Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds
aa r X i v : . [ m a t h . DG ] F e b EIGENVALUE BOUNDS FOR THE PANEITZ OPERATOR AND ITS ASSOCIATEDTHIRD-ORDER BOUNDARY OPERATOR ON LOCALLY CONFORMALLY FLATMANIFOLDS
MAR´IA DEL MAR GONZ ´ALEZ AND MARIEL S ´AEZ
Abstract.
In this paper we study bounds for the first eigenvalue of the Paneitz operator P and its associatedthird-order boundary operator B (see (1.1) and (1.12) for a precise definitions) on four-manifolds. We restrictto orientable, simply connected, locally confomally flat manifolds that have at most two umbilic boundarycomponents. The proof is based on showing that under the hypotheses of the main theorems, the consideredmanifolds are confomally equivalent to canonical models. This equivalence is proved by showing the injectivityof suitable developing maps. Then the bounds on the eigenvalues are obtained through explicit computationson the canonical models and its connections with the classes of manifolds that we are considering. The factthat P and B are conformal in four dimensions is key in the proof. Introduction
Let ( M , g ) be a 4-dimensional Riemannian manifold and denote by Ric and W the Ricci and Weyl tensorsof g , respectively. Define J to be the trace of the Schouten tensor A = (Ric − J g ) (actually J is a multipleof the scalar curvature R , this is, J = R ) and dv g the volume element for the metric g .The Paneitz operator P g on ( M, g ), first introduced in 1983 [57], is defined by(1.1) P g = ( − ∆ g ) + div g (cid:8) A g − J g (cid:9) d,P is a conformally covariant operator and, in particular, it satisfies that under a change of metric g f = e f g ,(1.2) P g f = e − f P g on M. This operator describes the transformation law for Branson’s Q -curvature [6], which is defined by Q g = ( − ∆ R g − | Ric g | + R g ) . Indeed, P g f + Q g f e f = Q g , for g f = e f g. There is an extensive bibliography on the Q -curvature equation in dimension four. Without being exhaustive,we mention [20, 63, 24, 48, 37].Now, if M is a compact 4-dimensional manifold without boundary, the Chern-Gauss-Bonnet formula [7]reads(1.3) Z M Q g dv g + 14 Z M | W g | g dv g = 8 π χ ( M ) . Note then that we may regard P as a generalization for 4-manifolds of the Laplace operator ∆ in twodimensions (that is also conformally covariant in that setting) and the curvature Q as a four-dimensionalanalog of the Gaussian curvature in the two-dimensional setting (which plays the same role as Q in theclassical Gauss-Bonnet formula).The study of eigenvalues of differential operators has an extensive history. In the particular case of theLaplacian in two dimensions it is possible to obtain bounds that only depend on the topology of the manifold(see [67]); more precisely, consider N to be a two-dimensional compact orientable Riemanniann manifoldwith no boundary and take ς to be the first non-zero eigenvalue of the Laplace-Beltrami on N . Let(1.4) Θ( N ) := sup { ς Vol( N ) } , where the sup is taken among all Riemannian metrics on N . It is well known that Θ( N ) < ∞ and(1.5) Θ( N ) ≤ π ( γ + 1) , where γ is the genus of the surface. See also [53] for a discussion of attainability. In fact, in that paper thediscussion is extended to the setting with boundary and a Neumann condition at that boundary.The first goal in our paper is to generalize the bound (1.5) for the Paneitz operator P on closed 4-manifolds.However, although it is well known that the spectrum of P g consists of a sequence of eigenvalues convergingto + ∞ , the principal eigenvalue λ g maybe negative. In consequence, one first imposes restrictions thatensure the positivity of the operator. With this objective, we recall two important conformal invariantquantities in four dimensions: Firstly, the total Q -curvature ,(1.6) κ g := Z M Q g dv g , and, secondly, the well known Yamabe invariant (1.7) Y [ g ] = inf g f = e f g R M R g f dv g f (cid:0)R M dv g f (cid:1) / . A key theorem by Gursky [36] yields that, if both the Yamabe invariant Y [ M ] and the total Q − curvature κ g are nonnegative, then λ g = 0 and the kernel of P g contains only the constant functions. Thus, the nexteigenvalue λ g is positive. A less restrictive condition was given in [38]: indeed, if M is a closed 4-manifoldwith positive scalar curvature and(1.8) Z M Q g dv g + 13 ( Y [ g ]) > , then the same conclusion holds. It is interesting to observe that (1.8) is a conformally invariant quantity.Now, the first eigenvalue of P g (that we denote as λ g ) can be computed through the Rayleigh quotient(1.9) λ g = inf R M u dv g =0 ,u =0 E Mg [ u ] R M u dv g , where(1.10) E Mg [ u ] = Z M (∆ g u ) dv g + Z M (cid:16) A gab − J g ab (cid:17) ∇ a u ∇ b u dv g . This is a conformal invariant quantity in 4-dimensions; indeed, if we have two metrics related by g f = e f g ,then E g f [ u ] = E g [ u ] . Our first theorem is a generalization of the bound (1.5) for the first eigenvalue of the Paneitz operator:
Theorem 1.1.
Let ( M, g ) be a compact, orientable, closed, locally conformally flat (l.c.f) Riemannian 4-manifold, and define λ g to be the first (non-zero) eigenvalue of the Paneitz operator P g . Then:i. If M is simply connected, then M is conformally equivalent to S . In this setting we have λ > with Ker( P g ) = { constants } and λ g Vol( M ) ≤ π . Equality holds if and only if M is diffeomorphic to S .ii. If Y [ g ] > and κ g > , and M is orientable, then M is conformally equivalent to S and the sameconclusions hold.iii. If Y [ g ] > and κ g = 0 , then M is conformally equivalent to a quotient R × S . Assume thatthe fundamental domain is contained in a region [0 , ̺ ] × S for some ̺ > . Then λ g > with Ker( P g ) = { constants } and λ g Vol( M ) ≤ π ̺. Equality holds if M is diffeomorhphic S ( ̺ ) × S , where S ( ̺ ) is the one-dimensional circle of length ̺ . IGENVALUE BOUNDS FOR PANEITZ WITH BOUNDARY 3
The jump from dimension 2 to dimension 4 is completely non-trivial, since in the two-dimensional case onecan use conformal invariance to map any manifold to (a cover of) the sphere. In dimension 4 the difficultyis to find such conformal immersion of M into a model manifold. Thus we restrict our study to locallyconformally flat (l.c.f.) manifolds, where the developing map plays the role of this immersion. In fact, wewill show that in the setting of Theorem 1.1 the developing map is injective.Finally, note that different bounds for λ g have been introduced in the literature. For instance, if M canbe conformally immersed into a unit sphere S K , then [66] gives a bound in terms of an K -conformal energyinspired in the conformal volume of Li-Yau [49]. In addition, [21, 22] showed some geometric bounds pro-vided that M is a compact submanifold of R K . A comparison theorem for this first eigenvalue was given in[62] for dimension K ≥ Manifolds with boundary.
