Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimension three
aa r X i v : . [ m a t h . A P ] J a n INTERIOR CURVATURE ESTIMATES FORHYPERSURFACES OF PRESCRIBING SCALARCURVATURE IN DIMENSION THREE
GUOHUAN QIU
Abstract.
We prove a priori interior curvature estimates for hyper-surfaces of prescribing scalar curvature equations in R . The method ismotivated by the integral method of Warren and Yuan in [22]. The newobservation here is that the “Lagrangian” submanifold constructed sim-ilarly as Harvey and Lawson in [6] has bounded mean curvature if thegraph function of a hypersurface satisfies the scalar curvature equation. Introduction
In this paper, we are studying the regularity thoery of a hypersurface M n ⊆ R n +1 with positive scalar curvature R g > . In hypersurface geome-try, the Gauss equation tells us R g = σ ( κ ) := X ≤ i , where ν is the normal of the given hypersurface as a graph over a ball B r ⊂ R n . This is the second order elliptic PDE depends on the graphfunction u . If n = 2 , it is Monge-Ampere equation(1.2) det( u ij ) = f ( x, u, ∇ u ) . Our study of the scalar curvature equation is motived by isometric embed-ding problems. A famous isometric embedding problem is the Weyl problem.It is the problem of realizing, in three-dimensional Euclidean space, a regu-lar metric of positive curvature given on a sphere. This is the Weyl problemwhich was finally solved by Nirenberg [15] and Pogorelov [16] independently.They solved the problem of Weyl by a continuity method where obtaining C estimate to the scalar curvature equation is important to the method.Also motived by the Weyl problem, E. Heinz [8] first derived a purelyinterior estimate for the equation (1.2) in dimension two. If u satisfies the Mathematics Subject Classification. nterior estimate for scalar curvature equations in dimension threeequation (1.2) in B r ⊆ R with positive f , then(1.3) sup B r | D u | ≤ C ( | u | C ( B r ) , | f | C ( B r ) , inf B r f ) . And this type of estimate turns out to be very useful when one study theisometric embedding problem for surfaces with boundary or for non-compactsurfaces. But Heinz’s interior C estimate is false for the convex solutionsto the equation det D u = 1 in B r ⊆ R n when n ≥ by Pogorelov [17] .The second motivation is from the studying of fully nonlinear partial dif-ferential equation theory itself. Caffarelli-Nirenberg-Spruck started to study σ k − Hessian operators and established existence of Dirichlet problem for σ k equations in their seminal work [2]. Here the σ k − Hessian operators are the k − th elementary symmetric function for ≤ k ≤ n . The key to the existenceof Dirichlet problem is by establishing C estimates up to the boundary. sup ¯Ω | D u | ≤ C ( | u | C (¯Ω) , f, ϕ, ∂ Ω) . Although there are C estimates to σ k − Hessian equations for boundary valueproblems, there are, in general, no interior C estimates to σ k − Hessian equa-tions. Because the Pogorelov’s counter-examples were extended in [21] to k ≥ . The best we can expect is the Pogorelov type interior C estimateswith homogeneous boundary data which were derived by Pogorelov [17] for k = n and by Chou-Wang for k < n [4] . So people in this field want to knowwhether the interior C estimate for σ equations holds or not for n ≥ . Infact, the interior regularity for solutions of the following σ -Hessian equationand prescribing scalar curvature equation is a longstanding problem,(1.4) σ ( ∇ u ) = f ( x, u, ∇ u ) > , x ∈ B r ⊂ R n . and σ ( κ ( x )) = f ( X ( x ) , ν ( x )) > . A major breakthrough was made by Warren-Yuan [22]. They obtained C interior estimate for the equation(1.5) σ ( ∇ u ) = 1 . x ∈ B ⊂ R Recently in [13], McGonagle-Song-Yuan proved interior C estimate for con-vex solutions of the equation σ ( ∇ u ) = 1 in any dimensions. Using a differ-ent argument, the interior C estimates for solutions of more general equa-tions (1.4) and (1.1) in any dimensions with certain convexity constraintswere also obtained by Guan-Qiu in [5]. Moreover, we proved interior curva-ture estimates for isometrically immersed hypersurfaces in R n +1 with positivescalar curvature in that paper [5].In this paper, we completely solve this problem for scalar equations indimension three. Theorem 1.
