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K-ENERGY ON POLARIZED COMPACTIFICATIONS OF LIE GROUPS YANLI BINZHOU∗ XIAOHUAZHU∗∗ 7 1 0 Abstract. Inthispaper,westudyMabuchi’sK-energyonacompactification 2 M of a reductive Lie group G, which is a complexification of its maximal compact subgroup K. We give a criterion for the properness of K-energy on n thespaceofK×K-invariantKählerpotentials. Inparticular,itturnstogive a J analternativeproofofDelcroix’stheoremfortheexistenceofKähler-Einstein metricsincaseofFanomanifoldsM. Wealsostudytheexistenceofminimizers 2 ofK-energyforgeneralKählerclassesofM. ] G D . h 1. Introduction t a ThefamousYau-Tian-Donaldson’sconjecturefortheexistenceofKähler-Einstein m metricsonFanomanifoldsassertsthattheexistenceisequivalenttotheK-stability. [ The conjecture has been recently solved by Tian [25]. Chen, Donaldson and Sun 1 also give an alternative proof [8]. The notion of K-stability was first introduced by v Tian by using special degenerations [23] and then reformulated by Donaldson in 6 0 algebraic geometry via test-configurations [14]. For both special degenerations and 3 test-configurations, one has to study an infinite number of possible degenerations 0 of the manifold. A natural question is how to verify the K-stability by reducing it 0 . to a finite dimensional progress. The answer is known for Fano surfaces by Tian 1 0 [22] and for toric Fano manifolds by Wang and Zhu [29] (see also [30]). In fact, in 7 both cases the existence is equivalent to the vanishing of Futaki invariant. 1 More recently, Delcroix extends Wang-Zhu’s result to a polarized compactifica- : v tionM ofareductiveLiegroupGwithc (M)>0[12]. WecallM a(bi-equivariant) 1 i X compactification of G if it admits a holomorphic G×G action on M with an open r and dense orbit isomorphic to G as a G×G-homogeneous space. (M,L) is called a a polarized compactification of G if L is a G×G-linearized ample line bundle on M. For more examples besides the toric manifolds, see [4, 12, 13]. Let TC be a r-dimensional maximal complex torus of G with dimension n and M its group of characters. Assume that Φ is the root system of (G,TC) in M and Φ is a chosen set of positive roots. Let P be the polytope associated to (M,L), + and P the part of P defined by Φ . Denote by 2P its dilation at rate 2. Let + + + 2000 Mathematics Subject Classification. Primary: 53C25;Secondary: 53C55,58J05,19L10. Key words and phrases. K-energy,Liegroup,Fanomanifolds,Kähler-Einsteinmetrics. *PartiallysupportedbyNSFC11571018and11331001. **PartiallysupportedbyNSFCGrants11271022and11331001. 