Now we turn our attention to the boundary case. If N is a compactsurface with boundary, one may ask the same questions for the Steklov eigenvalues, which are the eigenvalues ϑ of the following the boundary value problem (cid:26) ∆ g u = 0 in N, − ∂ η u = ϑu on ∂N. A good reference for this problem is [32]. Given N , is well known that there exists an increasing sequenceof eigenvalues 0 = ϑ < ϑ ≤ ϑ ≤ . . . and, moreover, ϑ = inf R ∂N u =0 ,u =0 R N |∇ u | dv R ∂N u dσ . The extremal problem for the Steklov eigenvalue analogous to (1.4) has been studied in a series of papers byFraser-Schoen [26, 27, 28]. If N is a surface of genus γ and k boundary components, they show the bound(1.11) ϑ ( N ) Lenght( ∂N ) ≤ π ( γ + k ) . For γ = 0 and k = 1 this result was obtained by Weinstock [64] and it is sharp, while if the boundary hastwo boundary components (i.e., an annulus), it is not attained. In addition, Weinstock showed that thebound is attained at a flat disk and the eigenfunctions can be identified with its coordinates. In the generalcase, Fraser-Schoen [27, 28] identified the eigenfunctions associated to maximal eigenvalues (with a giventopology and number of boundary components) with coordinate functions of free boundary minimal surfacesin the unit ball B K . In the particular case that N is homeomorphic to the annulus, in [26] and [28] it isshown that the quantity ϑ ( N ) Lenght( ∂N ) is maximized by the coordinate functions of a critical catenoid(in R ) which meets the boundary sphere orthogonally. This problem has also been studied in the higherdimensional setting [29], where conformal invariance is lost and the maximizer does not exist in the class ofsmooth metrics.In this paper we are interested in the analog question for a conformal third-order boundary operator as-sociated to the Paneitz operator, and which yields the natural 4-dimensional generalization of the Steklovproblem from the conformal geometry point of view. In addition, it contains strong topological informationthanks to the Chern-Gauss-Bonnet formula given in formula (1.16) (and the discussion above it). It wasintroduced in [18, 19] (see also the surveys: [16, 15], for instance), and fully generalized in [13]. We followthe presentation in the latter.Set ( M , g ) be a 4-dimensional compact Riemannian manifold with boundary Σ = ∂M . We keep the no-tation above for the interior quantities, while tilde will mean the corresponding quantities for the boundarymetric. Denote by h the restriction of the metric g to T Σ and by dσ h the volume form for h on Σ. Let η bethe outward-pointing normal, II = ∇ η | T Σ the second fundamental form, H = tr h ∇ η the mean curvature ofΣ, and II = II − H h the trace free part of the second fundamental form. M.D.M. GONZ ´ALEZ AND M. S ´AEZ
We set, on the boundary Σ: B g u = ηu,B g u = − ˜∆ u + D u ( η, η ) + 13 Hηu, and the third order operator(1.12) B g u = − η ∆ u − ηu + 2 h II , ˜ D u i − H ˜∆ u + 23 h ˜ ∇ H, ˜ ∇ u i + (cid:16) − H − A ( η, η ) + 2 ˜ J + 12 | II | (cid:17) ηu, These operators also satisfy a conformally covariance property coupled with (1.2), this is(1.13) B kg f = e − kf B kg on Σ , k = 1 , , . Define the bilinear form Q g ( u , u ) = Z M u P g u dv g + I Σ (cid:0) u B g ( u ) + B g ( u ) B g ( u ) (cid:1) dσ h for u , u ∈ C ∞ ( M ). The main theorem in [13] shows that Q g is symmetric. The corresponding energyfunctional E [ u ] = Q g ( u, u )is a conformal invariant. Indeed,(1.14) E g f [ u ] = E g [ u ] . The boundary operator B g operator is associated to the following curvature quantity(1.15) T g = ηJ −
23 ˜∆ H − h II , ˜ A i + 43 H ˜ J + 13 H | II | − H . For a conformal metric g f = e f g , the T -curvature equation is B g f + T g = T g f e f . In addition, the integral quantity κ g,h := Z M Q g dv g + Z Σ T g dσ h is a conformal invariant.It is well known that the mean curvature is the associated boundary curvature to the scalar curvature on M , and that the pair ( R, H ) is conformally covariant. From the PDE point of view, these arise from aboundary value problem for the conformal Laplacian (see (3.1) below). If one considers instead fourth-orderequations on manifolds with boundary, the couple (
Q, T ) is the natural generalization of the pair (
R, H ),and has been well studied: for the construction of constant Q -curvature metrics with vanishing T -curvature,see [54], while the constant T -curvature problem was considered in [55]. A Q -curvature flow on manifoldswith boundary was analyzed in [56]. In the particular case of ( B , S ) sharp Sobolev trace inequalities forthe curvature T were proved in [1].In addition, the pair ( Q, T ) controls topology in the 4-dimensional setting. More precisely, there is aChern-Gauss-Bonnet formula analogous to (1.3) for 4-manifolds with boundary [18]:(1.16) 8 π χ ( M ) = Z M (cid:16) | W | g Q g (cid:17) dv g + Z Σ (cid:16) T g −
23 tr II (cid:17) dσ h . If M is a l.c.f. manifold with umbilic boundary, this formula greatly simplifies:(1.17) 8 π χ ( M ) = Z M Q g dv g + Z Σ T g dσ h . Our first result in the boundary case is a classification statement based on the injectivity of the developingmap Φ : M → S for a l.c.f. manifold, thus partially generalizing the seminal work by Schoen-Yau [59], [60,Chapter VI] to manifolds with boundary. IGENVALUE BOUNDS FOR PANEITZ WITH BOUNDARY 5
Note also that statement b. below was proved by Raulot [58]; however, we include it here for completeness.Here Y [ g ] stands for the Yamabe invariant for manifolds with boundary. It is the natural generalization of(1.7) (thus, denoted by the same letter), and its precise expression will be written in (3.4). Theorem 1.2.
Let M be a compact, orientable,l.c.f. Riemannian 4-manifold with umbilical boundary Σ = ∂M . Then, a. If M is simply connected and Σ has one connected component, then M is conformally equivalent toa half-sphere S = { ( z , . . . , z ) ∈ R : | z | = 1 , z ≥ } . b. If M is not necessarily simply connected, but χ ( M ) = 1 and Y [ g ] > , then the same conclusionholds [58] . c. Assume that M is simply connected, Σ has exactly two connected components, R g > and Q g > .Then M is conformally equivalent to an annulus in R , that can be chosen as A ρ := { x ∈ R : ρ ≤ | x | ≤ } for some ρ ∈ (0 , . Note first that the umbilicity assumption in the Theorem is a natural one, since it is a conformal invariantproperty. Next, we observe that Part a. may be understood as a 4-dimensional version of the classicalRiemann mapping theorem in the plane. For the multiply connected case, part c. tells us that we cannotmap two double-connected regions M and M ′ unless they share the same ρ . This is a very similar behaviorto what happens in the 2-dimensional case, where two ring regions in the plane can only be mapped to oneanother unless they have the same extremal distance , which is a conformal invariant. This notion goes backto Ahlfors [2] (see also, for instance, the more modern exposition of [47]). Finally, we remark that, while a.follows by a simple doubling argument, part c. is partly inspired in the work of Chang, Hang and Yang [17]for closed manifolds of positive Q -curvature.The proof of Theorem 1.2 also relies in the study of the developing map. A conformally invariant quantity,that will be relevant in analyzing this developing map was defined by Escobar and it is the analogue of theYamabe invariant for manifolds with boundary. This invariant is crucial in the so-called Yamabe problemwith boundary, which seeks a conformal metric on M to a given one that has constant scalar curvature andzero mean curvature on the boundary. The Yamabe problem with boundary was solved in many cases byEscobar in [25] (in particular, in dimension four which suffices for our purposes). Related work can be foundin [23, 39, 4, 52, 9] from the variational point of view, and [8, 3] for flow-type methods.Note in addition that since the right hand side of (1.17) is conformally invariant, it is convenient to takeEscobar’s solution as a background metric in M and in this particular case, T ≡ B g . To this operator we need to associate a second boundary condition, so we will work on theclass of functions U = { u : M → R : u smooth , ∂ η u = 0 on Σ } . In this class the energy functional reduces to E Mg [ u ] = Z M (∆ g u ) dv g + Z M (cid:16) A gab − J g ab (cid:17) ∇ a u ∇ b u dv g + 23 Z Σ H | ˜ ∇ u | h dσ h − Z Σ ( II ) ij ˜ ∇ i u ˜ ∇ j u dσ h . (1.18)Thus we would like to study the boundary eigenvalue problem P g u = 0 in M, (1.19) B g u = 0 on Σ , (1.20) B g u = λu on Σ . (1.21)It is possible to show that there exists an increasing sequence of eigenvalues λ g ≤ λ g ≤ λ g ≤ . . . A straightforward calculation from the models yields a statement about the positivity of B g : M.D.M. GONZ ´ALEZ AND M. S ´AEZ
Corollary 1.3.
