WEAK CONTINUITY OF THE GAUSS-CODAZZI-RICCI SYSTEM FOR ISOMETRIC EMBEDDING

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1 WEAK CONTINUITY OF THE GAUSS-CODAZZI-RICCI SYSTEM FOR ISOMETRIC EMBEDDING GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG Abstract. We estabish the wea continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform L p -bounded soution sequence for p > 2, which impies that the wea imit of the isometric embeddings of the manifod is sti an isometric embedding. More generay, we estabish a compensated compactness framewor for the Gauss-Codazzi-Ricci system in differentia geometry. That is, given any sequence of approximate soutions to this system which is uniformy bounded in L 2 has reasonabe bounds on the errors made in the approximation (the errors are confined in a compact subset of H 1 oc ), then the approximating sequence has a weay convergent subsequence whose imit is a soution of the Gauss-Codazzi-Ricci system. Furthermore, a minimizing probem is proposed as a seection criterion. For these, no restriction on the Riemann curvature tensor is made. 1. Introduction The Gauss-Codazzi-Ricci system is a fundamenta system of noninear partia differentia equations in differentia geometry (cf. [2, 3, 10, 12, 13, 21, 23]). For exampe, the fundamenta theorem of the surface theory indicates that the existence of a oca or goba soution of the Gauss-Codazzi-Ricci system can yied a oca or goba higher dimensiona isometric embedding. Therefore, it is important to underst the behavior of this noninear system for soving isometric embedding probems other important geometric probems. In genera, the Gauss-Codazzi-Ricci system has no type, neither purey hyperboic nor purey eiptic. We are concerned with the wea continuity of the Gauss-Codazzi-Ricci system reated compensated compactness framewor for approximate soutions to this system. In Chen-Semrod-Wang [6], we noted that the Gauss-Codazzi equations for isometric embedding of M 2 into R 3 fa naturay within the formation of compensated compactness. In this paper, we first show that this is aso true in the genera case for the Gauss-Codazzi- Ricci system. One of our main observations here is that the Codazzi Ricci equations naturay have the Div-Cur structure. Based on this observation, we estabish the wee continuity of this system with respect to the uniform L p -bounded soution sequence for p > 2, which impies that the wea imit of the isometric embeddings of the manifod is sti an isometric embedding. This is reminiscent of the wea continuity of determinants which Date: Apri 22, Mathematics Subject Cassification. Primary: 53C42, 53C21, 53C45, 35L65, 35M20, 35B35; Secondary: 53C24, 57R40, 57R42,58J32. Key words phrases. Wea continuity, Gauss-Codazzi-Ricci system, isometric embedding, wea convergence, approximate soutions, compensated compactness, Div-Cur emma, minimization probem, seection criterion, Riemann curvature tensor. 1