1 2 YANLI BINZHOU∗ XIAOHUAZHU∗∗ ρ = 1(cid:80) α and Ξ be the relative interior of the cone generated by Φ . Then 2 α∈Φ+ + Delcroix proved Theorem 1.1. Let M be a polarized compactification of G with c (M)>0. Then 1 M admits a Ka¨hler-Einstein metric if and only if (1.1) bar ∈4ρ+Ξ, (cid:82) yπ(y)dy where bar = 2P+ is the barycentre of 2P+ with respect to the weighted (cid:82) π(y)dy measure π(y)dy2aPn+d π(y)=(cid:81) (cid:104)α,y(cid:105)2. α∈Φ+ ItispointedbyDelcroixthat(1.1)impliesthattheFutakiinvariantvanishesfor holomorphic vector fields induced by G×G, but the inverse is not true in general. Thus one may ask if (1.1) is related to the K-stability and is determined by a generalized Futaki invariant for some test-configurations. In the present paper, we willanswerthisquestion. Infact,motivatedbythestudyontoricmanifolds[14],we investigatetheK-energyonthespaceofK×K-invariantKählerpotentialsthrough the reduced K-energy K(·) via Legendre transformation. We show that condition (1.1) comes from our formula of K(·) naturally when c (M) > 0 (cf. Proposition 1 3.1, Proposition 3.4). Moreover, we give an alternative proof of Theorem 1.1 by showing the properness of the K-energy (cf. Section 4). The Kähler-Ricci solitons case can be discussed similarly (cf. Section 5). The main purpose of this paper is to give a criterion for the properness of the K-energy on a general polarized compactification (M,L) of G as done on a toric manifold in [33]. We divide ∂(2P )∩∂(2P) into several pieces {F }d0 such that + A A=1 for any A, F lies on an (r−1)-dimensional hyperplane defined by (cid:104)y,u (cid:105) = λ A A A for some primitive u ∈ N, where N is the Z-dual of M. Define a cone by E = A A (cid:91)d0 {ty| t∈[0,1], y ∈F } for any A. It is clear that 2P = E . Let A + A A=1 2 (1.2) Λ = (1+(cid:104)2ρ,u (cid:105)). A λ A A Then the average of scalar curvature S¯ of ω ∈2πc (L) is given by1 0 1 (cid:80) (cid:82) n Λ πdy (1.3) S¯= A(cid:82) A EA . πdy 2P+ Define a weighted barycentre b(cid:102)ar of 2P+ by (cid:80) (cid:82) Λ yπdy (1.4) b(cid:102)ar = (cid:80)A A (cid:82)EA . Λ πdy A A EA Notethatbothbarandb(cid:102)arareinthedualspacea∗ofa,whereaisthenon-compact part of Lie algebra tC of TC. Denote by barss and b(cid:102)arss the projections of bar and b(cid:102)ar on the semisimple part a∗ of a∗, respectively. We prove ss 1(1.3)willbeverifiedattheendofSection2. K-ENERGY ON POLARIZED COMPACTIFICATIONS OF LIE GROUPS 3 Theorem 1.2. Let (M,L) be a polarized compactification of G with vanishing Futaki invariant, and ω ∈2πc (L) a K×K-inariant Kähler metric. Suppose that 0 1 the polytope 2P satisfies the following conditions, + (cid:16) (cid:17) (1.5) minΛA·b(cid:102)arss−4ρ ∈Ξ, A (cid:16) (cid:17) (1.6) b(cid:102)arss−barss ∈Ξ¯, (1.7) (n+1)·minΛ −S¯>0. A A Then the K-energy µ (·) is proper on H (ω ) modulo Z(G), where ω0 K×K 0 √ H (ω )={φ∈C∞(M) | ω =ω + −1∂∂¯φ>0 and φ is K×K-invariant } K×K 0 φ 0 and Z(G) is the centre of G. IncasethatM isFanoandL=K−1,thenS¯=nandΛ =1forallA. Wehave M A b(cid:102)ar =bar,thus(1.6),(1.7)areautomaticallysatisfied. Moreover,(1.1)isequivalent to the vanishing of Futaki invariant and (1.5) (cf. Corollary 3.3). Consequently, µ (·) is proper modulo the action of Z(G). Hence we get