In all cases a., b., c. in Theorem 1.2 above we have λ g = 0 and the corresponding eigenspaceconsists only of constant functions. This implies, in particular, that λ g > λ g = min U : R Σ u =0 E Mg [ u ] Z Σ u dσ h . The question of positivity of B g has also been analyzed in other contexts, see for example the work in [14].Now we look at the min-max problem for λ g . Our main Theorem is the four-dimensional generalization of(1.11), which may be applied to manifolds satisfying the hypothesis of Theorem 1.2: Theorem 1.4.
We get the following bounds for λ g > :a. If M is conformally equivalent to a half-sphere S , λ g Vol(Σ) ≤ π . and it is attained at a flat disk.b. If M is conformally equivalent to an annulus A ρ , λ g Vol(Σ) ≤ c ( ρ ) , where c ( ρ ) is a constant that can be explicitly computed and it depends only on ρ . The bounds of the previous theorem are obtained by comparison with explicit computations in two typesof model manifolds: a 4-dimensional ball and 4-dimensional annuli (see Section 5). The computations inthe ball case are straightforward and provide optimal bounds. On the other hand, the computations for theannuli are based on the ideas in [26], and although they are elementary, there is a non-trivial dependence ofthe parameter ρ . Hence, our results do not provide optimal bounds even for special families of metrics. Inany case, the bound can be taken uniform in ρ as ρ → Acknowledgments: The authors would like to thank Jeffrey Case, Alice Chang, Gaven Martin, Vicente Mu˜ozand Riccardo Piergallini, for many useful discussions and suggestions. The closed case
In this section we give the proof of Theorem 1.1, which we summarize here: first, the positive curvatureassumptions allow us to control the topology of M and, either M is conformally equivalent to a sphere S ,or M is covered by a cylinder R × S . In the first case, we can obtain an upper bound for λ using thescheme in Yang-Yau [67] , which is based on a trick by Hersch [40]. Hersch’s idea is to use the coordinatefunctions of the embedding as test functions in the Rayleigh quotient (1.9). A modification of this strategy IGENVALUE BOUNDS FOR PANEITZ WITH BOUNDARY 7 yields the cylinder case too.We recall now some facts about locally conformally flat (l.c.f.) manifolds; for additional background, werefer to the book [60, Chapter VI]. A Riemannian metric g on a smooth manifold M is called l.c.f. if forevery point p ∈ M , there exists a neighbourhood U of p and a smooth function f on U such that the metric e f g is flat on U . Note that, in dimension 4, a Riemannian manifold is locally conformally flat if and onlyif the Weyl tensor W vanishes.We assume, to start with, that M is a simply connected, closed, compact, l.c.f. manifold of dimension n .Liouville’s theorem [60, Theorem 1.6 in Chapter VI] allows us to patch all these neighborhoods to obtain aglobally defined conformal immersion Φ : M → R n (or equivalently, Φ : M → S n by stereographic projec-tion), such that the locally conformally flat structure of M is induced by Φ. The function Φ is called the developing map and it is unique up to conformal transformations of S n .Note that a simple topological argument yields the well known characterization result by Kuiper [46] (seealso the notes [43] for remarks on regularity). Indeed, Φ( M ) is at the same time open and closed in S n .More precisely: Theorem 2.1 (Kuiper [46]) . Any n -dimensional closed simply-connected locally conformally flat manifoldis conformally equivalent to S n .Proof of Theorem 1.1.i. By our previous discussion, there is a bijective conformal embedding Φ : ( M , g ) → ( S , g S ), where g S is the canonical metric on the sphere. We denote by ( z , z , . . . , z ) the coordinates of S in R and by Φ i the i -th coordinate of the embedding Φ, i = 0 , , , , λ g > P g ) = { constants } since condition (1.8) is trivially satisfiedon the sphere.Let us check now that Φ i is an admissible function for the Rayleigh quotient (1.9). A standard calibrationargument (see Lemma 1.1 in [40] or page 107 in [33]) yields that we can choose the embedding satisfying(2.1) Z M Φ i dv g = 0 , i = 0 , . . . , . Moreover,(2.2) λ g Z M Φ i dv g ≤ E Mg [Φ i ] = E Φ( M ) g S [ z i ] , Adding on i we have λ g Vol( M ) ≤ X i =0 E Φ( M ) g S [ z i ] , here we have used that P i =0 Φ i = 1. Now recall that Φ : M → S is bijection and calculate, from theexpression of the energy (1.10), E S g S [ z i ] = µ Z S z i dv g S − Z S |∇ z i | dv S = (cid:0) µ − µ (cid:1) Z S z i dv g S . Here µ = 4 is the first non-zero eigenvalue of the (minus) Laplace-Beltrami operator on S . Thus weconclude X E Φ( M ) g S [ z i ] = 8 Vol( S ) = 643 π , and that this bound is sharp, since the coordinate functions are already eigenfunctions. This completes theproof. (cid:3) We next consider the non-simply connected case; in the l.c.f. 4-dimensional setting it turns out that posi-tive curvature gives information about the topology. Since the Weyl term vanishes for l.c.f manifolds, underthe assumption κ g ≥
0, the Gauss-Bonnet formula (1.3) implies that χ ( M ) ≥
0. The classification of such
M.D.M. GONZ ´ALEZ AND M. S ´AEZ manifolds according to the Euler characteristic was studied by Gursky in [34, Theorem A]: if M is a compact4- or 6-dimensional manifold which admits a l.c.f. of non-negative scalar curvature g , then χ ( M ) ≤
2. Fur-thermore, χ ( M ) = 2 if and only if ( M, g ) is conformally equivalent to the sphere with its canonical metric,and χ ( M ) = 1 if and only if ( M, g ) is conformally equivalent to projective space with its canonical metric.The remaining case χ ( M ) = 0 was characterized in [35, Corollary G]: if ( M, g ) is a compact, l.c.f. 4-manifoldwith Y [ g ] > χ ( M ) = 0, then ( M, g ) is conformal to a quotient of the cylinder R × S .A related result was proven by Chang, Hang and Yang [17, Corollary 1.2]. More precisely, if M has positivescalar curvature and positive Q -curvature, then M is conformally equivalent to a quotient of the sphere.Note that if we remove the positive Q -curvature assumption one may construct examples of manifolds thatare conformally equivalent to S \ { p , . . . , p N } (see [17, Theorem 1.3] and the discussion there). Proof of Theorem 1.1.ii.
