∗g0=det(dϕ)g0.For any u∈H1(S2),set

then it is a classical fact that Jρ(uϕ,0)=Jρ(u,0).

Consider the spinor bundle S→S2associated to the unique spin structure of S2and let/D=/Dg0be the Dirac operator.For basic material on spin geometry and Dirac operators,one may refer to[18,19,25,34].Recall that the spectrum of the Dirac operator is

and the eigenvalue±(k+1)has(real)multiplicity 4(k+1).In particular,there are no harmonic spinors on S2,and the first positive eigenvalue is 1 with eigenspinors having constant length(they are actually given by the Killing spinors).For more details we refer to[19,Chapter 2 and Appendix].

We give a brief description of the Sobolev spaces H1(S2)andwhich we will employ.For basic material on Sobolev spaces and fractional Sobolev spaces,see[4,20].Most recent papers on analysis of Dirac operators contain such an introductory part,and here we only collect some necessary material.

The Sobolev space H1(S2)is equipped with the inner product

For smooth functions(which are dense in H1(S2)),an integration by parts gives

where the last bracket denotes the dual pairing.Note that,in contrast to the case in[24],here the functional u7→ Jρ(u,0)is not coercive.At any u ∈ H1(S2)there are finitely-many negative directions of the Hessian HessuJρ(u,0).Moreover,the functional Jρ(u,0)does not admit local linking geometry around the trivial critical point u=0.

The fractional Sobolev space of the sections of the spinor bundle S can be defined via the L2-spectral decomposition.Recall that/D is a first-order elliptic operator which is essentially self-adjoint and has no kernel:counting eigenvalues with multiplicities,the eigenvalues{λk}k∈Z∗(where Z∗≡Z{0})are listed in a non-decreasing order:

Moreover,the spectrum is symmetric with respect the the origin.Let(ϕk)kbe the eigenspinors corresponding to λk,k∈Z∗with kϕkkL2(M)=1:they form a complete orthonormal basis of L2(S).For any spinor ψ∈Γ(S),we have

3 Conformal symmetry

which has the same form as(EL).

The automorphisms group of the Riemann sphere S2=C∪{∞}is a family of conformal maps that induce a natural action on Sobolev spaces of functions and spinors.Let ϕ∈PSL(2,C)=Aut(S2)be a conformal diffeomorphism with ϕ∗g0=det(dϕ)g0.For any(u,ψ),we set

where β:S→ϕ∗S denotes the isometry of the spinor bundles.Then,not only(uϕ,ψϕ)satisfies(EL),but also the functional on(S2,g0)stays invariant

This generalizes[12,Prop.2.1]in the classical Liouville case.

As consequences of such symmetries,on one hand,for any given metric on the sphere S2,we can use a conformal diffeomorphism to reduce the problem to the case where the metric on S2is the standard round metric g0with Kg0≡1;on the other hand,a critical point(u,ψ)of Jρis never isolated in.Since the elements in the orbits of the conformal transformations are geometrically the same,we will overcome this problem by picking those elements with centers of mass at the origin.

4 A natural constraint

Due to the above conformal symmetry,without loss of generality we may consider the problem with respect to the standard round metric g=g0.Then the functional becomes

In the functional Jρ,the part involving spinors is strongly indefinite while the remaining terms are invariant with respect to the M¨obius group:both these properties make the variational approach quite challenging.We therefore need to confine such defects.

For u∈H1(S2),the function e2ucan be considered as a mass distribution on S2,see[12].be the position vector.The center of mass of e2uis defined as

For any u∈H1(M),there exists a ϕ∈PSL(2,C)such that C.M.(e2uϕ)=0∈R3;moreover,the M¨obius transformation can be chosen to depend on u in a continuous way[12,Lemma 4.2].Note that such a ϕ is never unique:there is always the freedom of a SO(3)-action which leaves|C.M.(e2u)|invariant.See[12]for the argument and more information on the center of mass.We remark that C.M.(e2u)=0 means that the function e2uis orthogonal to the first eigenfunctions on S2with respect to the L2-inner product.Let

where αj,µk,τ ∈ R are the Lagrange multipliers††The right-hand side is the projection of the unconstrained gradient of Jρ on the normal space at(u,ψ)∈ρ,hence it is well-defined in the Hilbert space H1×.In particular, the series on the right-hand side converges.The same remark applies also in the sequel..In the equation for u the term δuϕk(u)denotes the variation of ϕk(u)with respect to u,which exists because of the analytic dependence of e−u/D on u,and

5 Convergence of bounded Palais-Smale sequences

In the last sections we will need to deal with bounded Palais-Smale sequences on the Nehari manifolds.Here we first show that any bounded(PS)csequence(i.e.Palais-Smale sequence at level c)admits a strongly convergent sub-sequence.We remark that,though we will not strictly use the result in this form,later on we will crucially rely on its proof.

6 Local geometry around the origin

By the above computation of the Hessian ofat θ,we see that

7 Local deformation of sub-levels around non-bifurcation points

8 Appendix:a conformal transformation

Acknowledgments

Andrea Malchiodi has been partially supported by the project Geometric problems with loss of compactness from Scuola Normale Superiore.Aleks Jevnikar and Andrea Malchiodi have been partially supported by MIUR Bando PRIN 2015 2015KB9WPT001.They are also members of GNAMPA as part of INdAM.Ruijun Wu is supported by Centro di Ricerca Matematica Ennio De Giorgi.