the an alternative proof ω0 for the sufficient part of Theorem 1.1 [10, 28]. Asmentionedabove,weproveTheorem1.2byusingthereducedK-energyK(·). One of the advantages of K(·) is that it can be defined on a complete space C˜ ∗ of convex functions on 2P . Following the argument in [34], we discuss the semi- + continuity property of K(·). As a consequence, we prove the following Theorem 1.3. K(u) is lower semi-continuous on C˜ . Furthermore, if µ (·) is ∗ ω0 proper on H (ω ) modulo Z(G), then there exists a minimizer of K(·) on C˜ . K×K 0 ∗ It is interesting to study the regularity of minimizers in Theorem 1.3. We guess that they are smooth in 2P if the dimension of the torus TC is less than two. In + case of toric surfaces, it is verified in [31, 32]. The paper is organized as following: In Section 2, we review some preliminaries on K×K-invariant metrics on M, and then we give a formula of scalar curvature of such metrics in terms of Legendre functions. The formula of K(·) is obtained in Section3. InSection4,weusetheideain[33]fortoricmanifoldstoproveTheorem 1.2,buttherearenewdifficultiesarisingfromenergyestimatesneartheWeylwalls to overcome. In Section 5, we focus on the Fano case, and prove the properness of modified K-energy provided a modified barycentre condition (5.2) (cf. Theorem 5.1). In Section 6, we prove Theorem 1.3. 2. Preliminaries In this section, we first recall some preliminaries for K ×K-invariant Kähler metrics on a polarized compactification (M,L) of G [11, 12, 13] and the associated Legendre functions, then we give a computation of scalar curvature in terms of Legendre functions. 4 YANLI BINZHOU∗ XIAOHUAZHU∗∗ 2.1. Polarized compactification. Let J be the complex structure of G and K be one of its maximal compact subgroup such that G=KC. Choose T a maximal torusofK. Denotebyg,k,tthecorrespondingLiealgebraofG, K, T,respectively. Then g=k⊕Jk. Set a=Jt and Lie algebra of Z(G) by z(g). We decompose a as a toric part and a semisimple part: a=a ⊕a , t ss where a :=z(g)∩a and a :=a∩[g,g]. Then for any x∈a, we have x=x +x t ss t ss with x ∈a and x ∈a . We extend the Killing form on a to a scalar product t t ss ss ss (cid:104)·,·(cid:105)onasuchthata isorthogonaltoa . Identifyaanditsduala∗ by(cid:104)·,·(cid:105). Then t ss a∗ also has an orthogonal decomposition a∗ =a∗⊕a∗ . t ss Denote by Φ and W the root system and Weyl group with respect to (G,TC), respectively. ChooseasystemofpositiverootsΦ . ThenitdefinesapositiveWeyl + chamber a ⊂a, and a positive Weyl chamber a∗ on a∗, where + + a∗ :={y| α(y):=(cid:104)α,y(cid:105)>0, ∀α∈Φ }, + + which is also called the relative interior Ξ of the cone generated by Φ . The Weyl + wall W is defined by W :={y| α(y)=0} for each α∈Φ . α α + 2.2. K ×K-invariant Kähler metrics. Let Z be the closure of TC in M. It is known that (Z,L| ) is a polarized toric manifold with a W-action, and L| is Z Z a W-linearized ample toric line