It follows as part i. taking into account that M is orientable. (cid:3) Now we deal with the remaining case in which M is (conformally) covered by a cylinder R × S . Thesemanifolds have been studied in [41], [42, Chapter 11]. It is known that a closed 4-manifold M is covered by R × S if and only if π = π ( M ) has two ends and χ ( M ) = 0. Its homotopy type is then determined by π and the first nonzero k -invariant k ( M ). While all the possible subgroups of π ( R × S ) are well known, thereis not a complete classification of which manifolds can be actually realized with such fundamental groups.In any case, since M is compact, the fundamental domain Ω := Φ( M ) will be contained in a region [0 , ̺ ] × S of R × S for some ̺ > Proof of Theorem 1.1.iii.
Note first that condition (1.8) for positivity of λ g is trivially satisfied by ourhypothesis.Now use the coordinate functions of S , given by ( y , y , y , y ) and denote by Ψ i , i = 0 , . . . ,
3, the corre-sponding coordinates of the embedding Φ : M → R × S . Again, a calibration argument as in (2.1) showsthat the Ψ i can be taken as are suitable test functions in the Rayleigh quotient (1.9). Thus we compute:(2.3) λ g Z M Ψ i dv g ≤ E Mg [Ψ i ] = E Φ( M ) g R × S [ y i ] , Adding on i we have λ g Vol( M ) ≤ X i =0 E Φ( M ) g R × S [ y i ] , here we used that P i =0 Ψ i = 1. Next, from (1.10) we have E Ω g R × S [ y i ] ≤ ˜ µ Z [0 ,̺ ] × S y i dv g R × S . Here ˜ µ = 3 is the first non-zero eigenvalue of − ∆ S . We conclude that(2.4) λ g Vol( X ) ≤ ̺ Vol( S ) = 18 ̺π , with equality if the fundamental domain Ω = [0 , ̺ ] × S . (cid:3) Preliminaries on the boundary case
Escobar’s problem.
Here we recall some background on the Yamabe invariant for manifolds withboundary. Here we use the notation from Subsection 1.1 in the Introduction. Let (
M, g ) be a compact, n -dimensional, Riemannian manifold with boundary Σ = ∂M , and let h be the restriction of the metric g tothe boundary. The first observation is that the conformal Laplacian on M can be associated to a boundaryoperator N g on Σ. We set(3.1) ( L g u := − ∆ g u + n − n − u in M,N g u := ∂ η u + n − H g u on Σ . IGENVALUE BOUNDS FOR PANEITZ WITH BOUNDARY 9
We remark that N plays the role of a Neumann (more precisely, Robin) condition. The most importantproperty for this system is that the couple ( L, N ) is conformally covariant. Indeed, for a conformal change g u = u n − g we have L g u ( u − φ ) = u − n +2 n − L g φ in M,N g u ( u − φ ) = u − nn − N g φ on Σ . (3.2)The Yamabe problem for manifolds with boundary asks to find a conformal metric to g with constant scalarcurvature on M and zero mean curvature on Σ. In PDE language we look for a positive solution to(3.3) ( L g u = cu n +2 n − in M,N g u = 0 on Σ . This problem was first studied by Escobar [25]. He solved it in many cases, including the 4-dimensional,l.c.f., umbilic boundary case which is the setting of this paper. More precisely, a solution may be foundusing variational methods for the following Yamabe invariant(3.4) Y [ g ] = inf {R g [ u ] : u ∈ W , ( M ) , u } where R g [ u ] = Z M uL g u dv g (cid:16) Z M u nn − dv g (cid:17) n − n = Z M (cid:16) |∇ u | g + n − n − R g u (cid:17) dv g + n − Z Σ H g u dσ h (cid:16) Z M u nn − dv g (cid:17) n − n . It is well known that a (positive) solution exists if Y [ g ] < Y [ g S n + ], and that if equality is attained then M is already diffeomorphic to the model S n + . In addition, the sign of Y [ g ] coincides with the sign of the firsteigenvalue for the conformal Laplacian on M (coupled with the boundary condition N g u = 0). Remark . In the following, we may assume without loss of generality that our background metric g hasconstant scalar curvature and zero mean curvature on the boundary.3.0.2. Raulot’s Theorem.
In this subsection we recall the following theorem of Raulot [58] in dimensions 4and 6:
Theorem 3.2 ([58]) . Let n be 4 or 6, and let M be an n -dimensional l.c.f. manifold with umbilical boundary.Assume that the Yamabe invariant satisfies Y [ g ] ≥ . Then the Euler characteristic satisfies the bound χ ( M ) ≤ . In addition, if χ ( M ) = 1 and Y [ g ] > , then ( M, g ) is conformally equivalent to the standardhalf-sphere S n + . Note that this immediatly yields part b. of Theorem 1.2.4.
Boundary case: Proof of Theorem 1.2
In this section we prove Theorem 1.2. We recall that in the locally conformally flat setting there exists aconformal map Φ : M → S (the developing map). The key ingredient of our result is the injectivity of suchmap Φ under hypotheses of Thereom 1.2.4.1. One boundary component.
We begin by proving the following result.
Proposition 4.1.
Let ( M, g ) be a simply connected, compact, l.c.f. 4-dimensional Riemannian manifoldwith umbilic boundary Σ . Assume that Σ has one connected component, then the developing map Φ : M → S is injective and thus, a diffeomorphism onto its image.Proof. In this situation the injectivity of the developing map relies on a classical doubling argument. Wefollow the presentation of Spiegel [61] to describe the construction.Consider the doubling of the manifold M defined as ˆ M = M ∪ ( − M ), where we write − M for a secondcopy of M that is distinguished from M itself (for instance by taking M × { } and M × {− } ). Here themanifold M and its copy − M are identified at their boundaries Σ and hence ˆ M is a closed manifold (see[65, Chapter 5] for more details). Since Σ = ∂M ⊂ ˆ M is umbilic, and this is a conformal invariant property,the image of Σ must be umbilic in S , thus it must be contained in a hypersphere of S . Now, since Φ | Σ is a local diffeomerphism from a compact manifold to a simply connected manifold, we have that Φ | Σ is actuallydiffeomorphism. Composing with a M¨obius transformation of S , if necessary, we may assume that Φ(Σ) isthe equator { z = ( z , . . . , z ) ∈ S ⊂ R ; z = 0 } .Now take the odd extension of Φ to ˆ M , as follows:ˆΦ( p ) := ( ˆΦ ( p ) , . . . , ˆΦ ( p )) , where ˆΦ i ( p ) = Φ i ( p ) for i = 1 , . . . , ( p ) = ( Φ ( p ) if p ∈ M, − Φ ( p ) if p ∈ − M. Now, by a straightforward connectedness argument again, we conclude that ˆΦ : ˆ M → S is a diffeomorphismand thus, Φ is injective, as desired. (cid:3) Proof of Theorem 1.2.a.
From the proof above we have (perhaps after a Moebius transformation) that Φ(Σ)is an equator. Since Φ : ˆ M → S is a bijective diffeormphism, then necessarily the restriction of Φ to M maps the manifold diffeomorphically into a hemisphere S . (cid:3) Two boundary components.
