Hemisphere rigidity under Q-curvature lower bound
aa r X i v : . [ m a t h . DG ] F e b HEMISPHERE RIGIDITY UNDER Q-CURVATURE LOWER BOUND
MIJIA LAI AND WEI WEI
Abstract.
Let g be a metric on hemisphere S n + ( n ≥
3) which is conformal to the standard roundmetric g . Suppose its Q -curvature Q g is bounded below by Q , we show that g is isometric to g , provided that the induced metric on ∂S n + coincides with g up to certain order. Introduction
In this paper, we prove two rigidity theorems for the standard hemisphere under Q -curvaturelower bound. Q -curvature is a fourth order curvature first introduced by Branson [B]. There isa conformal covariant operator P , the Paneitz operator, associated with the Q -curvature. Thepair ( P, Q ) plays an important role in conformal geometry. Even though Q -curvature is a higherorder curvature, it still reinforces considerable control on the geometry and topology of underlyingmanifolds, especially in dimension four (c.f. [C]). Our first theorem adds a new phenomenon to thisspeciality. Theorem A.
Let g be a smooth metric on S , which is conformal to the standard round metric g .Suppose (1) Q g ≥ Q , (2) the induced metric on ∂S n + coincides with g , (3) ∂S n + is totally geodesic.Then g = g . For other dimensions, we need to add an additional condition to obtain the rigidity of S n + . Theorem B.
Let g be a smooth metric on S n + , n = 4 , which is conformal to the standard roundmetric g . Suppose (1) Q g ≥ Q , (2) the induced metric on ∂S n + coincides with g , (3) ∂S n + is totally geodesic, (4) R g ≥ n ( n − on ∂S n + .Then g = g . Now let’s give detailed definitions of P and Q . Let ( M, g ) be a smooth Riemannian manifoldwith dimension n ≥ , the Q curvature is given by (we follow the notation in [HY]) Q = − ∆ J − | A | + n J , where J = R n − , A = 1 n − Ric − Jg ) . The Paneitz operator is defined as
P ϕ = ∆ ϕ + div (4 A ( ∇ ϕ, e i ) e i − ( n − J ∇ ϕ ) + n − Qϕ, (1)
M. Lai’s research is supported in part by National Natural Science Foundation of China No. 12031012. W. Wei’sresearch is supported in part by BoXin programme BX2019082. where e , · · · , e n is an orthonormal frame with respect to g .In dimension 4 , the Paneitz operator satisfies P e w g ϕ = e − w P g ϕ, and the Q curvature transforms as Q e w g = e − w ( P g w + Q g ) . In dimension n = 4 , the operator satisfies P ρ n − g ϕ = ρ − n +4 n − P g ( ρϕ ) , for any positive smooth function ρ. As a consequence of (1) we have Q ρ n − g = 2 n − P ρ n − g n − ρ − n +4 n − P g ρ. (2)Our work is inspired by the work of Hang and Wang [HW], in which they proved the followinghemisphere rigidity. Theorem 1.1 (Hang-Wang) . Let g be a metric on S n + conformal to the standard spherical metric g , suppose • R g ≥ n ( n − , • the induced metric on ∂S n + coincides with g ,then ( S n + , g ) is isometric to the standard hemisphere. The above theorem confirmed the Min-Oo conjecture within the conformal class of the roundmetric. The Min-Oo conjecture was motivated by the celebrated positive mass theorem. Such studyyields many interesting theorems on the