bundle on Z [2, 3, 4, 12]. Let ω ∈ 2πc (L) be a 0 1 K×K-invariantKählerforminducedfrom(M,L)andP bethepolytopeassociated to(Z,L| ),whichisdefinedbythemomentmapassociatedtoω . ThenP isaW- Z 0 invariantdelzentpolytopeina∗. BytheK×K-invariance,foranyφ∈H (ω ), K×K 0 the restriction of ω on Z is a toric Kähler metric. It induces a smooth strictly φ convex function ψ on a, which is W-invariant [5]. By the KAK-decomposition ([21], Theorem 7.39), for any g ∈ G, there are k , k ∈ K and x ∈ a such that g = k exp(x)k . Here x is uniquely determined 1 2 1 2 up to a W-action. This means that x is unique in a¯ . Then we define a smooth + K×K-invariant function Ψ on G by Ψ(exp(·))=ψ(·): a→R. Clearly Ψ is well-defined since ψ is W-invariant. We usually call ψ the function associated to Ψ. It can be verified that Ψ is a Kähler potential on G such that √ ω = −1∂∂¯Ψ on G (cf. Lemma 2.2 below). The following KAK-integral formula can be found in [20], Proposition 5.28 (see also [19]) Proposition 2.1. Let dV be a Haar measure on G and dx the Lebesgue measure G on a. Then there exists a constant C > 0 such that for any K ×K-invariant, H K-ENERGY ON POLARIZED COMPACTIFICATIONS OF LIE GROUPS 5 dV -integrable function Ψ on G, G (cid:90) (cid:90) Ψ(g)dV =C J(x)ψ(x)dx, G H G a+ where J(x)=(cid:81) sinh2(α(x)). α∈Φ+ Next we recall the local holomorphic coordinates on G used in [12]. By the standard Cartan decomposition, we can decompose g as g=(t⊕a)⊕(⊕ V ), α∈Φ α where V = {X ∈ g| ad (X) = α(H)X, ∀H ∈ t⊕a}, the root space of complex α H dimension 1 with respect to α. By [18], one can choose X ∈V such that X = α α −α −ι(X ) and [X ,X ]=α∨, where ι is the Cartan involution and α∨ is a dual of α α −α α bytheKillingform. LetE :=X −X andE :=J(X +X ). Denoteby α α −α −α α −α k , k the real line spanned by E , E , respectively. Then we have the Cartan α −α α −α decomposition of k, (cid:0) (cid:1) k=t⊕ ⊕ (k ⊕k ) . α∈Φ+ α −α Choosearealbasis{E0,...,E0}oft. Then{E0,...,E0}togetherwith{E ,E } 1 r 1 r α −α α∈Φ+ forms a real basis of k, which is indexed by {E ,...,E }. {E ,...,E } can also be 1 n 1 n regarded as a complex basis of g. For any g ∈ G, we define local coordinates {zi } on a neighborhood of g by (g) i=1,...,n (zi )→exp(zi E )g. (g) (g) i It is easy to see that θi| = dzi | , where θi is the dual of E , which is a right- g (g) g i (cid:16) (cid:17) invariant holomorphic 1-form. Thus ∧n dzi ∧dzi¯ | is also a right-invariant i=1 (g) (g) g (n,n)-form, which defines a Haar measure dV . G The complex Hessian of the K × K-invariant function Ψ in the above local coordinates was computed by Delcroix as follows [12, Theorem 1.2]. Lemma 2.2. Let Ψ be a K ×K invariant function on G, and ψ the associated function on a. Let Φ ={α ,...,α }. Then for x∈a , the complex Hessian + (1) (n−r) + 2 matrix of Ψ in the above coordinates is diagonal by blocks, and equals to 1HessR(ψ)(x) 0 0  4  0 Mα(1)(x) 0  (2.1) HessC(Ψ)(exp(x))= 0 0 ... ... ,  ... ... ... 