We now study the case with two boundaries. We first remark that, asexplained in Section 3, we may choose a conformal metric on M such that the scalar curvature R is constantin M and the boundary is umbilic and minimal. Then, we can again consider again the doubling ˆ M ofthe manifold M (following the same construction of Subsection 4.1) and using the result in the Appendixof [25] we have that ˆ M is smooth with Escobar’s metric (and actually has a well characterized Green’sfunction). Moreover, since ˆ M is a compact l.c.f. Riemannian manifold without boundary, the developingmap Φ : ˆ M → S exists.We remark that the metric in the doubling ˆ M , denoted by ˆ g , can be taken to be C ,α smooth. In addition, R ˆ g >
0. Thus we can apply the results in [60, Theorem 4.1] to conclude that ˆ M is conformally equivalentto a quotient Ω / Γ for some domain Ω in S n .Now we restrict to the image of M by the developing map, denoted by Ω ′ = Φ( M ). It is a subdomain inΩ. The boundary of Ω ′ can be written as Γ := Φ(Σ) ∪ B , where the latter is a set of branching points.We first analyze the boundary Σ = ∂M , which we recall is umbilic and hence the image of Σ must be umbilicin S , that is, each component of Σ must be contained in a hypersphere of S . Now take into account that,for each connected component Σ ′ of Σ, Φ | Σ ′ is a local diffeomorphism from a compact manifold to a simplyconnected manifold, so we must have that Φ | Σ ′ is actually diffeomorphism. This implies that one can find asmall neighborhood around Σ in M such that Φ is actually a local diffeomorphism and hence, no branchingpoints can occur in this set.To analyze B away from Φ(Σ), we consider Ω ′ with the metric induced by the original metric g in M (notEscobar’s). Since with the metric g we assumed that the Q − curvature is positive, we can use the argumentsin the proof of [17, Theorem 1.2] to conclude that the set B is empty. It is important to observe that this ispossible since the proof of [17] is local around each point x ∈ B and we argued in the paragraph above that B is at a positive distance of Γ.In summary, we conclude that Φ cannot have branching points and it is a local diffeomorphism. This inparticular implies that ˆ M can be identified with a quotient of S and thus, by restricting the developingmap Φ : ˆ M → S to M we have that the developing map of M is injective.We remark that, in fact, under our assumptions, the classical proof of injectivity for the developing map bySchoen-Yau [60] can be performed directly for manifolds with boundary since all the additional boundaryintegral terms that appear would vanish. Proof of Theorem 1.2.c.
Consider the developing map Φ : ˆ M → S . We have that Φ is a conformal diffeo-morphism onto its image (regular at all points). Recall again the boundaries are assumed to be umbilic andhence their images are umbilic in S . Now, since Φ : M → S is an immersion, three situations can occur: • Both components of Σ are mapped to the same great circle in S . • Each component of Σ is mapped to two different great circles in S with non-empty intersection. IGENVALUE BOUNDS FOR PANEITZ WITH BOUNDARY 11 • Each component of Σ is mapped to two different circles in S with empty intersection. Thus Φ( M )an annulus type region in S (or R by stereographic projection).The first and second situations are ruled out by the injectivity of the developing map, thus we conclude thatwe are the third case and this finishes the proof of Theorem 1.2.c. (cid:3) Explicit calculations in known models
In this section we provide the explicit solution to the eigenvalue problem (1.19)-(1.21) for a family ofrotationally symmetric metrics.In the particular case that M is a flat ball B r of radius r in R , and Σ = ∂M the a sphere S r with itscanonical metric, we can simply write: P = ( − ∆) ,B = ∂ r ,B = − ∂ r ∆ − ∂ r − r ˜∆ − r ∂ r . (5.1)However, for our purposes it is more convenient to rewrite these operators in cylindrical coordinates.5.1. Cylindrical coordinates.
First, we write the flat metric as(5.2) | dx | = dr + r dθ = e − t [ dt + dθ ] =: e − t g c , where r = e − t is the radial variable, dθ is the canonical metric on S , and g c the cylindrical metric on X = R × S . Consider the spherical harmonic decomposition of S . For this, let µ ℓ and Y mℓ be theeigenvalues and eigenfunctions for − ∆ S , respectively. This is, µ ℓ = ℓ ( ℓ + 2) and − ∆ S Y mℓ = µ ℓ Y mℓ , ℓ = 0 , , . . . . Then any function u on R × S can be written as u ( t, θ ) = P ℓ,m u ℓ ( t ) Y mℓ ( θ ), t ∈ R , θ ∈ S .In order to write the Paneitz operator P with respect to the metric g c , we observe that P g c diagonalizesunder this eigenfunction decomposition. Let P ( ℓ ) its projection over the eigenspace h Y mℓ i . Recalling againthe conformal property (5.2), we have that P ( ℓ ) u ℓ = e − t ( − ∆ R ) | h Y mℓ i u ℓ = r (cid:16) ∂ rr + 3 r ∂ r − µ ℓ r (cid:17)(cid:16) ∂ rr + 3 r ∂ r − µ ℓ r (cid:17) u ℓ = (cid:16) ∂ tt − (2 + ℓ ) (cid:17) (cid:0) ∂ tt − ℓ (cid:1) u ℓ , (5.3)after the change of variable r = e − t .Let us consider P g c when M is a ball of radius r . The corresponding boundary operators on the boundaryΣ = S r , which in t coordinates is Σ = { t = − log r } , will be denoted by B r and B r . From the conformalchange g c = e t | dx | we have B r u = e − t B u | r = r = − ∂ t u | t = − log r , and(5.4) B , ( ℓ ) r u ℓ = e − t B u ℓ | h Y mℓ i ,r = r = { ∂ ttt − (3 µ ℓ + 3) ∂ t } u ℓ | t = − log r , where we have denoted by B , ( ℓ ) r , ℓ = 0 , , . . . the projection over spherical harmonics of B r .We next take a new metric in M given by g f = e f g c for a radially symmetric conformal factor f . ThePaneitz operator with respect to the metric g f on M will be denoted by P f , and the corresponding boundaryoperators on Σ by B f , B f . By the conformal property of the operators, we have P f u = e − f P g c ,B f u = e − f B r u (cid:12)(cid:12) t = − log r B , ( ℓ ) f u ℓ = e − f { ∂ ttt − (3 µ ℓ + 3) ∂ t o u ℓ (cid:12)(cid:12) t = − log r , (5.5) Eigenvalues for the (unit) ball.
Let M be the unit ball in R , parameterized in cylindrical coordi-nates (here t ∈ [0 , + ∞ )). We take a conformally flat metric g f = e f g c , where f only depends on the radialvariable t . We normalize e f (0) = 1, which is the same as normalizing the volume of the boundary sphere to2 π . After projection over spherical harmonics we obtain, Lemma 5.1.
In this setting, eigenvalues for (1.19) - (1.21) are given by λ ℓ = 4( ℓ + 2) for ℓ ≥ , λ = 0 , with associated eigenfunctions (5.6) u ℓ ( t ) Y mℓ for ℓ = 1 , , . . . , and u ( t ) = 1 . where u ℓ ( t ) = ( ℓ + 2)2 ℓ e − ℓt − e − ( ℓ +2) t . The proof will be postponed to the Appendix.5.3.
Radially symmetric metrics in an annulus.