2 2 GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG pays an essentia roe in the theory of poyconvexity by Ba [1] in noninear easticity (aso see Dacorogna [7], Evans [11], Morrey [17], Müer [18]). More generay, we estabish a stronger compensated compactness framewor for the Gauss-Codazzi-Ricci system. That is, given any sequence of approximate soutions to this system which is uniformy bounded in L 2, has reasonabe bounds on the errors made in the approximation (the errors are confined in a compact subset of H 1 oc ), then the approximating sequence has a weay convergent subsequence whose imit is sti a soution of the Gauss-Codazzi-Ricci system. For these, no restriction on the Riemann curvature tensor is made. A ong-sting fundamenta probem in differentia geometry is the existence of oca ( if possibe goba) embeddings of a d-dimensiona Riemannian manifod M d, d 3, into the Eucidean space R N with optima dimension N. As noted in Han-Hong [15], the first goba existence of smooth embeddings was given by Nash [22], but the best resut as of this time is the foowing theorem of Günther [14]: Any smooth d-dimensiona compact Riemannian manifod admits a smooth (i.e. C ) isometric embedding in R N for N = 1 2 max{d(d+5), d(d+3)+10}. Needess to say, it is of considerabe interest to now if Günther s dimension N is optima. In a simiar vein, we coud try to formuate a seection or admissibiity criterion to choose one of the possiby infinite embeddings provided by Günther s theorem. Within the ream of surface theory eastic manifods, this has been recenty considered in [9, 26] where the seection is done by minimizing an integra of norm of the second fundamenta form. Indeed, this seems a natura approach for seection in the genera case is even in the same spirit of Dafermos s entropy rate criterion [8]. In Section 4, we propose a minimizing probem as a seection criterion show by the compensated compactness framewor that any minimizing sequence has a subsequence in L p, p > 2, which converges weay to a minimizer that satisfies the Gauss-Codazzi- Ricci system. Since any sequence of isometric embeddings of M d into R N (say given by Günther s theorem) must satisfy the equations exacty, this impies that the probem of minimizing the L p -norms of the second fundamenta form the connection form on the norma bunde (sometimes caed torsion coefficients [3]) does have a soution within the cass of wea soutions of the Gauss-Codazzi-Ricci system, hence yieding an isometric immersion of W 2,p cass for p > The Gauss-Codazzi-Ricci System for Isometric Embedding of M d into R N In this section, we use the foowing conventiona notation: g ij : given metric of the Riemannian manifod, Γ ij : R ij : h a ij : κ a b : Christoffe symbos, Riemann curvature tensor, Coefficients of the second fundamenta form, Coefficients of the connection form (torsion coefficients) on the norma bunde, where the indices a, b, c run from 1 to N, i, j,,, m, n run from 1 to d 3. For given metric g ij, the Christoffe symbos are Γ ij = 1 2 g ( j g i + i g j g ij ),

3 WEAK CONTINUITY OF THE GAUSS-CODAZZI-RICCI SYSTEM FOR ISOMETRIC EMBEDDING 3 which depend on the first derivatives of (g ij ), the Riemann curvature tensor is R ij = g m ( Γ m ij j Γ m i + Γn ijγ m n Γn i Γm nj), which depends on (g ij ) its first second derivatives, where (g ) denotes the inverse of (g ij ) j = xj. We denote g = det(g ij ) The Gauss-Codazzi-Ricci System. As is we-nown in Riemannian Geometry, the isometric embedding probems for d-dimensiona Riemannian manifods into the Eucidean space R N can be reduced as the sovabiity probems of the Gauss-Codazzi-Ricci system of noninear partia differentia equations with the foowing form: The Gauss equations: h a jih a ha i ha j = R ij; (2.1) The Codazzi equations: The Ricci equations: h a j x ha j x + Γ m j ha m Γm j ha m + κa b hb j κa b hb j = 0; (2.2) κ a b x κa ( ) b x g mn h a m hb n ha m hb n + κ a c κc b κa c κc b = 0. (2.3) Notice that the coefficients of the second fundamenta form are symmetric: h a ij = h a ji, (2.4) whie the coefficients of the connection form on the norma bunde are antisymmetric: In particuar, the antisymmetry of κ a b impies κ a b = κb a. (2.5) κ a a = κa a, so κ a a = 0. Thus, the ath coumn of the d d matrix κ a is zero. When d = 3, the Janet dimension N = d(d+1) 2 = 6 (cf. Janet [16]). Then κ 1 = 0 κ1 12 κ κ 1 22 κ 1 23, κ 2 = κ κ 2 13 κ 1 0 κ 1 32 κ κ 2 23, κ 3 = κ1 13 κ κ 1 33 κ κ 2 23 κ κ 1 33 κ