Suppose M is a smooth graph over B ⊂ R with positivescalar curvature and it is a solution of equation (1.1). Then we have (1.6) sup x ∈ B | κ ( x ) | ≤ C, uohuan Qiu where C depends only on || M || C ( B ) k f k C ( B × S ) , and k f k L ∞ ( B × S ) . Analogously we have
Theorem 2.
Let u be a solution to (1.4) on B ⊂ R . Then we have (1.7) sup B | D u | ≤ C, where C depends only on k f k C ( B × R × R ) , k f k L ∞ ( B × R × R ) and || u || C ( B ) .Remark . The proof of Theorem 2 is similar to Theorem 1. So we willomit its proof which can be found in a recent paper [18] by the author. Themethod given here is easier than to the previous one, so we will not submitthe previous paper [18].In order to introduce our idea, let us briefly review the ideas for attackingthis problem so far. In two dimensional case, Heinz used Uniformizationtheorem to transform this interior estimate for Monge-Ampere equation intothe regularity of an elliptic system and univalent of this mapping, see also[7, 12] for more details. Another interesting proof using only maximumprinciple was given by Chen-Han-Ou in [3]. Our new quantity in [5] can givea new proof of Heinz. The restriction for these methods is that we need someconvexity conditions which are not easily got in the higher dimension.In R , a key observation made in [22] is that equation (1.5) is exactly thespecial Lagrangian equation which stems from the special Lagrangian ge-ometry [6]. And an important property for the special Lagrangian equationis that the Lagrangian graph ( x, Du ) ⊂ R × R is a minimal submanifoldwhich has mean value inequality and sobolev inequality. So Warren-Yuanhave proved interior C estimate for the special Lagrangian submanifoldwhich in turn proved interior C estimate for the special Lagrangian equa-tion. Our new observation in this paper is that the graph ( X, ν ) , where X is position vector of the hypersurface whose scalar curvature satisfy equa-tion (1.4), can be viewed as a submanifold in R × R with bounded meancurvature. Then applying similar arguement of Michael-Simon [14], see alsoHoffman-Spruck [9], we have a mean value inequality in order to removethe convexity condition in [5]. Finally, we apply a modified argument ofWarren-Yuan in [22] to get the estimate.At last, we remark that the arguments are higher co-dimensional analo-gous to the original integral proof by Bombieri-De Giorgi-Miranda [1] for thegradient estimate for co-dimension one minimal graph and by Ladyzhenskayaand Ural’Tseva [10] for general prescribed mean curvature equations. Herewe use the method very similar to Trudinger’s simplified proof of gradientestimate for mean curvature equations in [19, 20].The higher dimensional cases for these equations are still open to us.2. Preliminary Lemmas
We first introduce some definitions and notations.nterior estimate for scalar curvature equations in dimension three
Definition 4.
For λ = ( λ , · · · , λ n ) ∈ R n , the k - th elementary symmetricfunction σ k ( λ ) is defined as σ k ( λ ) := X λ i · · · λ i k , where the sum is taken over for all increasing sequences i , · · · , i k of theindices chosen from the set { , · · · , n } . The definition can be extended tosymmetric matrices where λ = ( λ , · · · , λ n ) are the corresponding eigenval-ues of the symmetric matrices.For example, in R σ ( D u ) := σ ( λ ( D u )) = λ λ + λ λ + λ λ . Definition 5.
For ≤ k ≤ n , let Γ k be a cone in R n determined by Γ k = { λ ∈ R n : σ ( λ ) > , · · · , σ k ( λ ) > } . The following lemma is from [11].
Lemma 6.
Suppose λ ∈ Γ and σ ii ( λ ) := ∂σ ∂λ i , then there is a constant c > depending only on n such that for any i from to n , σ ii ( λ ) ≥ cσ ( λ ) σ ( λ ) . If we assume that λ ≥ · · · ≥ λ n , then there exist c > and c > dependingonly on n such that (2.1) σ ( λ ) λ ≥ c σ ( λ ) , and for any j ≥ (2.2) σ jj ( λ ) ≥ c σ ( λ ) . So the curvature estimates can be reduced to the estimate of mean cur-vature H due to the following fact(2.3) max | λ i | ≤ H = σ ( κ ) . In the rest of this article, we will denote C to be constant under control(depending only on k f k C , k f k L ∞ and k M k C ), which may change line byline.Suppose that a hypersurface M in R n +1 can be written as a graph over B r ⊆ R n . At any point of x ∈ B , the principal curvature κ = ( λ , λ , · · · , λ n ) of the graph M = ( x, u ( x )) satisfy a equation(2.4) σ ( κ ) = f ( X, ν ) > , where X is the position vector of M , and ν a normal vector on M .We choose an orthonormal frame in R n +1 such that { e , e , · · · , e n } aretangent to M . Let ν is a normal on M such that H > . We recall thefollowing fundamental formulas of a hypersurface in R n +1 : X ij = − h ij ν ( Gauss f ormula ) ν i = h ij e j ( W eingarten equation ) h ijk = h ikj ( Codazzi equation ) R ijkl = h ik h jl − h il h jk ( Gauss equation ) , uohuan Qiuwhere R ijkl is the curvature tensor. We also have the following commutatorformula: h ijkl − h ijlk = h im R mjkl + h mj R mikl . (2.5)Combining Codazzi equation, Gauss equation and (2.5), we have h iikk = h kkii + X m ( h im h mi h kk − h mk h ii ) . (2.6)For scalar curvature equation (1.1) with positive scalar curvature, we mayassume that M is admissible in the following definition without loss of gen-erality. Definition 7. A C surface M is called admissible if at every point X ∈ M ,its principal curvature satisfies κ ∈ Γ . Moreover, for any symmetric matrix h ij , it follows from Lemma 6 that σ ij := ∂σ ( λ ( h ij )) ∂h ij is positive definite if λ ( h ij ) ∈ Γ . Lemma 8.
Suppose the scalar curvature of hypersurface M satisfies equation(2.4). In orthonormal coordinate, we have the following equations (2.7) σ kl h kli = ∇ f ( e i ) , and σ kl h iikl + X k = l h kki h lli − X k = l h kli h kli − f X k h ki + ( f σ − σ ) h ii = ∇ f ( e i , e i ) . (2.8) If f is a form with gradient term, then there are estimates (2.9) |∇ f | ≤ C (1 + H ) , and (2.10) − C (1 + H ) + X k h kij d ν f ( e k ) ≤ ∇ f ( e i , e j ) ≤ C (1 + H ) + X k h kij d ν f ( e k ) . Proof.
Taking twice differential of the equation σ ( κ ) = f , we get (2.7) and σ kl h klii + X k = l h kki h lli − X k = l h kli h kli = ∇ f ( e i , e i ) . Then we obtain (2.8) by (2.6) and the following elementary identities σ ij h ij = 2 f, and X m σ kl h mk h ml = σ σ − σ . nterior estimate for scalar curvature equations in dimension threeMoreover, by (2.3), Codazzi equation and the following direct computa-tions ∇ f ( e i ) = d X f ( e i ) + h ki d ν f ( e k ) , and ∇ f ( e i , e j ) = d X f ( e i , e j ) + h kj d X,ν f ( e i , e k ) − h ij d X f ( ν ) + h ki d ν,X f ( e k , e j )+ h ki h lj d ν f ( e k , e l ) − h ki h kj d ν f ( ν ) + h kij d ν f ( e k ) , we get the estimates (2.9) and (2.10). (cid:3) We recall some elementary facts about hypersurface. Denoting W = p | Du | , the second fundamental form and the first fundamental formof the hypersurface can be written in local coordinate as h ij = u ij W and g ij = δ ij + u i u j . The inverse of the first fundamental form and the Wein-garten Curvature are g ij = δ ij − u i u j W and h ji = D i ( u j W ) . Definition 9.
The Newton transformation tensor is defined as [ T k ] ji := 1 k ! δ ii ··· i k jj ··· j k h i j · · · h i k j k , and the corresponding (2 , -tensor is defined as [ T k ] ij := [ T k ] ik g kj . From this definition one can easily show a divergence free identity X j ∂ j [ T k ] ji = 0 . Lemma 10.
There is a family of elementary relations between σ k opertatorsand Newton transformation tensors [ T k ] ji = σ k δ ji − [ T k − ] li h jl . (2.11)or(2.12) [ T k ] ji = σ k δ ji − [ T k − ] jl h li . Moreover, the (2 , -tensor of T k is symmetry such that(2.13) [ T k ] ij = [ T k ] ji . Proof.
We only prove the first one, because the second one is similar. FromDefinition 4, it is easy to check that(2.14) σ k ( κ ) = 1 k ! δ i ··· i k j ··· j k h i j · · · h i k j k . By definition and (2.14), we obtain (2.11) as follows: [ T k ] ji = 1 k ! δ ii ··· i k jj ··· j k h i j · · · h i k j k = 1 k ! δ i ··· i k j ··· j k h i j · · · h i k j k δ ji − k − δ ii ··· i k j j ··· j k h jj h i j · · · h i k j k = σ k δ ji − [ T k − ] ki h jk . uohuan QiuFor k = 1 , the symmetry of the (2 , -tensor of T is obviously from thesymmetry of h . Inductively, we assume the symmetry of (2 , -tensor T k istrue when k = m . From (2.11), we have [ T m +1 ] ij = [ T m +1 ] il g lj = σ m +1 δ il g lj − [ T m ] pl h ip g lj = σ m +1 g ij − [ T m ] pj h ip . On the other hand, by (2.12) we have [ T m +1 ] ji = [ T m +1 ] jl g li = σ m +1 δ jl g li − [ T m ] jp h pl g li = σ m +1 g ji − [ T m ] jp h pi = σ m +1 g ji − [ T m ] jp h ip . So from the symmetry of g and T m , we have proved (2.13). (cid:3) Lemma 11. If u satisfies the scalar equation (1.1), then the following inte-gral is bounded Z B r ( x ) ( σ f − σ ) dx ≤ C, where C depends only on k f k C ( B r +1 ( x )) and k u k C ( B r +1 ( x )) .Proof. For a non-negative function φ ∈ C ∞ ( B r +1 ( x )) with |∇ φ | + |∇ φ | ≤ C , we assume that φ ≡ in B r ( x ) and ≤ φ ≤ in B r +1 ( x ) . It is obviousthat the first part of the integral is bounded as follows Z B r ( x ) f σ dx ≤ C Z B r +1 ( x ) φ σ dx = C Z φ div ( DuW ) dx = C Z − X i ( φ ) i u i W dx ≤ C. Then we estimate the second part Z B r ( x ) − σ dx ≤ − Z B r +1 ( x ) φ σ dx = − Z φ [ T ] ji D j ( u i W ) dx = 2 Z φ [ T ] ji φ j u i W dx.
Using (2.11), we continue our estimate Z φ [ T ] ji φ j u i W dx = Z φφ i u i W σ dx − Z φ [ T ] ki φ j u i W D k ( u j W ) dx ≤ C + Z [ T ] ki ( φφ j ) k u i W u j W dx + Z [ T ] ki D k ( u i W ) φφ j u j W dx ≤ C, where we have used (2.3) and the scalar curvature equation (1.1) in thelast inequality. (cid:3) nterior estimate for scalar curvature equations in dimension three3. An important differential inequality
Let us consider the quantity of b ( x ) := log σ . In dimension three, wehave a very important differential inequality. Lemma 12.
For admissible solutions of the equation (1.1) in R , we have (3.1) σ ij b ij ≥ σ ij b i b j − C ( f σ − σ ) + g ij b i d ν f ( e j ) , where C depends only on k f k C , k f k L ∞ and k u k C .Remark. We do not know whether the corresponding higher dimensionalinequalties (3.1) hold or not. This is one of the difficulty to generalize ourthereom in higher dimensions.
Proof.
The calculation was done in Lemma 3 of [18]. We give its details inthe appendix. (cid:3) Mean value inequality.
In this section we prove a mean value type inequality. So we can transformthe pointwise estimate into the integral estimate which is easier to deal with.It is unclear for higher dimensional scalar curvature equations. This is thesecond difficulty to generalize our theorem in higher dimensions.
Theorem 13.
Suppose u are admissible solutions of equation (1.1) on B ⊂ R , then we have for any y ∈ B (4.1) sup B b = b ( y ) ≤ C Z B ( y ) b ( x )( σ f − σ ) dx, where C depends only on k f k C , k f k L ∞ and k u k C . Proof.
Because the graph X Σ := ( X, ν ) = ( x , x , x , u, u W , u W , u W , − W ) where u satisfies equation (1.1) can be viewed as a three dimensional smoothsubmanifold in ( R × R , f ( X, ν ) i =4 P i =1 dx i + i =4 P i =1 dy i ) .When f = 1 , we shall see it is a submanifold with bounded mean curva-ture. This is the key observation in the paper.In fact, we have X Σ i = ( X i , ν i ) = ( X i , h ki X k ) , and G ij = < X Σ i , X Σ j > R × R = g ij + h ki h kj . If we denote σ ij := ∂σ ∂h ij , we can verify that G ij = σ ij σ − σ . uohuan QiuThen we can prove that the mean curvature is bounded as follows: | H | ≤ | G ij X Σ ij | = | σ ij σ − σ ( − h ij ν, h kij X k − h ki h kj ν ) |≤ | ( − σ ν, − ( σ − σ ) ν ) σ − σ | ≤ C. By Michael-Simon’s mean value inequalities we get the estimate (6.10) forscalar curvature equations.When f = f ( X, ν ) , we write down the details of this proof in the appendix. (cid:3) Proof of the theorem 1
Proof.
From Theorem 13, we have at the maximum point x of ¯ B (0) b ( x ) ≤ Z B ( x ) b ( σ f − σ ) dx. (5.1)We shall estimate the first part R B ( x ) bσ f dx in the above integral atfirst. Recalling that(5.2) σ ij b ij ≥ σ ij b i b j − C ( σ f − σ ) + g ij d ν f ( e i ) b j , we have an integral version of this inequality for any r < , Z B r +1 − σ ij φ i b j dM ≥ c Z B r +1 φσ ij b i b j dM (5.3) − C [ Z B r +1 ( σ f − σ ) φdx + Z B r +1 g ij d ν f ( e i ) b j φdM ] . for all non-negative φ ∈ C ∞ ( B r +1 ) . We choose different cutoff functions.They are all denoted by ≤ φ ≤ , which support in larger ball B r +1 ( x ) and equals to in smaller ball B r ( x ) with | Dφ | + | D φ | ≤ C . Z B ( x ) bσ dx ≤ Z B ( x ) φbσ dx ≤ C ( Z B ( x ) bdx + Z B ( x ) | Db | dx ) ≤ C (1 + Z B ( x ) | Db | dx ) . (5.4)We only need to estimate R B ( x ) | Db | dx . We use σ σ ij ≥ cδ ij , to get Z B ( x ) | Db | dx ≤ C Z B ( x ) q σ ij b i b j √ σ dx. nterior estimate for scalar curvature equations in dimension threeBy Holder inequality, Z B ( x ) | Db | dx ≤ ( Z B ( x ) σ ij b i b j dx ) ( Z B ( x ) σ dx ) ≤ C Z B ( x ) φ σ ij b i b j dx. (5.5)Then using (5.3) and Lemma 11, we get Z B ( x ) φ σ ij b i b j dx ≤ C Z B ( x ) φ σ ij b i b j dM ≤ C [ − Z B ( x ) φσ ij φ i b j dM + Z B ( x ) φ ( σ f − σ ) dx + Z B ( x ) φ | Db | dM ] ≤ C ( Z B ( x ) q φ σ ij b i b j q σ kl φ k φ l dx + 1 + Z B ( x ) φ q σ ij b i b j √ σ dx ) . By Cauchy-Schwarz inequality Z B ( x ) φ σ ij b i b j dx ≤ C ( ǫ Z B ( x ) φ σ ij b i b j dx + Z B ( x ) σ ij φ i φ i dx + 1 ǫ ) ≤ Cǫ Z B ( x ) φ σ ij b i b j dx + Cǫ .
We choose ǫ small such that Cǫ ≤ , Z B ( x ) φ σ ij b i b j dx ≤ C. (5.6)So far we have obtained the estimate for the first part of (5.1), by combining(5.4), (5.5), and (5.6). We have(5.7) Z B ( x ) bf σ dx ≤ C. The second part is to estimate R B ( x ) − bσ dx . Thanks to the divergencefree property, we integral by parts as follows − Z B ( x ) φ bσ dx = − Z φ b [ T ] ji D j ( u i W ) dx = Z [ T ] ji ( φ ) j b u i W | {z } I dx + Z [ T ] ji φ b j u i W | {z } II dx. (5.8)uohuan QiuWe estimate I by applying (2.11), I = Z ( σ δ ji − [ T ] ki h jk )( φ ) j b u i W dx ≤ Z bdx − Z [ T ] ki D k ( u j W )( φ ) j b u i W dx ≤ C + Z [ T ] ki u j W ( φ ) jk b u i W dx + Z [ T ] ki u j W ( φ ) j b k u i W dx +2 Z σ u j W ( φ ) j bdx ≤ C + Z σ bdx + Z [ T ] kl b k g li u i W u j W ( φ ) j dx. (5.9)The seconde term of (5.9) can be estimated by the same argument asbefore. We only need to estimate the last term of (5.9). By Cauchy-Schwarzinequality and (5.6), we have Z [ T ] kl b k g li u i W X j ( u j W )( φ ) j dx ≤ Z φ [ T ] ij b i b j dx + 4 Z [ T ] ij g ik u k W g jl u l W ( X p u p φ p W ) dx ≤ C. (5.10)From (5.9) and (5.10) we obtain(5.11) I = Z [ T ] ji ( φ ) j b u i W dx ≤ C. Now we deal with II by using (2.12) II ≤ Z ( σ δ ji − [ T ] jk h ki ) φ b j u i W dx ≤ C Z | Db | dx − Z [ T ] jk h ki φ b j u i W dx. (5.12)As before, the first term of (5.12) is already estimated by (5.5) and (5.6).We compute the second term of (5.12) − Z [ T ] jk h ki φ b j u i W dx ≤ Z φ [ T ] ji b j b i dx + 2 Z [ T ] ij h ik u k W h jl u l W φ dx ≤ Z φ [ T ] ji b j b i dx + 2 Z σ u i u j W h ij φ dx − Z σ | Du | W φ dx. By (5.6) and Lemma 11, we get the estimate for II ,(5.13) II = Z [ T ] ji φ b j u i W dx ≤ C. With the estimate (5.11) and (5.13) for I and II , we get(5.14) Z B ( x ) − bσ dx ≤ C. Finally, combining (5.7) and (5.14), we get the estimate(5.15) log σ ( x ) ≤ C. (cid:3) nterior estimate for scalar curvature equations in dimension three6. Appendix
Proof of the Lemma 12.
Proof.
We may choose an orthonormal frame and assume that { h ij } is diag-onal at the point. The differential equation of b by using Lemma 8 is A := σ ij b ij − ǫσ ij b i b j ≥ P i ( P k = p h kpi − P k = p h kki h ppi ) σ − (1 + ǫ ) σ ii ( P k h kki ) σ − C ( f σ − σ ) + g ij b i d ν f ( e j ) . We use (2.7) to substitute terms with h iii in A , A ≥ h σ + 2 P k = p h kpp σ + X k = p h kkp σ ( P i = p σ ii h iip − f p σ pp ) − h h + 2 h h + 2 h h σ − C ( f σ − σ ) + g ij b i d ν f ( e j ) − (1 + ǫ ) σ ii ( P k = i h kki − P k = i σ kk h kki σ ii + f i σ ii ) σ . Due to symmetry, we only need to give the lower bound of the terms whichcontain h and h . We denote these terms by A . A := 2( σ + σ ) h σ σ + 2( σ + σ ) h σ σ − h + h ) f σ σ + 2( σ + σ − σ ) h h σ σ − (1 + ǫ )[( λ − λ ) h + ( λ − λ ) h + f ] σ σ . Then we use Cauchy-Schwarz inequality and Lemma 6, − (1 + ǫ )[( λ − λ ) h + ( λ − λ ) h + f ] σ σ ≥− (1 + 2 ǫ )[( λ − λ ) h + ( λ − λ ) h ] σ σ − Cǫ σ . (6.1)Similarly, we have − h + h ) f σ σ ≥ − ǫ σ ( h + h ) σ σ − f ǫ σ σ ≥ − ǫ σ ( h + h ) σ σ − Cǫ σ . (6.2)uohuan QiuThen we substitute (6.1) and (6.2) into A to get A ≥ σ + 2 σ σ σ h + 2 σ + 2 σ σ σ h + 4 λ σ σ h h − ǫ σ ( h + h ) σ σ − (1 + 2 ǫ )[( λ − λ ) h + ( λ − λ ) h ] σ σ − Cǫ σ . We will show the Claim 14 and Claim 15 in the below. Then we choose ǫ = , such that δ ≥ ǫ − ǫ , where δ is small constant in the Claim 15.In all, we shall get A = X i =1 A i − C ( f σ − σ ) + g ij b i d ν f ( e j ) ≥ − C ( f σ − σ ) + g ij d ν f ( e i ) b j . (cid:3) Claim . For any ǫ ≤ , we have σ + 2 σ σ σ h + 2 σ + 2 σ σ σ h + 4 λ σ σ h h ≥ ǫσ ( h + h ) σ σ . Proof.
This claim follows from the following elementary inequality ( σ + σ − ǫσ )( σ + σ − ǫσ ) − ( λ − ǫσ ) = (1 − ǫ ) σ + (1 − ǫ ) σ ( λ + λ ) + λ λ − ( λ − ǫσ ) = 3(1 − ǫ ) f + (2 − ǫ )( λ + λ λ + λ ) . If we assume − ǫ ≥ , we have above inequality nonnegative. (cid:3) Claim . For any δ ≤ , we have σ + 2 σ σ σ h + 2 σ + 2 σ σ σ h + 4 λ σ σ h h ≥ (1 + δ )[( λ − λ ) h + ( λ − λ ) h ] σ σ . Proof.
We can compute the coefficient in front of h σ σ σ + σ ) σ − (1 + δ )( λ − λ ) = (1 − δ ) λ + (1 − δ ) λ + 4 λ + 6 f + 2 δλ λ = (1 − δ )( λ + δ − δ λ ) + 1 − δ − δ λ + 4 λ + 6 f. nterior estimate for scalar curvature equations in dimension threeAnd similarly, the coefficient in front of h σ σ is (1 − δ )( λ + δ − δ λ ) + 1 − δ − δ λ + 4 λ + 6 f. We also compute the coefficient of h h σ σ λ σ − (1 + δ )( λ − λ )( λ − λ ) = (1 − δ ) λ + (3 + δ ) f − δ ) λ λ = (1 − δ )( λ + δ − δ λ )( λ + δ − δ λ ) − − δ − δ λ λ + 3 f. It is easy to see that for any small δ [ 1 − δ − δ λ + 4 λ ][ 1 − δ − δ λ + 4 λ ] ≥ [ − − δ − δ λ λ ] and (6 f ) ≥ (3 f ) . We have proved that the coefficient matrix in front of h , h and h h is positive definite. So we complete the proof of this claim. (cid:3) Proof of the theorem 13.
Proof.
We prove this theorem similar to Michael-Simon [14]. First fromLemma 12, we have σ ij b ij ≥ − C ( f σ − σ ) + g ij b i d ν f ( e j ) . Let χ be a non-negative, non-decreasing function in C ( R ) with support inthe interval (0 , ∞ ) and set ψ ( r ) = Z ∞ r tχ ( ρ − t ) dt where < ρ < , and r = f ( X ( x ) , ν ( x )) | X ( x ) − X ( y ) | +2 − ν ( x ) , ν ( y )) .Let us denote B ρ = { x ∈ B ( y ) : f ( X ( x ) , ν ( x )) | X ( x ) − X ( y ) | +2 − ν ( x ) , ν ( y )) ≤ ρ } . We may assume that ( X ( y ) , ν ( y )) = (0 , E ) .By direct computation, we have(6.3) rr i = f i | X | + 2 f ( X, e i ) − h ki ( e k , E ) , and r i r j + 2 rr ij = f ij | X | + 2 f i ( X, e j ) + 2 f j ( X, e i ) + 2 f δ ij − f h ij ( X, ν ) − h kij ( e k , E ) + 2 h ki h kj ( ν, E ) . (6.4)Because λ + λ > , we may assume λ < , and λ ≥ λ ≥ λ , f σ − σ ≥ f λ + λ λ ≥ − cλ λ ≥ − cλ λ . (6.5)uohuan QiuBy equation we also have(6.6) λ λ = f − λ λ − λ λ ≤ C ( f σ − σ ) . We then have from (6.3), (6.4), (6.5), (6.6) and Lemma 8 σ ij ψ ij = σ ij ( − r i rχ ( ρ − r )) j = − σ ij r ij rχ ( ρ − r ) − σ ij r i r j χ ( ρ − r ) + σ ij r i r j rχ ′ ( ρ − r )= − χ ( ρ − r ) σ ij [ f δ ij − f h ij ( X, ν ) + f ij | X | f i ( X, e j )]+ χ ( ρ − r ) σ ij [ h kij ( e k , E ) − h ki h kj ( ν, E )]+ σ ij r i r j rχ ′ ( ρ − r ) ≤ − σ f − σ ) χ + C ( r χ + rχ )( f σ − σ )+ σ ij r i r j rχ ′ . (6.7)We claim that(6.8) σ ij r i r j ≤ ( f σ − σ )(1 + Cr ) . In fact, σ ii [ f i | X | + 2 f ( X, e i ) − P k h ki ( e k , E )] r ≤ C ( f σ − σ ) r + σ ii [ f ( X, e i ) − h ii ( e i , E )] r . Moreover, we have following elementary properties(6.9) ( f σ − σ ) δ ij − f σ ij = σ kl h ki h lj . and f σ ii h ii ( X, e i ) + 2 f σ ii ( X, e i )( e i , E ) h ii + f σ ii ( e i , E ) ≥ f σ ii [ h ii ( X, e i ) + ( e i , E )] . Then we obtain (6.8) by σ ii [ f ( X, e i ) − h ii ( e i , E )] r ≤ ( f σ − σ ) . We obtain from (6.8) and (6.7) that σ ij ψ ij ≤ ( σ f − σ )[ − χ + C ( r χ + rχ ) + (1 + Cr ) rχ ′ ] . Then we mutiply both side by b and take integral on the domain B , Z B bσ ij ψ ij dM ≤ ρ ddρ ( Z B bχ ( ρ − r ) ρ ( σ f − σ ) dM )+ C Z B br χ ′ ( σ f − σ ) dM + C Z B rbχ ( ρ − r )( σ f − σ ) dM. (6.10)nterior estimate for scalar curvature equations in dimension threeBy (3.1), we have(6.11) − C Z B ( σ f − σ ) ψdM + Z B g ij d ν f ( e i ) b j ψdM ≤ Z B bσ ij ψ ij dM. Inserting (6.11) into (6.10), we get − ddρ ( R B bχ ( ρ − r ) ρ ( σ f − σ ) dM ) ≤ C R B rbχ ( ρ − r )( σ f − σ ) dMρ + C R B br χ ′ ( σ f − σ ) dMρ + C R B ( σ f − σ ) ψdMρ − R B g ij d ν f ( e i ) b j ψ dMρ . Because χ , χ ′ and ψ are all supported in B ρ , we deal with right hand sideof above inequality term by term. For the first term, we have(6.12) C R B rbχ ( ρ − r )( σ f − σ ) dMρ ≤ C R B bχ ( ρ − r )( σ f − σ ) dMρ . Then for the second term, we integrate from δ to R , R Rδ R B br χ ′ ( σ f − σ ) dMρ dρ ≤ R Rδ R B bχ ′ ( σ f − σ ) dMρ dρ ≤ R B bχ ( σ f − σ ) dMρ | Rδ + R Rδ R B bχ ( σ f − σ ) dMρ dρ. (6.13)For the third term, we use the definition of ψ to estimate C R B b ( σ f − σ ) ψdMρ ≤ C R B bχ ( ρ − r )( σ f − σ ) dMρ . (6.14)For the last term, we use (6.3) and the definition of ψ to get − R B g ij d ν f ( e i ) b j ψdMρ ≤ C R B bσ ψdM − R B g ij d ν f ( e i ) r j brχdMρ ≤ C [ R B bσ ψdM + R B ( σ f − σ ) brχdM ] ρ ≤ C R B bχ ( ρ − r )( σ f − σ ) dMρ . (6.15)Combining (6.12), (6.13), (6.14) and (6.15), we integrate from ≤ δ to R ≤ , then use Grönwall’s inequality to get Z B b ( σ f − σ ) χ ( δ − r ) δ dM ≤ C Z B b ( σ f − σ ) χ ( R − r ) R dM. uohuan QiuLetting χ approximate the characteristic function of the interval (0 , ∞ ) ,in an appropriate fashion, we obtain,(6.16) R B δ b ( σ f − σ ) dMδ ≤ C R B R b ( σ f − σ ) dMR . Because the graph ( X, ν ) where u satisfied equation (1.4) can be viewedas a three dimensional smooth submanifold in ( R × R , f ( P i =1 dx i ) + P i =1 dy i ) with volume form exactly ( σ f − σ ) dM . Moreover, for a sufficient small δ > , the geodesic ball with radius δ of this submanifold is comparable with B δ . Then let δ → , we finally get b ( y ) ≤ C R B R (¯ y ) b ( σ f − σ ) dMR ≤ C R B R ( y ) b ( σ f − σ ) dxR . (cid:3) Acknowledgement.
The author would like to express gratitude to ProfessorPengfei Guan for supports and many helpful discussions when he did post-doctoral research at McGill University.
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Department of Mathematics, The Chinese University of Hong Kong, Shatin,N.T., Hong Kong, China.
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