We now let A ρ = { ρ ≤ r ≤ } be an annulus in R .In cylindrical coordinates we have t ∈ [0 , τ ], where τ = − log ρ . Take a conformally flat metric g f = e f g c ,where f = f ( t ) is a radially symmetric function. In addition, we impose the normalization(5.7) e f (0) + e f ( τ ) = 1 . This again fixes the volume of the boundary to be 2 π . We set α = e f (0) , for α ∈ (0 , u ℓ ( t ) Y mℓ ( θ ), ℓ = 0 , , . . . : Lemma 5.2.
Fix α ∈ (0 , . Let λ be the an eigenvalue for the ℓ -th projection, ℓ ≥ . Then λ satisfies thequadratic equation (5.8) a ( ℓ ) λ + b ( ℓ ) λ + c ( ℓ ) = 0 , where a ( ℓ ) = − ℓ ( ℓ + 2) + 2( ℓ + 1) cosh(2 τ ) − ℓ + 2) τ ) ,b ( ℓ ) = 1 α (1 − α ) 4 ℓ ( ℓ + 1)( ℓ + 2)[( ℓ + 1) sinh(2 τ ) + sinh((2 ℓ + 2) τ )] ,c ( ℓ ) = − α (1 − α ) 8 ℓ ( ℓ + 1) ( ℓ + 2) [cosh((2 ℓ + 2) τ ) − cosh(2 τ )] . For each ℓ ≥ , equation (5.8) has exactly two solutions λ − ℓ < λ + ℓ . For each λ that solves (5.8) , thecorresponding eigenfunctions are the form u ± ℓ ( t ) Y mℓ ( θ ) for u ± ℓ ( t ) := u ℓ + λ ± ℓ α ℓ ( ℓ + 2)( ℓ + 1) u ℓ , where u ℓ ( t ) =( ℓ + 2) sinh(( ℓ + 2) τ ) cosh( ℓt ) − ℓ sinh( ℓτ ) cosh(( ℓ + 2) t ) ,u ℓ ( t ) =[ ℓ sinh( ℓτ ) − ( ℓ + 2) sinh(( ℓ + 2) τ )][( ℓ + 2) sinh( ℓt ) − ℓ sinh(( ℓ + 2) t )] − ℓ ( ℓ + 2)[cosh( ℓτ ) − cosh(( ℓ + 2) τ )][cosh( ℓt ) − cosh(( ℓ + 2) t )] . For ℓ = 0 there are two eigenvalues: λ − = 0 (with just constant eigenfunctions) and λ +0 = 4 α (1 − α ) sinh(2 τ )1 − cosh(2 τ ) + τ sinh(2 τ ) > , with eigenfunction u +0 ( t ) = sinh(2 τ )(sinh(2 t ) − t ) + (1 − cosh(2 τ )) cosh(2 t ) + 2(1 − α )(1 − cosh(2 τ ) + τ sinh(2 τ )) − τ ) . IGENVALUE BOUNDS FOR PANEITZ WITH BOUNDARY 13
The proof is just computational and it is also postponed to the Appendix.If write λ − ℓ = − b ( ℓ )2 a ( ℓ ) − s b ( ℓ ) a ( ℓ ) − c ( ℓ ) a ( ℓ )it is clear that: Corollary 5.3.
All the non-trivial eigenvalues associated to the eigenvalue problem (1.19) - (1.20) - (1.21) in ( A ρ , g f ) are strictly positive. In addition, the eigenspace corresponding to the zero eigenvalue consists onlyof constant functions. Unfortunately, it is not possible to characterize the spectral gap for the operator since the calculations fora general τ are too complicated (even if elementary). However, we have strong numerical evidence for thefollowing: Conjecture.
For each τ >
0, the sequence { λ − ℓ } is increasing in ℓ .Note that a similar computation is performed in [26] for Steklov eigenvalues in 2 dimensions and in theirsituation the conjecture holds. In addition, they prove that for each α there is a value τ ∗ ( α ) such that for τ ≤ τ ∗ ( α ) the smallest non-trivial eigenvalue is given by λ − , while for τ ≥ τ ∗ ( α ) we have that the smallestnon-zero eigenvalue is λ +0 . In the result of [26] there is an α ∗ such that for τ ∗ ( α ∗ ) that the smallest eigenvalueis maximized and in that case λ − = λ +1 = λ +0 . We prove a partial result in that direction. Set β = α (1 − α ). Proposition 5.4.
Given β ∈ (cid:0) , (cid:1) , there are values τ − , τ ∗ and τ + such that: for τ − it holds λ +0 > λ − ,for τ − we have λ − > λ +0 , and for τ ∗ we encounter λ − = λ +0 .Proof. For ℓ = 1 we have a (1) = − τ ) − ,b (1) = 48 α (1 − α ) sinh(2 τ )[1 + cosh(2 τ )] ,c (1) = − α (1 − α ) [cosh(4 τ ) − cosh(2 τ )] . Then, the associated eigenvalue is λ − = 1 α (1 − α ) h τ )(1 + cosh(2 τ ))(cosh(2 τ ) − − s sinh (2 τ )(1 + cosh(2 τ )) (cosh(2 τ ) − − α (1 − α ) [cosh(4 τ ) − cosh(2 τ )](cosh(2 τ ) − i , and it has multiplicity four (this is the number of spherical harmonics in 3 dimensions associated to ℓ = 1).Now we take the quotient λ − /λ +0 , denoted by F ( β, τ ) := λ − λ +0 = 32 h sinh(2 τ )(1 + cosh(2 τ ))(cosh(2 τ ) − − s sinh (2 τ )(1 + cosh(2 τ )) (cosh(2 τ ) − − β [cosh(4 τ ) − cosh(2 τ )](cosh(2 τ ) − i · (cid:18) sinh(2 τ )1 − cosh(2 τ ) + τ sinh(2 τ ) (cid:19) − . A tedious, but straightforward computation reveals that • F ( β, τ ) → τ → • F ( β, τ ) → ∞ as τ → ∞ By the continuity of F ( β, τ ) we conclude that for each β there are values of τ − , τ ∗ and τ + for which F ( β, τ − ) < F ( β, τ + ) > F ( β, τ ∗ ) = 1, which concludes the proof. (cid:3) Remark . Note that for each value of β we have two values of α . This is because the problem is symmetric. Remark . Numeric computations strongly suggest that the function F ( β, τ ) in the proof above is increasingand the value τ ∗ is unique. Remark . If the conjecture above holds, then the smallest eigenvalue is either λ − or λ +0 .6. (Boundary) eigenvalue problems In this section we study the eigenvalue problem (1.19)-(1.20)-(1.21), which we recall here:(6.1) P g u = 0 in M,B g u = 0 on Σ ,B g u = λu on Σ . for M a manifold with boundary satisfying the hypothesis in Theorem 1.2. For (6.1), one can show thatthere exists a non-decreasing sequence of eigenvalues { λ g , λ g , λ g , . . . } . Corollary 1.3 characterizes the zero-eigenspace and yields positivity of the operator if Σ has either one or two connected components. In addition,recall that λ g is characterized by the Rayleigh quotient (1.22). Since the energy E Mg [ u ] is conformallyinvariant, the quotient (1.22) remains strictly positive when conformal transformations of M are applied.In what follows we prove Corollary 1.3 and Theorem 1.4, by considering separately the cases that, either M is conformally equivalent to a half-sphere S , or an annulus A ρ by Theorem 1.2. This allows to reducethe study of problem (6.1) to the model cases from the previous section.6.1. One boundary component. Proof of Corollary 1.3 and Theorem 1.4.