0    0 0 M (x) α(n−2r) where √ (cid:18) (cid:19) 1 cothα (x) −1 M (x)= (cid:104)α ,∇ψ(x)(cid:105) √(i) . α(i) 2 (i) − −1 cothα (x) (i) 6 YANLI BINZHOU∗ XIAOHUAZHU∗∗ By (2.1) in Lemma 2.2, we see that ψ is convex on a. The complex Monge- √ Ampére measure is given by ωφn =( −1∂∂¯Ψ)n =MAC(Ψ)dVG, where 1 1 (cid:89) (2.2) MAC(Ψ)(exp(x))= MAR(ψ)(x) (cid:104)α,∇ψ(x)(cid:105)2. 4r+p J(x) α∈Φ+ 2.3. Legendre functions. By the convexity of ψ on a, the gradient ∇ψ defines a diffeomorphism from a to the interior of the dilated polytope 2P2. Let P := + P ∩a¯∗, then by the W-invariance of ψ and P, the restriction of ∇ψ to a is a + + diffeomorphism to the interior of 2P . We note that one part of ∂(2P ) lies on + + ∂(2P) (which we call ”outer faces”) and the other part lies on Weyl walls {W }. α For simplicity, we may assume that 2P contains the origin O in its interior. Then 2P can be described as the intersection of l (y):=−ui y +λ >0, A˜=1,...,d, A˜ A˜ i A˜ where λ >0 and u are primitive vectors in N. A˜ A˜ Recall that Guillemin’s function of 2P is given by 1(cid:88) (2.3) u = l (y)logl (y). 0 2 A˜ A˜ A˜ Set C ={v| v is strictly convex, v−u ∈C∞(2P) and v is W-invariant} ∞,W 0 and C ={v | v ∈C }. ∞,+ |2P+ ∞,W By [17], the Legendre function u of ψ belongs to C . The inverse is also true. ∞,W This means that any u ∈ C corresponds to a Kähler potential in H (ω ) ∞,W K×K 0 (cf. [4, Proposition 3.2]). By a direct computation, we have (2.4) u =−1(cid:88)(logl (y)+1)ui , u = 1(cid:88)uiA˜ujA˜. 0,i 2 A˜ A˜ 0,ij 2 l (y) A˜ A˜ A˜ Note that uijν →0 as y →F , where (uij)=(u )−1 and ν =(ν ,...,ν ) is the 0 i A˜ 0 0,ij A˜ 1 r unit normal vector of face F = {y| l (y) = 0}. Similarly −uij ν → 2 (cid:104)y,ν (cid:105), A˜ A˜ 0,j i λA˜ A˜ where uij = ∂ui0j. Thus we get 0,k ∂yk Lemma 2.3. If u∈C , then for any A˜, as y →F , ∞,W A˜ 2 (2.5) uijν →0 and uijν → (cid:104)y,ν (cid:105), i ,j i λ A˜ A˜ where (uij)=(u )−1 and uij = ∂uij. ,ij ,k ∂yk 2Weremarkthatthemomentmapisgivenby 1∇ψ,whoseimageisP. 2 K-ENERGY ON POLARIZED COMPACTIFICATIONS OF LIE GROUPS 7 2.4. The scalar curvature. We compute the Ricci curvature of ω . Clearly it φ is also K ×K-invariant. As in Lemma 2.2, in the local coordinates in Sect. 2.2, Ric(ω ) can be expressed as φ −HessC(logdet(∂∂¯Ψ))(exp(x)) 1HessR(ψ˜)(x) 0 0  4  0 M˜α(1)(x) 0  =− 0 0 ... ...     ... ... ... 0    0 0 M˜ (x) α(n−2r) for any x∈a , where + ψ˜ = logdet(∇2ψ)+2 (cid:88) logα(∇ψ)+χ(x), α∈Φ+ (cid:88) χ(x) = −logJ(x)=−2 logsinhα(x), α∈Φ+ √ 1(cid:68) (cid:69)(cid:18)cothα(x) −1 (cid:19) M˜ (x) = α,∇ψ˜ √ . α 2 − −1 cothα(x) Then the scalar curvature (2.6) S(ω )| =tr(cid:16)∇2ψ)−1∇2ψ˜(cid:17)+ (cid:88) (cid:104)α,∇ψ˜(cid:105). φ exp(x) (cid:104)α,∇ψ(cid:105) α∈Φ+ By using the Legendre function u, we get Lemma 2.4.   