4 4 GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG 2.2. The Div-Cur Structure of the Codazzi Ricci Equations. In this section we present one of our main observations on the features of the Codazzi Ricci equations: the Div-Cur structure, which eads to the wea continuity of the system. For w = (w 1, w 2,, w d ), is a d d matrix fied. or cur w := ( j w i i w j ) 1 i,j d From the Codazzi equations (2.2), for <, they possess the form: h a j x ha j x +.o.t = 0, div( 0,, h a j, 0,, ha j, 0,, 0) +.o.t = 0, (2.6) cur(h a 1j, h a 2j,, h a dj ) +.o.t = 0, (2.7) where.o.t represents the ower-order terms without invoving derivatives in the equation. Simiary, we observe that the identica form of the Ricci equations (2.3) can aso be written as div( 0,, 0, κ a b, 0,, κa b, 0,, 0) +.o.t = 0, (2.8) cur(κ a 1b, κa 2b,, κa db ) +.o.t = 0. (2.9) Now repacing a by b, j by i in the Codazzi equations (2.6) (2.7), we obtain div( 0,, h b i, 0,, hb i, 0,, 0) +.o.t = 0, (2.10) cur(h b 1i, h b 2i,, h b di ) +.o.t = 0. (2.11) Simiary, repacing a by b b by c in the Ricci equations (2.8) (2.9), we have div( 0,, 0, κ b c, 0,, κb c, 0,, 0) +.o.t = 0, (2.12) cur(κ b 1c, κ b 2c,, κ b dc ) +.o.t = 0. (2.13) One of our main observations is that the scaar product of the two vector fieds in the rewritten forms (2.6) (2.13) yied the noninear quantities in the ower-order terms in the Gauss-Codazzi-Ricci system (2.1) (2.3): Forms (2.6) (2.11) yied h a j hb i ha j hb i ; (2.14)

5 WEAK CONTINUITY OF THE GAUSS-CODAZZI-RICCI SYSTEM FOR ISOMETRIC EMBEDDING 5 forms (2.8) (2.13) yied κ a b κb c κa b κb c ; (2.15) forms (2.9) (2.10) yied κ a b hb i κa b hb i. (2.16) This observation is essentia for us to estabish the wea continuity of the Gauss-Codazzi- Ricci system in Wea Continuity Compensated Compactness Framewor In this section we estabish the wea continuity of the Gauss-Codazzi-Ricci system reated compensated compactness framewor for approximate soutions to the system via the Div-Cur emma (see Murat [19] Tartar [24]). The Div-Cur emma is a basic resut in the compensated compactness theory for the wea continuity of the scaar product of two vector fieds (cf. [7, 11, 19, 20, 24, 25]) is cosey reated with the Hodge decomposition. Theorem 3.1 (Div-Cur Lemma). Let Ω R d, d 2, be open bounded. Let p, q > 1 such that 1 p + 1 q = 1. Assume that, for any ε > 0, two fieds uε L p (Ω; R d ) v ε L q (Ω; R d ) satisfy the foowing: (i) u ε u weay in L p (Ω; R d ) as ε 0; (ii) v ε v weay in L q (Ω; R d ) as ε 0; (iii) div u ε are confined in a compact subset of W 1,p oc (Ω; R); (iv) cur v ε are confined in a compact subset of W 1,q oc (Ω; R d d ). Then the scaar product of u ε v ε are weay continuous: in the sense of distributions. u ε v ε u v Based on our observation of the Div-Cur structure of the Codazzi Ricci equations, we empoy the Div-Cur emma to formuate the foowing compensated compactness framewor. Let a sequence of vector fieds (h a,ε ij, κa,ε b )(x), defined on an open bounded subset Ω Rd, satisfy the foowing Framewor (A): (A.1) (h a,ε ij, κa,ε b ) L 2 (Ω) C for some C > 0 independent of ε > 0; (A.2) ha,ε j κa,ε x b κa,ε x b x (A.3) There exist o ε j (1), j = 1, 2, 3, with oε j such that h a,ε j x κ a,ε b x ha,ε j x are confined in a compact set in H 1 oc (Ω); (1) 0 in the sense of distributions as ε 0 h a,ε j x + Γ m j ha,ε m Γm j ha,ε m + κa,ε b hb,ε j κa,ε b hb,ε ) κa,ε b x ( g mn h a,ε m hb,ε n ha,ε m hb,ε n j = oε 1(1), + κ a,ε c κc,ε b κa,ε c κc,ε b = oε 2(1), (3.1) h a,ε ji ha,ε h a,ε i ha,ε j = R ij + o ε 3(1). (3.2)