Let M be conformallyequivalent to a half-sphere S . By stereographic projection, we can assume that M = B , Σ = ∂ B = S ,with a conformal metric g = e w | dx | .We consider first the trivial setting, namely w ≡
0. Lemma 5.1 implies that λ g = 0 with Ker( B ) = { csts } and λ g >
0. Nevertheless, this result can be proved directly by a simple integration by parts argument.Indeed, take the model M = R , Σ = R , with coordinates ( x , x , x , y ), y > x , x , x ∈ R . In thisparticular case we have P = ∆ , B = − ∂ y , B = − ∂ y ∆ . Let ψ be any smooth solution to (6.1). Integrating by parts we explicitly see that λ Z Σ ψ dx = Z Σ ψB ψ dx = Z M (∆ ψ ) dxdy ≥ , and it is zero if and only if ∆ ψ = 0. Since we also have that ∂ y u = 0 at Σ, we conclude that ψ is constantup to the boundary, as desired.We also remark that existence such solution ψ can be proved by a standard minimization argument, whilethe uniqueness follows from the previous proof (since the constant obtained above would be 0).For the case of a general manifold M in Corollary 1.3 we recall again that the energy E Mg [ u ] in (1.22) isconformally invariant. In particular, if ˜ u is a minimizer of (1.22) with respect to the metric g = e w | dx | ,we would have that λ g = E Mg [˜ u ] R Σ ˜ u dv g ≥ λ R Σ ˜ u dv R Σ ˜ u dv g > , where λ g is the first non-zero eigenvalue with respect to the metric g , λ is the first non-zero eigenvalue withrespect to the flat metric and dv , dv g are volume elements with respect to the flat metric in the ball andthe metric given by g , respectively.To prove Theorem 1.4 we proceed as in the proof of Theorem 1.1. We use the unit ball model and assumethat there exists a conformal embedding Φ : M → B satisfying Φ(Σ) = ∂ B = S . In the two-dimensionalcase, the bound for the Steklov eigenvalue is obtained by using the coordinate functions of the embeddingas test functions in the Rayleigh quotient (1.22). In our setting we will use instead the (four) eigenfunctionsfor the first non-zero eigenvalue in the ball model and calculated in (5.6) for ℓ = 1. The precise expression is U m ( r, θ ) = u ( − log r ) Y m ( θ ) , m = 1 , , , . IGENVALUE BOUNDS FOR PANEITZ WITH BOUNDARY 15
Next, on M we set U m = U m ◦ Φ.Denote by Φ m , m = 1 , , , Z Σ Φ m dσ h = 0 . In addition, B g U m = 0 thanks to the covariance property (1.13) and the construction of u . Finally, notingthat we have u (0) = 1, this implies that U m = Φ m along Σ. We conclude that U m is an admissible testfunction in the Rayleigh quotient (1.22).We thus calculate(6.2) λ g Z Σ U m dσ g ≤ E Mg [ U m ] = E Φ( M ) g B [ U m ] . Adding on m = 1 , , , P m =1 ( Y m ) ( θ ) = 1 we have λ g Vol(Σ) ≤ X m E Φ( M ) g B [ U m ] . Next, since U m is an eigenfunction in the ball model, E B g B [ U m ] = λ Z S U m dσ g S . Adding on m , taking into account that λ = 12, we conclude λ g Vol(Σ) ≤
12 Vol( S ) = 24 π . Two boundary components. Proof of Corollary 1.3 and Theorem 1.4.
In the light of Theorem1.2, it is enough to take M to be conformally equivalent to the annulus A ρ = { ρ ≤ | x | ≤ } in R . Denoteby Σ , Σ ρ the boundaries at | x | = 1, | x | = ρ , respectively.First, Corollary 1.3 follows from Lemma 5.2 and Corollary 5.3.As in the case of one boundary component, we will use the eigenfunctions corresponding to ℓ = 1 as testfunctions. These are calculated in Lemma 5.2. More precisely, we take the one corresponding to λ − . Fromthe discussion above, it will be clear that this choice may not yield the sharpest bound. In any case, let U m ( r, θ ) := u − ( − log r ) u (0) Y m ( θ ) , m = 1 , , , , and set U m := U m ◦ Φ. Let c ( ρ ) := u − ( τ ) u − (0) , where τ = − log ρ . Again, a calibration argument implies that U m is an admissible test function in the Rayleigh quotient (1.22), and(6.3) λ g Z Σ U m dσ g ≤ E Mg [ U m ] = E Φ( M ) g flat [ U m ] . Adding on m = 1 , , , P m =1 ( Y m ) ( θ ) = 1 we have, on the one hand, for the left handside X m Z Σ U m dσ g = Vol(Σ ) + c ( ρ ) Vol(Σ ρ ) ≥ min { , c ( ρ ) } Vol(Σ) . On the other hand, for the left hand side in (6.3), X m E A ρ g flat [ U m ] = λ − Z ∂ A ρ U m dσ g flat = λ − X m "Z { r =1 } U m dσ S + Z { r = ρ } U m dσ g S ρ = λ − X m (cid:2) Vol( S ) + c ( ρ ) Vol( S ρ ) (cid:3) = 2 λ − π (1 + c ( ρ ) ρ ) . We conclude λ g Vol (Σ) ≤ λ − π c ( ρ ) ρ min { , c ( ρ ) } . It is possible to construct different test functions using any of the eigenfunctions given by Lemma 5.2. Wedescribe separately the bound for ℓ = 0 and ℓ > U := U ◦ Φ, where U ( r, θ ) := u +0 ( − log r ) and u +0 is given by Lemma 5.2.This yields λ g Vol (Σ) ≤ λ +0 π c ( ρ ) ρ min { , c ( ρ ) } , where c := u ( τ ) u (0) . Note that R Σ u +0 dσ h = 0 because u +0 should be orthogonal to the zero eigenspace, thus u +0 is an admissible test function.A bound for general ℓ can be obtained similarly, recalling that the spherical harmonics satisfy ([5, Chapter2.2]) X m =1 ( Y mℓ ) ( θ ) = N ℓ π , where N ℓ = (2 l + 2)( l + 1)!2 l ! . Then we obtain λ g Vol (Σ) ≤ λ − ℓ N ℓ π c ℓ ( ρ ) ρ min { , c ℓ ( ρ ) } . where c ℓ := u ℓ ( τ ) u ℓ (0) .We conclude the result by taking the infimum of these bounds. Remark that the bound can be takenuniform in ρ as ρ →
0. 7.
Appendix
Proof of Lemma 5.1.
This is a straightforward calculation that we detail below. Taking into account thatthe Paneitz operator P is conformally invariant, in the interior of the ball equation (1.19) reduces to (cid:16) ∂ tt − (2 + ℓ ) (cid:17) (cid:0) ∂ tt − ℓ (cid:1) u ℓ = 0In addition we require(7.1) u ′ ℓ (0) = 0(which corresponds to the condition (1.20) B ( u ℓ ) = 0), and u is finite at infinity. Let v ℓ = ( ∂ tt − ℓ ) u ℓ , then (cid:16) ∂ tt − (2 + ℓ ) (cid:17) v ℓ = 0so v ℓ ( t ) = Ae − ( ℓ +2) t . Hence, for ℓ ≥ u ℓ ( t ) = Ce − ℓt + A ℓ + 4 e − ( ℓ +2) t . Imposing the boundary condition (7.1) we have u ℓ ( t ) = − A ( ℓ + 2)4 ℓ ( ℓ + 1) e − ℓt + A ℓ + 4 e − ( ℓ +2) t . The eigenvalues of B are given by B u ℓ u ℓ (cid:12)(cid:12)(cid:12) t =0 , this is, recalling (5.5), λ ℓ = ( ℓ + 2) ℓ − ( ℓ + 2) − ( ℓ + 2) + 1 = 4( ℓ + 2) , for ℓ ≥ . For ℓ = 0 the calculation is slightly different, since u = C + A e − t . Imposing the boundary condition implies that A = 0, this is, u is constant and λ = 0, as expected. (cid:3) IGENVALUE BOUNDS FOR PANEITZ WITH BOUNDARY 17
Proof of Lemma 5.2.