rigidity of manifolds with nonnegative scalar curvature,e.g., [M], [ST]. As revealed by the counterexample to the Min-Oo conjecture given by Brendel-Marques-Neves [BMN], now we know the rigidity phenomenon is more intriguing when scalar curva-ture has positive lower bound and underlying manifold is modeled on spherical geometry. Thereforeone usually confines within the conformal class of the round metric as the first attempt to investigaterigidity of certain spherical domains.There are several generalizations of Hang-Wang’s theorem on locally conformlly flat manifolds,e.g., [R], [S]. More recently, Barbosa-Cavalcante-Espinar [BCE] proved a fully nonlinear version ofHang-Wang’s theorem, where the scalar curvature is replaced by some fully nonlinear operator oneigenvalues of the Schouten tensor. Gursky-Zhang [GZ] also proved a rigidity theorem for S underthe assumption that the metric is Bach-flat. Our main theorem can be viewed as a higher ordergeneralization of Hang-Wang’s theorem.From analytical point of view, the key to our work relies on proving the uniqueness of certainbiharmonic equations on the unit ball subject to geometric meaningful boundary conditions. Theuniqueness of metrics on S n conformal to g with constant Q -curvature was proved by Chang-Yang [CY], Lin [L], Wei-Xu [WX]. The analysis is reduced to the study of the entire solutionof ∆ u = e u for n = 4; ∆ u = u n +4 n − for n ≥ . It is also known that there exists a continuous family of conformal metrics on the standard hyperbolicspace having constant Q -curvature thanks to the work of Grunau-Ould Ahmedou-Reichel [GOR].Finally, we would like to mention that we don’t know if condition (4) in Theorem B is necessaryor not.The organization of the paper is as follows: in Section 2, we prove Theorem A. Four dimensiondistinguishes itself from others due to the fact that the integral of the Q -curvature plus a boundaryterm is conformal invariant; in Section 3, we prove Theorem B. EMISPHERE RIGIDITY UNDER Q-CURVATURE LOWER BOUND 3 Rigidity of S In this section, we prove Theorem A. We first establish a uniqueness result for a fourth orderdifferential inequality. The theorem then follows from a spherical average method on the conformalfactor.
Proposition 1.
Let v : [0 , ∞ ) → R be a smooth function satisfying v ′′′′ ( t ) − v ′′ ( t ) ≥ e v (3) with v (0) = 0 , v ′ (0) = 0 , v ′′′ (0) = 0 and lim t →∞ v ′ ( t ) = − . Then v = − ln cosh( t ) .Proof. Since ( v ′′′ ( t ) − v ′ ( t )) ′ ≥ , and v ′′′ (0) − v ′ (0) = 0, we infer that v ′′′ ( t ) − v ′ ( t ) ≥ t ∈ R + . By the maximum principle, v ′ ( t ) cannot attain any positive local maximum. In view of v ′ (0) = 0 and lim t →∞ v ′ ( t ) = −
1, itfollows that v ′ ( t ) ≤ ∀ t ∈ R + . Consequently v ′′ (0) ≤