S(ωφ)=−(cid:88)ui,ijj +4 (cid:88) ααi(uyi,)jj +4 (cid:88) αα(iyβ)jβu(iyj) −2 (cid:88) α(αiα(yj)u)i2j i,j α∈Φ+ α,β∈Φ+ α∈Φ+ (2.7) −(cid:88)u,ik ∂x∂i2∂χxk(cid:12)(cid:12)(cid:12)(cid:12) −2(cid:88) (cid:88) ∂∂xχi(cid:12)(cid:12)(cid:12)(cid:12) αα(yi), i,k x=∇u i α∈Φ+ x=∇u where y ∈2P , uij = ∂2uij and α are the components of α. + ,kl ∂yk∂yl i Proof. By the relations (∇2u)−1|y =(∇2ψ)|x=∇u, ∂xi∂∂x3ψj∂xk(cid:12)(cid:12)(cid:12)(cid:12)x = ∂∂xi (cid:0)ujk|y=∇ψ(cid:1)= uj,lkuli(cid:12)(cid:12)(cid:12)y=∇ψ, we have ∂∂xψ˜p(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x=∇u =u,ijui,kjukp+2α(cid:88)∈Φ+ ααl(uyl)p + ∂∂xχp(cid:12)(cid:12)(cid:12)(cid:12)x=∇u, ∂x∂p2∂ψ˜xq(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)x=∇u =(u,ijui,kjukp),susq+2α(cid:88)∈Φ+(cid:18)ααl(uyl)p(cid:19),susq+ ∂x∂p2∂χxq(cid:12)(cid:12)(cid:12)(cid:12)x=∇u. Substituting them into (2.6), we obtain (2.7) immediately. (cid:3) 8 YANLI BINZHOU∗ XIAOHUAZHU∗∗ Note π(y)=(cid:81) (α(y))2. Since α∈Φ+ ∂π(y)=2π(y) (cid:88) αi , ∂y α(y) i α∈Φ+   (2.8) ∂2π (y)=π(y)4 (cid:88) αiβj −2 (cid:88) αiαj , ∂y ∂y α(y)β(y) (α(y))2 i j α,β∈Φ+ α∈Φ+ we can rewrite S as π π S(ω )=−uij −2uij ,i −uij ,ij φ ,ij ,j π π (2.9) −u,ik ∂x∂i2∂χxk(cid:12)(cid:12)(cid:12)(cid:12) − ∂∂xχi(cid:12)(cid:12)(cid:12)(cid:12) ππ,i. x=∇u x=∇u By Proposition 2.1, it follows (cid:90) (cid:90) (cid:90) (cid:89) Sωn =C Sdet(∇2ψ) (cid:104)α,∇ψ(cid:105)2dx=C Sπdy. φ H H M a+ α∈Φ+ 2P+ Since π ≡0 on each W , by integration by parts on (2.9), we get α (cid:90) Sπdy 2P+ (cid:90) (cid:90) (cid:90) =− uijν πdσ − uijπ dy− uijπ dy ,j i 0 ,j ,i ,ij ∂(2P+) 2P+ 2P+ (cid:90) ∂ (cid:18) ∂χ (cid:12)(cid:12) (cid:19) (cid:90) ∂χ (cid:12)(cid:12) − ∂y ∂xj(cid:12)(cid:12) πdy− ∂xj(cid:12)(cid:12) π,idy 2P+ i x=∇u 2P+ x=∇u (cid:88)(cid:90) (cid:18) 2 (cid:19) (cid:88) (cid:90) = (cid:104)y,ν (cid:105)+4(cid:104)ρ,ν (cid:105) πdσ = Λ (cid:104)y,ν (cid:105)πdσ λ A A 0 A A 0 A FA A A FA (cid:90) (cid:88) =n Λ πdy. A A EA HereweusedLemma2.3andthefactthat ∂χ(x)→−4ρ asx→∞. Ontheother ∂xi i hand, by Proposition 2.1, the volume of (M,ω ) is given by φ (cid:90) (cid:90) (cid:89) VM := ωφn = CH MAR(ψ) (cid:104)α,∇ψ(cid:105)2dx M a+ α∈Φ+ (cid:90) = C πdy. H 2P+ Hence, combining the above two relations, we get (1.3). 3. Reduction of the K-Energy Let(M,L)andω ∈2πc (L)beasbefore. DenotebyH(ω )thespaceofKähler 0 1 0 potentials in [ω ]. Mabuchi’s K-energy is defined on H(ω ) by 0 0 1 (cid:90) 1(cid:90) (3.1) µ (φ)=− φ˙ (S(ω )−S¯)ωn ∧dt, ω0 V t φt φt M 0 M K-ENERGY ON POLARIZED COMPACTIFICATIONS OF LIE GROUPS 9 whereV =(cid:82) ωn,S¯istheaverageofS(ω )and{φ }isapathofKählerpotentials M M 0 0 t joining0andφinH(ω ). Inthissection,wegiveaformulaofµ (·)onH (ω ) 0 ω0 K×K 0 in terms of the Legendre function u. 3.1. Reduced K-energy. Define (cid:90) (cid:90) K(u)=(cid:88) Λ (cid:104)y,ν (cid:105)uπdσ − S¯uπdy A A 0 A FA 2P+ (cid:90) (cid:90) − logdet(u )πdy+ χ(∇u)πdy, ij 2P+ 2P+ (cid:80) where χ(x)=−logJ(x)=−2 logsinhα(x) for any x∈a. Then we have α∈Φ+ Proposition3.1. Letφ∈H (ω )andubetheLegendrefunctionofψ =ψ +φ. K×K 0 0 Then 1 µ (φ)= K(u)+const., ω0 V (cid:82) where V = πdy. 2P+ Proof. Note φ˙ =−u˙ . By (2.2), it is easy to see t t 1 (cid:90) 1(cid:90) (cid:90) S¯φ˙ ωn ∧dt= S¯uπdy. C t φt H 0 M 2P+ Then by (2.9), it suffices to compute the part 1 (cid:90) 1(cid:90) I := − φ˙ S(ω )ωn ∧dt C t φt φt H 0 M (cid:90) 1(cid:90) = − φ˙tS(ωφt)|exp(x)MAR(ψt)|x (cid:89) (cid:104)α,∇ψt(cid:105)2|xdx∧dt. 0 a+ α∈Φ+ (cid:90) 1(cid:90) = u˙ (−u ij )πdy∧dt t t,ij 0 2P+ (cid:90) 1(cid:90) (cid:90) 1(cid:90) + u˙ (−2u ijπ )dy∧dt+ u˙ (−uijπ )dy∧dt t t,j ,i t t ,ij 0 2P+ 0 2P+   (cid:90) 1(cid:90) ∂2χ (cid:88) ∂χ + u˙t−ut,ik∂xi∂xk|x=∇utπ− ∂xi|x=∇utπ,i dy∧dt. 0 2P+ α∈Φ+ By integration by parts, it follows (cid:90) 1(cid:90) (cid:90) 1(cid:90) I = u˙ (−u ijν )πdσ ∧dt+ u˙ u ijπdy∧dt t t,j i 0 t,i t,j 0 ∂(2P+) 0 2P+ (cid:90) 1(cid:90) (cid:90) 1(cid:90) − u˙ (u ijπ )dy∧dt− u˙ (uijπ )dy∧dt t t,j ,i t t ,ij 0 2P+ 0 2P+ (cid:90) 1(cid:90) ∂ (cid:32) ∂χ(cid:12)(cid:12) (cid:33) (3.2) − u˙t∂y ∂xi(cid:12)(cid:12) π dy∧dt. 0 2P+ i x=∇ut 10 YANLI BINZHOU∗ XIAOHUAZHU∗∗ Note that ∂χ(x) → −4ρ as x → ∞ in a and is away from Weyl walls, and π ∂xi i + vanishes quadratically along any Weyl wall. Then the last term in (3.2) becomes (cid:90)2P+χ(∇ut)πdy(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)10−(cid:90)01(cid:90)∂(2P+)u˙t ∂∂xχi(cid:12)(cid:12)(cid:12)(cid:12)x=∇utνiπdσ0∧dt (cid:90) (cid:90) (3.3) = χ(∇u)πdy+4 (cid:104)ρ,ν(cid:105)uπdσ +const. 0 2P+ ∂(2P+) On the other hand, by the second relation in Lemma 2.3, we have (cid:90) 1(cid:90) (cid:90) 1(cid:90) u˙ u ijπdy∧dt− u˙ (u ijπ )dy∧dt t,i t,j t t,j ,i 0 2P+ 0 2P+ (cid:90) 1(cid:90) (cid:90) 1(cid:90) = u˙ u ijν πdσ ∧dt− u˙ u ijπdy∧dt t,i t j 0 t,ij t 0 ∂(2P+) 0 2P+ (cid:90) 1(cid:90) (cid:90) 1(cid:90) + u˙ u ijπ dy∧dt− u˙ u ijν π dσ ∧dt t t ,ij t t j ,i 0 0 2P+ 0 ∂(2P+) (cid:90) 1(cid:90) d (cid:90) 1(cid:90) (3.4) =− [logdet(u )]πdy∧dt+ u˙ u ijπ dσ ∧dt. dt t,ij t t ,ij 0 0 2P+ 0 ∂(2P+) Thus combining (3.4) and (3.3), we get from (3.2), (cid:90) 1(cid:90) (cid:90) I = u˙ (−u ijν )πdσ ∧dt+4 (cid:104)ρ,ν(cid:105)uπdσ t t,j i 0 0 0 ∂(2P+) ∂(2P+) (cid:90) (cid:90) − logdet(u )πdy+ χ(∇u)πdy+const. ,ij 2P+ 2P+ By Lemma 2.3, we see (cid:90) (cid:88)(cid:90) 2 u˙ (−u ijν )πdσ = u˙ (cid:104)y,ν (cid:105)πdσ . t t,j j 0 tλ A 0 ∂(2P+) A FA A Hence, we obtain (cid:90) (cid:90) (cid:88) I = Λ (cid:104)y,ν (cid:105)uπdσ − [logdet(u )−χ(∇u)]πdy+const. A A 0 ij A FA 2P+ Recall that V =C ·V, the proof is finished. (cid:3) M H For convenience, we write K(u) as K(u)=L(u)+N(u), where (cid:90) (cid:90) (cid:90) (3.5) L(u)=(cid:88) Λ (cid:104)y,ν (cid:105)uπdσ − S¯uπdy− 4(cid:104)ρ,∇u(cid:105)πdy, A A 0 A FA 2P+ 2P+ (cid:90) (cid:90) (3.6) N(u)=− logdet(u )πdy+ [χ(∇u)+4(cid:104)ρ,∇u(cid:105)]πdy. ,ij 2P+ 2P+ By integration by parts, we can rewrite L(u) as (cid:90) (3.7) L(u)=(cid:88) (cid:2)(cid:104)Λ y−4ρ,∇u(cid:105)+(Λ n−S¯)u(cid:3)πdy, A A A EA