Fix ℓ ≥
1. Using (5.3) and the conformal property (1.2) we observe that the generalsolution u ℓ can be written as u ℓ ( t ) = A sinh( ℓt ) + B cosh( ℓt ) + C sinh(( ℓ + 2) t ) + D cosh(( ℓ + 2) t ) . Imposing that B u ℓ = 0 at the boundary of A ρ , it is easy to see that solutions are spanned by two functionsthat can be chosen as u ℓ ( t ) =( ℓ + 2) sinh(( ℓ + 2) τ ) cosh( ℓt ) − ℓ sinh( ℓτ ) cosh(( ℓ + 2) t ) u ℓ ( t ) =[ ℓ sinh( ℓτ ) − ( ℓ + 2) sinh(( ℓ + 2) τ )][( ℓ + 2) sinh( ℓt ) − ℓ sinh(( ℓ + 2) t )] − ℓ ( ℓ + 2)[cosh( ℓτ ) − cosh(( ℓ + 2) τ )][cosh( ℓt ) − cosh(( ℓ + 2) t )] . Then, eigenfunctions can be written as Au ℓ + Bu ℓ .We can directly compute ∂ ttt u ℓ ( t ) =( ℓ + 2) ℓ sinh(( ℓ + 2) τ ) sinh( ℓt ) − ℓ ( ℓ + 2) sinh( ℓτ ) sinh(( ℓ + 2) t ) ∂ ttt u ℓ ( t ) = ℓ ( ℓ + 2)[ ℓ sinh( ℓτ ) − ( ℓ + 2) sinh(( ℓ + 2) τ )][ ℓ cosh( ℓt ) − ( ℓ + 2) cosh(( ℓ + 2) t )] − ℓ ( ℓ + 2)[cosh( ℓτ ) − cosh(( ℓ + 2) τ )][ ℓ sinh( ℓt ) − ( ℓ + 2) sinh(( ℓ + 2) t )] . The eigenfunction condition at t = 0 can be written from (5.4) as e − f (0) ( A∂ ttt u ℓ (0) + B∂ ttt u ℓ (0)) = λ ( Au ℓ (0) + Bu ℓ (0)) , which implies e − f (0) B ([ ℓ sinh( ℓτ ) − ( ℓ + 2) sinh(( ℓ + 2) τ )][( ℓ + 2) ℓ − ℓ ( ℓ + 2) ]= λA (( ℓ + 2) sinh(( ℓ + 2) τ ) − ℓ sinh( ℓτ )) , or equivalently,(7.2) 4 Bℓ ( ℓ + 2)( ℓ + 1) = λAe f (0) . For the eigenvalue condition at τ we need to take into account that the outward normal is reversed, so wehave − e − f ( τ ) ( A∂ ttt u ℓ ( τ ) + B∂ ttt u ℓ ( τ )) = λ ( Au ℓ ( τ ) + Bu ℓ ( τ )) . Multiplying by λ and using (7.2) we obtain − e − f ( τ ) (4 ℓ ( ℓ + 2)( ℓ + 1) e − f (0) ∂ ttt u ℓ ( τ ) + λ∂ ttt u ℓ ( τ )) = λ ℓ ( ℓ + 2)( ℓ + 1) e − f (0) u ℓ ( τ ) + λ u ℓ ( τ )Or equivalently(7.3) a ( ℓ ) λ + b ( ℓ ) λ + c ( ℓ ) = 0 , where a ( ℓ ) = u ℓ ( τ ) ,b ( ℓ ) =4 ℓ ( ℓ + 1)( ℓ + 2) e − f (0) u ℓ ( τ ) + e − f ( τ ) ∂ ttt u ℓ ( τ ) ,c ( ℓ ) =4 ℓ ( ℓ + 1)( ℓ + 2) e − f ( τ )+ f (0)) ∂ ttt u ℓ ( τ ) . From the previous computations we have, at t = τ , u ℓ ( τ ) =( ℓ + 1) sinh(2 τ ) + sinh((2 ℓ + 2) τ ) ,u ℓ ( τ ) = − ℓ ( ℓ + 2) + 2( ℓ + 1) cosh(2 τ ) − ℓ + 2) τ ) , and for the derivatives ∂ ttt u ℓ ( τ ) = − ℓ ( ℓ + 1)( ℓ + 2)[cosh((2 ℓ + 2) τ ) − cosh(2 τ )] ,∂ ttt u ℓ ( τ ) =4 ℓ ( ℓ + 1)( ℓ + 2)[( ℓ + 1) sinh(2 τ ) + sinh((2 ℓ + 2) τ )] . To conclude, recall our normalization (5.7) and set α = e f (0) , for α ∈ (0 , e − f ( τ ) = (1 − α ) − .Let us have a closer look at these eigenvalues now. By differentiating twice in τ , we can easily check that a ( ℓ ) < τ >
0. In addition, b ( ℓ ) > c ( ℓ ) <
0. After some calculation, one can explicitly see that the discriminant associated to (7.3) is strictly positive for α ∈ (0 , < λ − ℓ < λ + ℓ .Now we compute the eigenfunction for ℓ = 0. In that case we that the particular solutions are u ( t ) = sinh(2 τ )(sinh(2 t ) − t ) + (1 − cosh(2 τ )) cosh(2 t ) ,u ( t ) = 1 , so the general solution can be written as Au + Bu . The eigenvalue condition at t = 0 is equivalent to(7.4) 8 e − f (0) A sinh(2 τ ) = λ ( A (1 − cosh(2 τ )) + B ) . On the other hand, at t = τ we have − e − f ( τ ) A (cid:8) sinh(2 τ ) cosh(2 τ ) + (1 − cosh(2 τ )) sinh(2 τ ) (cid:9) = λ (cid:8) A sinh(2 τ )(sinh(2 τ ) − τ ) + A (1 − cosh(2 τ )) cosh(2 τ ) + B (cid:9) . Subtracting both equations, we obtain a simple non-trivial eigenvalue: λ +0 = 4 α (1 − α ) sinh(2 τ )1 − cosh(2 τ ) + τ sinh(2 τ ) . Substituting in (7.4) we obtain B = { − α )(1 − cosh(2 τ ) + τ sinh(2 τ )) − τ ) } A, and this finishes the proof of Lemma 5.2. (cid:3) Acknowledgements:
M.d.M. Gonz´alez is supported by the Spanish government grant and MTM2017-85757-P and the Severo Ochoa program at ICMAT. M. S´aez is supported by the grant Fondecyt Regular1190388.
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Mar´ıa del Mar Gonz´alezUniversidad Aut´onoma de MadridDepartamento de Matem´aticas, and ICMAT, 28049 Madrid, Spain
Email address : [email protected] Mar´ıa del Mar Gonz´alezP. Universidad Cat´olica de ChileDepartamento de Matem´aticas, Santiago, Chile
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