0. Multiplying v ′ ( t ) to (3), we find( v ′′′ v ′ − v ′′ − v ′ − e v ) ′ ≤ . Hence ( v ′′′ v ′ − v ′′ − v ′ − e v )(0) ≥ lim t →∞ ( v ′′′ v ′ − v ′′ − v ′ − e v )( t ) . It follows that v ′′ (0) ≤ Case 1: v ′′ (0) = − . We then have( v ′′′ v ′ − v ′′ − v ′ − e v )(0) = lim t →∞ ( v ′′′ v ′ − v ′′ − v ′ − e v )( t ) . Thus ( v ′′′ v ′ − v ′′ − v ′ − e v ) ′ ≡ , and consequently v ′′′′ ( t ) − v ′′ ( t ) = 6 e v ( t ) . With such initial data the solution is unique locally. Since w := − ln cosh( t ) is a global solutionhaving same set of initial data with v , thus v = w . We would like to mention that a detailed analysisof above O.D.E was carried out [FM]. Case 2: − < v ′′ (0) ≤ . We shall show this case does not occur. Suppose not, we first claim v ≥ w = − ln cosh( t ) , ∀ t ∈ R + , (4)and then derive a contradiction.Since v (0) = w (0) = v ′ (0) = w ′ (0) = 0 and v ′′ (0) > w ′′ (0), we have ( v − w )( t ) > t small. Suppose (4) does not hold, then there exists t such that ( v − w )( t ) ≥ t ∈ [0 , t ] and( v − w )( t ) = 0. Thus there exists an interior maximum of v − w in (0 , t ), say at t . It follows that( v − w ) ′′ ( t ) ≤
0. Hence there exists t ≤ t such that( v − w ) ′′ ( t ) ≥ , ∀ t ∈ [0 , t ] and ( v − w ) ′′ ( t ) = 0 . Since v ( t ) ≥ w ( t ) for t ∈ [0 , t ], we have v ′′′′ ( t ) − v ′′ ( t ) ≥ e v ≥ e w = w ′′′′ ( t ) − w ′′ ( t ) . Setting f := ( v − w ) ′′ , we then have f ′′ ( t ) − f ( t ) ≥ , t ∈ [0 , t ] . MIJIA LAI AND WEI WEI
Let g ( t ) = ae t + b . Choose a, b such that g (0) = a + b = f (0) > , g ( t ) = ae t + b = f ( t ) = 0 . It is easy to see that a < , b >
0. Direct computation shows that( f − g ) ′′ − f − g ) ≥ . In view of ( f − g )(0) = 0 and ( f − g )( t ) = 0, we infer that f ( t ) ≤ g ( t ) for t ∈ [0 , t ], from which weget f ′ (0) ≤ g ′ (0) = 2 a < , a contradiction.It follows from (4) that v ′′′′ ( t ) − v ′′ ( t ) ≥ e v ≥ e w = w ′′′′ ( t ) − w ′′ ( t ) , ∀ t ∈ R + . Consequently, f := ( v − w ) ′′ cannot attain nonnegative local maximum. Since f (0) > t →∞ f ( t ) = 0, we have only two possibilities for the behavior of f : Case 2.a: f is nonnegative and decreasing. Sincelim t →∞ ( v − w ) ′ ( t ) = Z ∞ f ( t ) dt + ( v − w ) ′ (0) , we necessarily have f ≡
0, a contradiction.
Case 2.b: there exists T such that f ≥ on [0 , T ] with f ( T ) = 0 . In this case, a similar barrierfunction construction ( g = ae t + b , with g (0) = f (0) > g ( T ) = f ( T ) = 0) yields a contradictiontoo. (cid:3) Proof of Theorem A.
Under the cylindrical coordinates, the standard metric is g = e w (cid:0) ( dt ) + g S (cid:1) , with w = − ln cosh( t ). Note t ∈ [0 , ∞ ) and ∂S corresponds to t = 0.Assuming g = e u (cid:0) ( dt ) + g S (cid:1) and using (1), the condition Q g ≥ Q g is equivalent to( ∂ ∂t + ∆ S ) ( u ) − ∂ u∂t ≥ e u . (5)We consider the spherical average of u : v ( t ) = 1v Z S u ( t, x ) dx, where v = vol( S ), the volume of standard 3-sphere. We then have v ′′′′ ( t ) = 1v Z S ∂ u∂t ( t, x ) dx ≥ Z S − (∆ S ) ( u ) − S ( ∂ u∂t ) + 4 ∂ u∂t + 6 e u dx = 4 v ′′ ( t ) + 6v Z S e u dx ≥ v ′′ ( t ) + 6 e v . (6)We have used (5) and the Jensen’s inequality R S e u dx ≥ e R S udx . Hence v ( t ) satisfies thedifferential inequality (3).We next verify the boundary conditions of v . In view of boundary conditions of the metric g , wehave u (0 , x ) ≡ ∂u∂t (0 , x ) ≡ . Thus v (0) = 0 and v ′ (0) = 0. Moreover since t = ∞ corresponds to a smooth interior point, itfollows that v ′ ( t ) → − t → ∞ . Consequently v ′′ ( t ) → v ′′′ ( t ) → t → ∞ . EMISPHERE RIGIDITY UNDER Q-CURVATURE LOWER BOUND 5 v ′′′ (0) is a hidden information due to the conformal invariance of the integral of the Q -curvature.On a compact 4-manifold M with boundary, the Gauss-Bonnet-Chern formula is4 π χ ( M ) = Z M ( |W| Q ) dv + Z ∂M ( L + T ) ds, where |W| dv and L ds are pointwise conformal invariants and T is a boundary curvature term firstintroduced by Chang and Qing in their study of zeta functional determinants on manifolds withboundary [CQ]. As a consequence Z M Qdv + Z ∂M T ds is conformally invariant. In the special case that M has totally geodesic boundary (c.f. [C]) T = 112 ∂∂ν R, where ν is the unit outward normal of the boundary w.r.t g . Thus for the standard metric g , wehave Z M Qdv + Z ∂M T ds = Z S Q g dv = Z ∞ Z S Q g e w dxdt = 12 v Z ∞ w ′′′′ ( t ) − w ′′ ( t ) dt = 12 v ( w ′′′ − w ′ ) | ∞ = 2v . (7)By the transformation law of the scalar curvature, we have e u R g = 6(∆ ( e u ) − e u ) . Thus 3 e u ∂u∂ν R g + e u ∂R g ∂ν = 6 (cid:18) ∂∂ν ∆ ( e u ) − e u ∂u∂ν (cid:19) . Since u ≡ ν is the same as the unit normal w.r.t g , which coincides with − ∂∂t on the boundary. Also note ∆ is w.r.t the product metric ( dt ) + g S .Hence ∂∂ν ∆ ( e u ) = ∂ ∂ν ( e u ) + ∂∂ν (∆ ( e u )) = ∂ ∂ν ( e u ) + ∆ ∂∂ν ( e u ) . Since u = 0 and ∂u∂ν = 0 on the boundary, we obtain that ∂R g ∂ν = 6 ∂ u∂ν = − ∂ u∂t .Hence for g = e u (cid:0) ( dt ) + g S (cid:1) , Z S Q g dv g + Z ∂S T ds = Z ∞ Z S Q g e u dxdt − Z S ∂ u∂t (0 , x ) dx = 12 v (cid:18)Z ∞ v ′′′′ ( t ) − v ′′ ( t ) dt − v ′′′ (0) (cid:19) = 12 v (cid:0) ( v ′′′ − v ′ ) | ∞ − v ′′′ (0) (cid:1) . (8)Comparing (8) with (7), we infer v ′′′ (0) = 0. By Proposition 1, v ≡ w . Hence all inequalities in(6) are equalities. It follows that u is radially symmetric and thus u = v = w . (cid:3) As an application of Theorem A, we prove
Theorem 2.1.
Let ( M , g ) be a locally conformal flat manifolds with totally geodesic boundary suchthat • the first Yamabe constant Y ( M, ∂M, [ g ]) = inf ˜ g ∈ [ g ] vol (˜ g ) − ( R M ˜ Rdv ˜ g + 2 R ∂M H ˜ g dσ ˜ g ) > , • Q g ≥ Q g , • ∂M is isometric to S (1) ,then ( M , g ) is isometric to the standard hemisphere S . MIJIA LAI AND WEI WEI
Proof.
Since ∂M is totally geodesic, we may form the double manifold: ˆ M = M ⊔ ∂M ( − M ), witha C , metric ˆ g . One can smooth ˆ g to get a conformally related smooth metric, say ˜ g with positive˜ Q . Since the first Yamabe constant Y is positive, by the work of Escobar [E], there exists metricconformal to g with positive scalar curvature and totally geodesic boundary, thus the Yamabeconstant of ˆ M is also positive. In view of the famous conformal sphere theorem of Chang-Gursky-Yang [CGY], we infer that ˆ M is diffeomorphic to S or RP . Since χ ( ˆ M ) = 2 χ ( M ) − χ ( ∂M ) = 2 χ ( M )must be an even number, RP is ruled out. Hence ˆ M is diffeomorphic to S . Since g is locallyconformally flat, it follows by Kuiper’s theorem [K] that ( ˆ M , g ) is indeed conformally equivalent tothe standard round sphere ( S , g ).Let ψ : ˆ M → S be a conformal diffeomorphism. Since umbilicity is preserved by conformal maps,we may assume that ψ maps ∂M onto ∂S . Since g | ∂M is isometric to ∂S , after composing with aconformal diffeomorphism which is an isomorphim on ∂S , we may assume that ( ψ − ) ∗ ( g ) = e u g ,with u = 0 on ∂S . Now the theorem follows directly from Theorem A. (cid:3) Rigidity of S n + , n = 4In this section, we prove Theorem B.Since the hemisphere is conformal to the Euclidean ball in R n , we use the Euclidean metric asthe background metric. Assume that g = u n − dx , then by (2), we shall be concerned with thefollowing differential inequalities: n ≥ : ∆ u ≥ Qu n +4 n − in B u = u s = 1 on ∂B ∂u∂n = ∂u s ∂n = − n − on ∂B , (9) where Q = ( n − n − n ( n +2)16 , u s = ( | x | ) n − and n is the unit outer normal vector on ∂B . n = 3 : ∆ u + u − ≤ B u = 1 on ∂B ∂u∂n = on ∂B . (10)Note u s is the corresponding conformal factor of the standard metric on the hemisphere and itsatisfies ∆ u = Qu n +4 n − . Since Q < n = 3, the inequality in (10) is reversed. This is the onlydifference as far as the proof is concerned. So from now on, we may assume n ≥ n = 3 to some obvious direction-reversions of inequalities.Under cylindrical coordinates, we assume g = ˜ u n − ( dt + g S n − ), then ˜ u ( t, x | x | ) = | x | n − u ( x ),where t = − ln | x | . (9) is then transformed to ( ( ∂ ∂t + ¯∆) ˜ u − A ∂ ∂t ˜ u + 2 ¯∆˜ u + B ˜ u ≥ Q ˜ u n +4 n − ˜ u (0) = 1 , ∂ ˜ u∂t (0) = 0 , (11)where A = ( n − +42 , B = ( n − n and ¯∆ is the Laplacian on S n − .Using (11) and the Jensen inequality, we similarly have Lemma 1.
Let ˜ v ( t ) = v n − R S n − ˜ u ( t, x ) dx be the spherical average of ˜ u , and let v be the corre-sponding radial function related to ˜ v by the relation ˜ v ( t ) = | x | n − v ( x ) , where t = − ln | x | . Then v satisfies (9) ( n ≥ ) or (10) ( n = 3 ). EMISPHERE RIGIDITY UNDER Q-CURVATURE LOWER BOUND 7
Lemma 2.
Let u be a radial solution of ∆ u = Qu n +4 n − in B u = c on ∂B ∂u∂n = c on ∂B ∆ u = c on ∂B , (12) where c , c , c are constants. Then u is unique.Proof. Suppose there are two solutions u , u of (12). We claim that u and u are ordered. Ifnot, first of all, as radial solutions, the zeros of u − u do not accumulate. So let r ∗ be the largestzero of u − u in (0 , u ( r ) ≥ u ( r ) on [ r ∗ , w := u − u , then we have w ( r ∗ ) = 0 , w ( r ) > ∀ r ∈ ( r ∗ , . Let r ∗∗ ∈ ( r ∗ ,
1) be such that ( u − u )( r ∗∗ ) = max [ r ∗ , ( u − u ) . It follows that w ′ ( r ∗∗ ) = 0 and ∆ w ( r ∗∗ ) ≤
0. On B \ B r ∗∗ , ∆ ( u − u ) = Q ( u n +4 n − − u n +4 n − ) ≥ . By the maximum principle, ∆ w ≤ max ∂ ( B \ B r ∗∗ ) ∆ w ≤ on B \ B r ∗∗ , and then w ≥ min ∂ ( B \ B r ∗∗ ) w = min { w | ∂B , w | ∂B r ∗∗ } . But we know ∂w∂n | ∂B r ∗∗ = 0 = ∂w∂n | ∂B , which contradicts to the Hopf Lemma. The claim thus isproved.Without loss of generality, we assume u ≥ u in B . Thus we have ∆ u ≥ ∆ u in B . By∆ u = ∆ u on ∂B , we have ∆( u − u ) ≤ B and ∂ ( u − u ) ∂n = 0 on ∂B which contradicts tothe Hopf Lemma again. (cid:3) Corollary 1.
There are at most two positive solutions satisfying ∆ u = Qu n +4 n − in B u = 1 on ∂B ∂u∂n = − n − on ∂B . (13) Proof.
By the work of Berchio-Gazzola-Weth [BGW], the solution of (13) is radial symmetric. ThePohozaev identity for the radial solution of ∆ u = Qu n +4 n − (c.f. [GOR]) implies that at r = 10 = r n − (∆ u ) ′ (cid:18) ru ′ + n − u (cid:19) + n r n − u ′ ∆ u − r n (cid:18)
12 (∆ u ) + n − n Qu nn − (cid:19) . Plugging the boundary conditions, we infer that n u ′ (1)∆ u (1) − (cid:18)
12 (∆ u (1)) + n − n Qu (1) nn − (cid:19) = 0 , from which we find two possible values for ∆ u (1): a = − ( n − n + 2)4 , a = − ( n − n − . The proof is thus completed in view of Lemma 2. (cid:3)
MIJIA LAI AND WEI WEI
It is desirable to prove the uniqueness of the solution (13). For the sake of the proof of Theorem B,we just need the following lemma.
Lemma 3.
Let u i be the solution of (13) satisfying ∆ u i (1) = a i . Then u ≤ u . ( u ≥ u , if n = 3 )Proof. We present the proof for n ≥
5. The proof n = 3 can be modified. We construct a sequenceof functions u k satisfying ∆ u k = Qu n +4 n − k − in B u k = 1 on ∂B ∂u k ∂n = − n − on ∂B , with u = tu + (1 − t ) u . Notice∆ u = Q ( tu + (1 − t ) u ) n +4 n − ≤ Q (cid:18) tu n +4 n − + (1 − t ) u n +4 n − (cid:19) = ∆ u . It follows from the comparison theorem (Theorem 5.6 in [GGS]) that u ≤ u . Inductively, we have a monotone decreasing sequence of functions u ≥ u ≥ · · · . Clearly we have a uniform lower barrier u k ( x ) ≥ n − n − | x | . Using the representation of Green’s function for biharmonic equation and monotone convergencetheorem, there exists a limit ˜ u ( x ) = lim k →∞ u k ( x ) , which solves (13). Therefore we have tu + (1 − t ) u ≥ u or tu + (1 − t ) u ≥ u . It follows u ≥ u since ∆ u (1) < ∆ u (1). By Lemma 2, we also have u = u s . (cid:3) Proposition 2.
Let u be a positive function satisfying (9), then u ≥ u s in B . ( u ≤ u s , if n = 3 )Proof. We construct a sequence of functions u k satisfying ∆ u k = Qu n +4 n − k − in B u k = 1 on ∂B ∂u k ∂n = − n − on ∂B , where u is set to be u . Like before, we have a sequence of monotone decreasing functions u ≥ u ≥ · · · , and u k ≥ n − n − | x | , ∀ k .Hence there exists a positive function ˜ u such that ˜ u ( x ) = lim k →∞ u k ( x ) which satisfies (13) andthus by Lemma 3, we have u s ≤ ˜ u ≤ u . (cid:3) Finally we are in position to prove Theorem B.
Proof of Theorem B.
By Lemma 1 and Proposition 2, we know v ≥ u s . Appealing to the conformalchange of the scalar curvature: 4( n − n − u n − n − ) + R g u n +2 n − = 0 , EMISPHERE RIGIDITY UNDER Q-CURVATURE LOWER BOUND 9 we have that ∆ u (1) ≤ − ( n +2)( n − . By the definition of the spherical average, we may find by directcomputation that ∆ v (1) ≤ − ( n +2)( n − = ∆ u s (1). Hence∆ ( v − u s ) ≥ Qv n +4 n − − Qu n +4 n − s ≥ . Since ∆( v − u s ) ≤ ∂B , it follows from the maximum principle that ∆( v − u s ) ≤ B ,however v − u s = ∂v − u s ∂ν = 0 on ∂B . It follows from the Hopf lemma that v ≡ u s , and consequentlythat u ≡ u s . (cid:3) References [BCE] E. Barbosa, M. Cavalcante and J. Espinar,
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