6 6 GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG Then we have Theorem 3.2 (Compensated compactness framewor). Let a sequence of vector fieds (h a,ε ij, κa,ε b ) satisfy Framewor (A). Then there exists a subsequence (sti abeed) (ha,ε ij, κa,ε b ) that converges weay in L 2 (Ω) to (h a ij, κa b ) as ε 0 such that (i) (h a ij, κa b ) L 2 (Ω) C; (ii) the quadratic terms in (2.1) (2.3) are weay continuous with respect to the subsequence (h a,ε ij, κa,ε b ) that converges to (ha ij, κa b ) weay in L2 (Ω) as ε 0; (iii) the imit vector fied (h a ij, κa b ) satisfies the Gauss-Codazzi-Ricci system (2.1) (2.3). That is, the imit vector fied (h a ij, κa b ) is a wea soution to the Gauss-Codazzi-Ricci system (2.1) (2.3). Proof. By assumption (A.1), there exists a subsequence (sti denoted) (h a,ε ij, κa,ε b ) a vector fied (h a ij, κa b ) L2 (Ω) such that (h a,ε ij, κa,ε b ) (ha ij, κ a b ) in L2 (Ω), (3.3) (h a ij, κ a b ) L 2 (Ω) C. (3.4) By the Div-Cur structure, observed in 2.2, assumption (A.2) impies that div( 0,, h a,ε j, 0,, ha,ε j, 0,, 0), cur(h a,ε 1j, ha,ε div( 0,, 0, κ a,ε b, 0,, κa,ε b, 0,, 0), cur(κ a,ε are confined in a compact set in H 1 oc (Ω). By exchanging the indices, we aso have div( 0,, h b,ε i, 0,, hb,ε i, 0,, 0), cur(h b,ε 2j,, ha,ε dj ) (3.5) 1b, κa,ε 1i, hb,ε div( 0,, 0, κ b,ε c, 0,, κb,ε c, 0,, 0), cur(κ b,ε are confined in a compact set in H 1 oc (Ω). 2b,, κa,ε db ) (3.6) 2i,, hb,ε di ) (3.7) 1c, κb,ε 2c,, κb,ε dc ) (3.8) Using the Div-Cur emma, Theorem 3.1, we concude that the wea continuity of the noninear quadratic quantities in the Gauss-Codazzi-Ricci system with respect to the

7 WEAK CONTINUITY OF THE GAUSS-CODAZZI-RICCI SYSTEM FOR ISOMETRIC EMBEDDING 7 sequence (h a,ε ij, κa,ε b ): h a,ε j hb,ε i ha,ε j hb,ε i h a j hb i ha j hb i, (3.9) κ a,ε b κb,ε c κa,ε b κb,ε c κ a b κb c κa b κb c, (3.10) κ a,ε b hb,ε i κ a,ε b hb,ε i κ a b hb i κa b hb i (3.11) in the sense of distributions as ε 0. Combining (3.3) (3.4) with (3.9) (3.11), we concude that the wea imit vector fied (h a ij, κa b ) of the sequence (ha,ε ij, κa,ε b ) satisfy the Gauss-Codazzi-Ricci system (2.1) (2.3) in the sense of distributions, that is, the imit vector fied (h a ij, κa b ) is a wea soution of (2.1) (2.3). As a coroary, we concude the wea continuity of the Gauss-Codazzi-Ricci system with respect to the uniform L p -bounded soution sequence for p > 2. Theorem 3.3 (Wea Continuity). Let (h a,ε ij, κa,ε b ) be a sequence of soutions to the Gauss- Codazzi-Ricci system (2.1) (2.3), which is uniformy bounded in L p, p > 2. Then the wea imit vector fied (h a ij, κa b ) of the sequence (ha,ε ij, κa,ε b ) in Lp is sti a soution to (2.1) (2.3). Proof. Since the soution sequence (h a,ε ij, κa,ε b ) is uniformy bounded in Lp, p > 2: (h a,ε ij, κa,ε b ) L p (Ω) C, (3.12) for some C > 0 independent of ε, then there exists a subsequence (sti denoted) (h a,ε ij, κa,ε b ) a vector fied (h a ij, κa b ) Lp (Ω) such that (h a,ε ij, κa,ε b ) (ha ij, κ a b ) (h a ij, κ a b ) L p (Ω) C. in Lp (Ω), Then we concude from (3.12) that a the ower-order terms for the soution sequence (h a,ε ij, κa,ε b ) in the Gauss-Codazzi-Ricci system (2.1) (2.3) are uniformy bounded in Lp/2, p > 2. This impies that h a j x ha j x, κ a b x κa b x are confined in a compact set in H 1 oc (Ω). (3.13) Since the domain Ω R d is bounded, the uniform bound in (3.12) impies the uniform bound of (h a,ε ij, κa,ε b ) in L2 (Ω). By the compensated compactness framewor (Theorem 3.2), we concude that the imit vector fied is a wea soution of (2.1) (2.3), which impies the wea continuity of the system. Remar 3.1. The wea continuity of the Gauss-Codazzi-Ricci system impies that, for p > 2, the wea imit of a sequence of isometric embeddings of the d-dimensiona manifod M d into R N as surfaces with corresponding uniform L p -bounded sequence (h a,ε ij, κa,ε b ) is sti an isometric embedding as a surface in R N. The requirement p > 2 is to ensure the H 1 -compactness in (3.13) to dea with the nonhomogeneous terms.

8 8 GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG 4. Minimization Probem In this section, as an exampe, we show that the soution sequence (h a,ε wea continuity in Theorem 3.3 can be obtained from a seection criterion. ij, κa,ε b ) for the Theorem 4.1. There exists a minimizer (h a ij, κa b ) for the minimization probem: min (h, κ) p ( ) S L p (Ω) := min g (h ij h ij ) p 2 + (κb κ b ) p 2 dx, (4.1) S Ω where S is the set of wea soutions to the Gauss-Codazzi-Ricci system (2.1) (2.3). Proof. Ceary, S is non-empty by Günther s theorem in [14] (aso see the statement in 1 above). A minimizing sequence provides the desired L p -norm for the wea continuity theorem (Theorem 3.3). Since the L p -norm is convex, which is weay ower semicontinuous, any minimizing sequence has a subsequence in L p (Ω) that converges weay to a minimizer which satisfies the Gauss-Codazzi-Ricci system (2.1) (2.3). Notice that any sequence of isometric embeddings of M d into R N as surfaces (say, given by Günther s theorem) must satisfy the Gauss-Codazzi-Ricci equations (2.1) (2.3). This impies that the probem of minimizing the L p -norms of the second fundamenta form the connection form on the norma bunde does have a soution within the cass of wea soutions of the Gauss-Codazzi-Ricci system (2.1) (2.3), hence yieding an isometric immersion of W 2,p cass for p > 2 for M d into R N as a surface. Acnowedgments. Gui-Qiang Chen s research was supported in part by the Nationa Science Foundation under Grants DMS , DMS , DMS , the Natura Science Foundation of China under Grant NSFC Marsha Semrod s research was supported in part by the Nationa Science Foundation under Grant DMS Dehua Wang s research was supported in part by the Nationa Science Foundation under Grant DMS , by the Office of Nava Research under Grant N This paper was written as part of the Internationa Research Program on Noninear Partia Differentia Equations at the Centre for Advanced Study at the Norwegian Academy of Science Letters in Oso during the academic year ; It was finaized when the authors participated in the SQuaRE on Isometric Embedding of Higher Dimensiona Riemannian Manifods, which was hed at the American Institute of Mathematics, Pao Ato, Caifornia, March 16 20, 2009.

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10 10 GUI-QIANG CHEN, MARSHALL SLEMROD, AND DEHUA WANG G.-Q. Chen, Schoo of Mathematica Sciences, Fudan University, Shanghai , China; Department of Mathematics, Northwestern University, Evanston, IL 60208, USA. E-mai address: M. Semrod, Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. E-mai address: D. Wang, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA. E-mai address: