2 BRANSON AND GLKEY coframe element dx m by N [. The (second) fundamental form L of the boundary embedding is a symmetric 2-tensor dened by L ab := 1

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1 Proceedings of the 24 th Editura Mirton, Timisoara, 1996, pp National Conference on Geometry and Topology (Romania), THE FUNCTONAL DETERMNANT N THE STANDARD CONFORMAL CLASS ON FOUR DMENSONAL BALLS AND SPHERCAL SHELLS Thomas P. Branson and Peter B. Gilkey Abstract. Let (A; B) be a natural, formally self-adjoint, elliptic, conformally covariant boundary problem on four-dimensional manifolds with boundary. f g[0] is the standard Riemannian metric on a four-dimensional ball or spherical shell (i.e., the one inherited from R 4 ), and g[!] = e 2! g[0] is a conformal metric, we develop a formula for (det AB)[!]=(det AB)[0], where \det" is the functional determinant, AB is the operator dened by the problem (A; B), and the sux [!] denotes evaluation in the metric g[!]. For certain!, g[!] is the standard metric on a hemisphere or cylinder; this allows us to check our results against some of our previous results, in which the hemisphere and cylinder metrics play the role of the background g[0]. n particular, we give the result when A is the conformal Laplacian, and B gives Dirichlet or Robin (conformally covariant Neumann) conditions. x1. Local differential geometry Consider the category of smooth, compact, m-dimensional Riemannian manifolds (M; g) with smooth On such a manifold, the pullback of the metric g under the M is a Riemannian metric ~g Let R be the Riemann curvature tensor of g, with the sign convention that makes R positive on standard spheres. We adopt the convention that letters i; j; : : : run from 1 to m, and index a local coordinate frame and coframe on M. We raise and lower indices using the metric tensor, and sum over repeated indices. The Ricci tensor of M has ij = R k ikj, and the scalar curvature of M is = i i. Additional invariants describe the embedding and are dened as tensor elds (as opposed to M). Let N be the inward unit geodesic normal in a collar in M, and consider local coordinates (x i ) in a neighborhood of a point for m = N, and for which the x a, a = 1; : : : m 1 are local coordinates Letters a; b; : : : will run from 1 to m 1, and index coordinate frames and coframes of this type The subscript N will be interchangeable with m in this setting, and will serve to indicate that we are working in such a coordinate system. We denote the coordinate Research of T. Branson partially supported by the NSF, the Danish Research Council, and the Erwin Schrodinger nstitute (Vienna). Research of P. Gilkey partially supported by the NSF, the Max Planck nstitute for Mathematics (Bonn), and HES. 1

2 2 BRANSON AND GLKEY coframe element dx m by N [. The (second) fundamental form L of the boundary embedding is a symmetric 2-tensor dened by L ab := 1 2 Ng ab : The trace H := L a a of L is a multiple of the mean curvature. Here we have used ~g to raise the boundary index. Repeated boundary indices are summed from 1 to m 1. A symmetric 2-tensor G is dened by G a b := R a NbN ; and we let F := G a a. The symmetric 2-tensor T is dened by T ab := R c acb : Note that (T + G) ab = ab, and that T a a = R ca ca = 2F. We use g and its pullback to dene quantities like jj 2 = ij ij, jlj 2 = L ab L ab, hl; Gi = L ab G ab, etc. Any intrinsic objects which is built from ~g will be denoted by oversetting a tilde to the notation for the analogous object built from g on M. For example, r and r ~ are the Levi-Civita connections on M respectively;, and ~ are the Laplacians on functions. We write dx and dy for the Riemannian measures on M respectively. Our sign convention for the Laplacian gives = d 2 =dx 2 on R 1. We shall sometimes use a standard abbreviation in which indices after a bar indicate covariant dierentiations with respect to r, for example ' ijjkl = r l r k ' ij := (rr') lkij ; and indices after a colon similarly indicate covariant dientiations with respect to r. ~ Let J = =2(m 1); V = ( Jg)=(m 2); C i jkl = R i jkl + V jk i l V jl i k + V i lg jk V i kg jl : C is the Weyl conformal curvature tensor; V and J are renormalizations of the Ricci and scalar curvatures which are particularly adapted to conformal variational calculations. Given a conformal class fg[!] = e 2! g[0] j! 2 C 1 (M)g with a distinguished base point (background metric) g[0], and given a metric-dependent quantity T, we shall always denote by T [!] the evaluation of T in the metric g[!]. We dene the Paneitz quantity and the Paneitz operator Q = 2jV j 2 + m 2 J 2 + J; P = 2 + f(m 2)J 4V gd + m 4 Q: 2 Here d is the exterior derivative, is the formal adjoint of d, and V is the bundle endomorphism ' = (' i ) 7! (V i j ' j ) on the cotangent bundle T M. We shall state our general results in the setting of tensor bundles over (M; g); that is, bundles of the form V = F V associated to the bundle F of orthonormal frames by a nite-dimensional representation (V; ) of O(m).

3 THE FUNCTONAL DETERMNANT x2. Boundary problems and heat invariants We need to impose several assumptions on the boundary problems (A; B) to be considered. 2.1 Analytic Assumptions. A is a dierential operator of positive order on sections of a tensor bundle V over M. A is formally self-adjoint and has positive denite leading symbol lead (A); that is, lead (A)(x; ) is positive denite in End Vx for all x 2 M and 0 6= 2 T x M. B is an operator on the bundle of Cauchy data for A with the property that the pair (A; B) is elliptic. Formal self-adjointness and the assumption on the leading symbol make sense because tensor bundles over a Riemannian manifold come equipped with Riemannian vector bundle structures. Since lead (A)(x; ) = (1) ord(a) lead (A)(x; ), the assumption of positive denite leading symbol forces the order of A to be even. We shall always denote ord(a) by 2` > 0, so that lead (A) = 2`(A). The bundle W of order 2` Cauchy data for sections of V has a natural grading by subbundles (2.1) W = W 0 : : : W 2`1 where W j holds the j th Cauchy datum. The boundary operator B for an elliptic boundary value problem is valued in an auxiliary bundle W 0 which admits a similar grading W 0 = W 0 0 : : : W 0 2`1 but which has dim W 0 = 1 2 dim W. Let A B be the restriction of A to the subspace C 1 (M; V) B = ff 2 C 1 (M; V) j B(CD 2` F ) = 0g: Here CD 2` : C 1 (M; V)! C 1 (@M; W ) is the operator which assigns the order 2` Cauchy data. Under the analytic assumptions, if f 2 C 1 (M), there is an asymptotic expansion (2.2) Tr L 2 f exp(ta B ) where a n (f; A; B) = M 1X n=0 fa n (x; A)dx + a n (f; A; B)t (nm)=2`; t # 0; n1 X (N f)a n; (y; A; B)dy: The a n (x; A) and a n; (y; A; B) are locally computable from the total symbols of A and B in local coordinates; see [G] for details. The auxiliary function f is a device which allows us to observe the distributional behavior of the heat kernel at the boundary. We need to deal with this extra information because conformal deformation of Tr exp(ta B ) and of det A B along the curve of metrics g["!] (! 2 C 1 (M), " a real parameter) lead to the asymptotics of Tr! exp(ta B ). Here and below, we write simply \Tr" for Tr L 2, and use the notation \tr" for traces over vector bundle bers.

4 4 BRANSON AND GLKEY 2.2 Naturality Assumptions. Suppose that A and B are given locally by universal, polynomial formulas in the jets of the Riemannian metric g and its inverse g ] (the induced metric on the cotangent bundle). Suppose that, with respect to uniform dilations of the metric, A has homogeneity degree ord A, and the boundary condition does not change: (2.) g = 2 g ( E = m E; = 1 ) ) A = 2`A; N ( BCD 2`) = N (B CD 2`); where N is the null space. Suppose further that A satises the analytic assumptions 2.1 categorically; that is, the realization of (A; B) on any Riemannian manifold with boundary satises the analytic assumptions. Since a scaling g = 2 g of the metric induces a scaling N = 1 N of the inward unit normal, the operator CD 2` is sensitive to uniform dilation; thus we had to speak of CD 2` in (2.). Under the naturality assumptions, in view of the remarks following the analytic assumptions, the heat invariants a n (x; A) and a n; (y; A; B) are built polynomially from the same ingredients as A and B, and have homogeneity degrees n and n 1 respectively under uniform dilation of the metric. x. The conformal index and the functional determinant We retain the previous notation, and assume that our boundary problem (A; B) satises the analytic and naturality assumptions 2.1, 2.2. The analytic assumptions guarantee that (A; B) has real eigenvalue spectrum 0 1 : : : " +1, with corresponding eigensections in C 1 (M; V) B. We dene the zeta function of the problem (A; B) by A;B (s) = X j 6=0 j j j s : There exist " > 0 and j 0 2 N such that j j " whenever j j 0, so A;B (s) is manifestly well-dened and holomorphic for large Re s. Since there are only nitely many nonpositive j, the heat expansion (2.2) gives (.1) X j 6=0 e tj jj = q(a; B) + 2 X j <0 = NX n=0 sinh t j + a n (A; B)t (nm)=2` + O NX n=0 t Nm+1 2` a n (A; B)t (nm)=2` + O ; t Nm+1 2` where a n (A; B) = a n (1; A; B), q(a; B) is the multiplicity of 0 as an eigenvalue of (A; B), and 8 a n (A; B) q(a; B); n = m; a n (A; B) = >< >: a n (A; B) + 2 X j <0 k j =k!; n = m + 2`(1 + 2k); k 2 N; a n (A; B) otherwise:

5 THE FUNCTONAL DETERMNANT 5 Applying the Mellin transform, we get a meromorphic continuation of A;B (s) to C : A;B (s) = 1 NX s m n 1 a n (A; B) (s) 2` + n=0 1 0 t s1 O t Nm+1 2` 1 X! dt + t s1 e tjjj dt ; 1 j 6=0 where O(t (Nm+1)=2`) is the error term from (.1). n particular, A;B (s) is regular at s = 0, and we dene the functional determinant of the problem (A; B) by det A B = (1) #f j<0g exp( 0 A;B(0)): t is important to note that the functional determinant is not invariant under uniform dilation of the metric. Suppose that g = 2 g, 0 < 2 R. Then A; B(0) = A;B (0); det A B = 2` A;B(0) det A B : That is, the quantity A;B (0) is scale-invariant, and the functional determinant is homogeneous of degree 2` A;B (0). Thus the functional P(A; B; g) = vol(g) 2` A;B(0)=m det A B is a scale-invariant \version" of the determinant. An added advantage of P(A; B; g) is that, like the determinant, it is a spectral invariant, since a 0 (A; B) = C vol(g); where C is a constant depending only on 2`(A). (The number 2`=m can be recovered from the spectrum because m=2` is the leading exponent in the heat asymptotics (2.2).) We e f m > 1 6= ;, we can multiply P by a power of the isoperimetric functional to get a new scale-invariant functional (.2) P (A; B; g) = vol(g) 2`=m vol(~g) 2`( A;B(0))=(m1) det A B ; 2 R: The new ingredient, vol(~g), is generically a spectral invariant: a 1 (A; B) has the form C vol(~g) for some constant C which depends on (A; B) but not on M. Thus vol(~g) is determined by the spectrum when C 6= 0. We shall now impose some additional assumptions..1 Conformal Assumptions. Suppose that A is a positive integer power of a natural dierential operator D, A = D h, which is conformally covariant in the sense that given a conformal class hg[0]i, e a+2`=h D[!] = D[0](e a! );! 2 C 1 (M);

6 6 BRANSON AND GLKEY for some a 2 R. condition Suppose that B is the operator arising from the iterated boundary b CD 2`=h ' = b CD 2`=h (D') = : : : = b CD 2`=h (D h1 ') = 0; where (D; b) is an elliptic boundary value problem. Suppose that the conformal behavior of b is compatible with that of D in the sense that N ((bcd 2`=h )[!]) = e a! N ((bcd 2`=h )[0]): For example, let A = D be the conformal Laplacian, or Yamabe operator Y = + m 2 4(m 1) : Y is conformally covariant of bidegree ((m 2)=2; (m + 2)=2): Y [!]' = e m+2 2! Y [0](e m2 2! '); ' 2 C 1 (M): Dirichlet conditions for Y are obtained by letting W 0 be a trivial line bundle setting W 1 = 0, and setting B 0;0 = d; B 0;1 = B 1;0 = B 1;1 = 0 in the block decomposition corresponding to (2.1). Dirichlet conditions are, of course, conformally compatible. There is also a conformally compatible Neumann condition, sometimes called the Robin condition by physicists. By [BG1, Appendix], N m 2 2(m 1) H is conformally compatible with Y. n more detail, let W 0 1 be a trivial line bundle, W 0 0 = 0, B 1;1 = d, B 0;1 = (m 2)H=2(m 1), and B 0;0 = B 1;0 = 0..2 Conformal ndex Theorem [B1, BG]. f (A; B) satises 2.1, 2.2, and.1, then the quantities q(a; B), #f j < 0g, a m (A; B), and A;B (0) are constant on each conformal class.. Theorem [B2, BG]. Suppose (A; B) satises 2.1, 2.2, and.1. Let (M; g[0]) be a particular manifold with boundary together with a conformal class on which N (A B ) = 0, and let! 2 C 1 (M). Then (d=d")j "=0 0 A["!];B["!] (0) = 2`a m(!; A[0]; B[0]):

7 THE FUNCTONAL DETERMNANT 7 x4. Determinant quotients We introduce abbreviations for local scalar invariants, some depending on an auxiliary function f 2 C 1 (M). For convenience, we write all indices down; the convention is that one index in each pair (it doesn't matter which) is raised before summing. Abbreviation nvariant ndex expression X 1 N R ijijjn X 2 H R ijij L aa X F H R anan L bb X 4 hg; Li R anbn L ab X 5 ht; Li R cacb L ab X 6 H L aa L bb L cc X 7 HjLj 2 L aa L bc L bc X 8 tr L L ab L bc L ca Y 1 (f) (Nf) f jn R ijij Y 2 (f) (N 2 f)h f jnn L aa Y (f) ( ~f)h f :aa L bb Y 4 (f) (Nf)H 2 f jn L aa L bb Y 5 (f) (Nf)F f jn R anan Y 6 (f) h r ~ rf; ~ Li f:ab L ab Y 7 (f) (Nf)jLj 2 f jn L ab L ab Y 8 (f) N()f f jiin 1 (f; f) (Nf)N 2 f f jn f jnn 2 (f; f) (Nf)( ~)f f jn f :aa (f; f) (Nf) 2 H f jn f jn L aa 4 (f; f) j dfj ~ 2 H f :a f :a L bb 5 (f; f) h df ~ df; ~ Li f :a f :b L ab 6 (f; f) h df; ~ d(nf)i ~ f :a (f jn ) :a E 1 (f; f; f) (Nf)j dfj ~ 2 f jn f :a f :a E 2 (f; f; f) (Nf) f jn f jn f jn Table 4.1

8 8 BRANSON AND GLKEY Let S = 1 X X 6 2 X 4 + 1X 9 6 1X 8; m = 4; S(!) = 1Y 1(!) + Y (!) Y 5 (!) Y 6 (!) 1Y 2 8(!); m = 4; L 4 = 1 m 1 X 2 + X (m )X 4 + X 5 ; 2 L 5 = (m 1) X 6 + X 7 m 1 X 8 ; `1(!) = Y 4 (!) + (m 1)Y 7 (!); `2(!) = Y 1 (!) (m )Y 2 (!) + Y (!) 1 m 1 Y 4(!) + (m 1)Y 5 (!); `(!) = m2 m 2 Y 2 (!) 2m m 1 m 1 Y (!) m2 5m + 2 Y (m 1) 2 4 (!) (m 4)Y 5 (!) + (m 2)Y 6 (!) + Y 8 (!); A = X 6 (m 1) 2 X 8 ; q 1 (!) = Y (!) (m 1)Y 6 (!); q 2 (!) = (m )Y 1 (!) + (m )(m 2)Y 2 (!) 2(m 2)Y (!); q (!) = (m 1)Y 1 (!) (m 1)(m 2)Y 2 (!) + 2(m 2)Y 4 (!): We note the identity f jiin = f jnnn + (f jn ) :aa + 2(L ab f :b ) :a H :a f :a F f jn jlj 2 f jn Hf jnn ; and the local expression (M) = (2 2 ) 1 R, and use H M (jcj 2 + 4Q)dx + (4 2 ) (S L 4 L 5 )dy; m = 4; for the Euler characteristic H R in dimension four. n what follows, we denote M simply or simply for the integral The following is a consequence of the Conformal ndex Theorem. 4.1 Theorem [BG]. Let m = 4, and suppose (A; B) satises 2.1, 2.2, and.1. Then a 4 (f; A; B) has the form a 4 (f; A; B) = 1 + fq + fs fx fl fjcj fj 1 6 fl 5 + 8X =1 Y (f); where the constants, = 1; : : : ; 5 and, = 1; : : : ; 8 depend only on the formal functorial expression for (A; B). n particular, a 4 (A; B) = 1 jcj Q + S L L 5 : We can now integrate along conformal curves g "! and use Theorem. to get a determinant quotient formula.

9 THE FUNCTONAL DETERMNANT Theorem [BG]. Under the assumptions of Theorem., (2`) 1 log det(a B)[!] det(a B )[0] = !(jCj 2 dx)[0]!(p [0]!)dx[0] + 1 2!(Qdx)[0] +!(Sdy)[0] 1 (Y (!)dy)[0] 1!(S(!)dy)[0] + (( )(!;!)dy)[0] f(j 2 dx)[!] (J 2 dx)[0]g + 4!(L 4 dy)[0] + 5!(L 5 dy)[0] X + i (`i(!)dy)[0] i=1 + 1 (q 1 (!)dy)[0] + (( )(!;!)dy)[0] + 2 (q 2 (!)dy)[0] + (( )(!;!)dy)[0] + (q (!)dy)[0] + ((9 2 4 )(!;!)dy)[0] +c (Y (!)dy)[0] + (( )(!;!)dy)[0] (E 1 (!;!;!)dy)[0] +c 4 (Y 4 (!)dy)[0] ( (!;!)dy)[0] + (E 2 (!;!;!)dy)[0] : This formula becomes a little more manageable when we assume that the background metric g 0 satises some local symmetry assumptions: 4. Denition. (M; g[0]) is a model background of type if (rr)[0] = is totally geodesic, and M is connected. (M; g[0]) is a model background of type if g[0] is at and ( ~ rl)[0] = 0. Recall the functionals P of (.2). 4.4 Theorem [BG]. Under the assumptions of Theorem., if (M; g[0]) is a background

10 10 BRANSON AND GLKEY of type and = a 4 (A; B), or if (M; g[0]) is a background of type and = 0, (2`) 1 log P (A; B; g[!]) P (A; B; g[0]) = E 1 ( or ) + 2!(P [0]!)dx[0] 1 (Y 4 (!)dy)[0] 1!(S(!)dy)[0] + (( )(!;!)dy)[0] X f(j 2 dx)[!] (J 2 dx)[0]g + i (`i(!)dy)[0] i=1 + 1 (( )(!;!)dy)[0] + 2 (q 2 (!)dy)[0] + (( )(!;!)dy)[0] + (q (!)dy)[0] + ((9 2 4 )(!;!)dy)[0] +c (Y (!)dy)[0] + (( )(!;!)dy)[0] (E 1 (!;!;!)dy)[0] +c 4 (Y 4 (!)dy)[0] ( (!;!)dy)[0] + (E 2 (!;!;!)dy)[0] ; where E = a 4(A; B) 4 R e 4(!!) dx[0] log v[0] = f 2 2 (M) ( )jCj 2 [0]v[0]g; and E = a 4(A; B) H e (!~!) dy[0] log ~v[0] = f (M) + 1 ( )L 4 [0]~v[0] + 1 ( )L 5 [0]~v[0]g: Here! := ( R!dx[0])=v[0] is the mean value of! over M, and ~! := ( H!dy[0])=~v[0] is the mean value x5. Balls and shells There need not be a model background of type or in a given conformal class, of course. t can happen, however, that there are model backgrounds of both types in the same conformal class. For example, the round metric on the closed hemisphere H 4 (type ) is conformal to the at metric on the closed ball B 4 (type ). The standard metric on the cylinder C 4 h = [0; h] S of height h (type ) is conformal to the at metric on the spherical shell A 4 s = fx 2 R 4 j 1 jxj sg, s = e h (type ). n [BG], we worked things out completely for hemispherical and cylindrical backgrounds. mplicit in these results, of course, is an answer for the conformally equivalent ball and shell backgrounds. n this section, we shall work things out explicitly and directly

11 THE FUNCTONAL DETERMNANT 11 for ball and shell backgrounds. This will provide checks on our formulas. An important motivational point is that these calculations are very dierent in the type case; they are not just a disguised form of the type calculations. The following results were originally distributed in preprint form as part of [BG2]. For the moment, let m be unrestricted, and consider the ball B m. Since all interior invariants vanish in a at metric, jcj 2 = J 2 = Q = 0; P = 2 : The fundamental form and normalized mean curvature are so L = ~g; H = m 1; X 6 = (m 1) ; X 7 = (m 1) 2 ; X 8 = m 1: Because the Riemann tensor R vanishes, As a result of the formulas for the X i, X 1 = X 2 = X = X 4 = X 5 = 0: L 4 = L 5 = 0: An elementary but tedious calculation on manifolds with boundary gives f iin = f jnnn + (f jn ) :aa + 2 ~ r a (L ab ~ rb f) H :a f :a F f jn jlj 2 f jn Hf jnn : n our situation, this reads Furthermore, and Y 8 (f) = N f + ( ~)(Nf) + 2( ~)f (m 1)N 2 f (m 1)Nf: Y 1 (f) = Y 5 (f) = 0 Y 2 (f) = (m 1)N 2 f; Y (f) = (m 1)( ~)f; Y 4 (f) = (m 1) 2 Nf; Y 6 (f) = ( ~ )f; `1(f) = 0; Y 7 (f) = (m 1)Nf; `2(f) = (m )(m 1)N 2 f + (m 1)( ~)f (m 1)Nf; `(f) = N f + ( ~)(Nf) + (m 2 4m 1)N 2 f m( ~)f (m 2 4m + 1)Nf; q 1 (f) = 0; q 2 (f) = (m )(m 2)f(m 1)N 2 f 2( ~)fg; q (f) = (m 1) 2 (m 2)fN 2 f + 2Nfg; (f; f) = (m 1)(Nf) 2 ; 4 (f; f) = (m 1)j ~ d!j 2 ; 5 (f; f) = j ~ d!j 2 : f m = 4, S = 2; S(f) = 1 2 N f 1 2 ( ~ )(Nf) + 2 N 2 f + ( ~ )f + 2 Nf: By the second formula of Theorem 4.1, the conformal index is a 4 (A; B) = Specializing Theorem 4.4, we have:

12 12 BRANSON AND GLKEY 5.1 Theorem. Suppose that (A; B) satises 2.1, 2.2, and.1, and that N (A B ) = 0 on (B 4 ; g[0]), where g[0] is the standard at metric. Then for! 2 C 1 (B 4 ), (2`) 1 log P 0(A; B; g[!]) P 0 (A; B; g[0]) = 2 42 log + 4 e (!~!) dy[0] c 2 ((N!)( B 4! 2 [0]!dx[0] + 1 4!((N 2!)dy)[0] 4!((N!)dy)[0] (J 2 dx)[!] + ( ) ((N 2!)dy)[0] B +( c 4 ) ((N!)dy)[0] ((N!)(N 2!)dy)[0] + (6 2 9c 4 ) c +c 4 ((N!) 4 (j ~ d!j 2 dy)[0] 4 ((N!) 2 4 ((N!)j ~ d!j 2 dy)[0] A formula for the determinant functional (2`) 1 log det(a B)[!] det(a B )[0] is obtained by replacing 42 2 log 4 e (!~!) dy[0] 2 2 by ~! in the formula for (2`) 1 log P 0(A; B; g[!]) P 0 (A; B; g[0]) : To check against the hemispherical result of [BG], note that the round hemisphere metric is g[] for 2 = log 1 + r : 2 We can use the above to compute (2`) 1 log det(a B)[] det(a B )[0] ;

13 THE FUNCTONAL DETERMNANT 1 and, if all is well, the answer will be the additive inverse of the quantity calculated in [BG, Corollary 5.4], where the hemisphere is the background and the ball is the perturbation. We note that = N 2 = 0; N N = 1 on the boundary: By the conformal covariance relation for the Paneitz operator [B], 2 [0] = Q[]e 4 Q[0] = 6e 4 : To evaluate R B 4 e 4 dx[0], note that e 4 dx[0] = dx[] is the measure on the round hemisphere. Since vol(h m ) = (4)m=2 (m=2) 2(m) ; vol(b m ) = 2m=2 m(m=2) ; we have 1 ( 2 [0])dx[0] = e 4 dx[0] = 4 B 4 2 B ( 8 log 2 1): 9 Thus (2`) 1 log P 0(A; B; g[]) P 0 (A; B; g[0]) = (2`)1 log det(a B)[] det(a B )[0] = 2 2 ( 8 log ) + 2 vol(h 4 ) + vol(s )f c 4 g = 2 f(4 log ) c 4 g: This is indeed exactly minus the number computed in [BG, Corollary 5.4]. Having worked out the ball case, we have a head start in working out the case of the spherical shell A 4 s, s > 1. We use the case of the unit ball B 4 1 treated above, plus the case of the ball B 4 s of radius s, which is of course a rescaling of the B 4 1 case. Besides the scale factor, we have to keep track of the direction of the unit normal. Our setup is in terms of Riemannian measures rather than volume elements, so dy is not signed. 5.2 Theorem. Suppose that (A; B) satises 2.1, 2.2, and.1, and that N (A B ) = 0 on (A 4 s ; g[0]), where g[0] is the standard at metric. Let X r, where r is the radial

14 14 BRANSON AND GLKEY spherical coordinate in R m. Then for! 2 C 1 (A 4 s), (2`) 1 log det(a B)[!] det(a B )[0] = ( c ) (~!(S s) ~!(S 1)) s 1 4 S A 4 s S s A 4 s! 2 [0]!dx[0] S s!!((x 2!)dy)[0] 4 s 2 (J 2 dx)[!] + ( ) s 1 +( c 4 ) + S s +(6 2 9c 4 ) S1 s 2 Ss S1! ((X!)dy)[0] 6 2 s 1 +( c ) +c 4 S s S1! S s! S1 s 1 S s ((X!) dy)[0]: S s ((X!) 2 dy)[0] S1!! ((X!)( ~!)dy)[0] S1! S s S s S s ((X!)dy)[0] S1!! (j ~ d!j 2 dy)[0] c S 1 S 1 S 1!! ((X!)(X 2!)dy)[0] S s!((x!)dy)[0]!((x!)dy)[0] ((X 2!)dy)[0]! ((X!)j d!j ~ 2 dy)[0] S1 Here S r is the sphere of radius r centered at the origin in R4, and ~!(S r ) is the mean value of! over this sphere. To check against the cylindrical result of [BG], let g[0] be the at A 4 s metric, and let g[] be the standard metric on the cylinder C 4 h = [0; h]s with h = log s; then = log r. Note that the term in Theorem 5.2 (with in place of!) which involves 2 [0] vanishes, since J[] = 1 and 2V [] = dt 2 + d 2, so that Q[] = 0. (Alternatively, we could use the fact that log r is a constant multiple of the fundamental solution of 2 in R 4.) The fact that J[] = 1 also evaluates the term in Theorem 5.2 as 1 2 h vol(s ). To evaluate the other terms, note that X = r 1 ; X 2 = r 2 ; X = 2r ; we need this at r = s and r = 1. No boundary integrals except the mean value of over S s survive the computation, and the result, using vol(s ) = 2 2, is (2`) 1 log det(a B)[] det(a B )[0] = 22 h( ): This is exactly minus the quantity computed in [BG, (5.1)], as desired.

15 THE FUNCTONAL DETERMNANT 15 x6. The conformal Laplacian with Dirichlet and Robin boundary conditions The determinant quotient formulas of Theorem 4.2 involve coecients (1 5), i (i = 1; 2; ), j (j = 1; 2; ), c k (k = ; 4) that depend only on the universal formula for (A; B), and not on the particular manifold M. n this section, we write down constants for the two boundary value problems described in x. Let Y be the conformal Laplacian, and let (Y; D), (Y; R) be the corresponding Dirichlet and Robin problems. We compute for the moment without restricting the dimension. We separate the computation of the interior and boundary terms; of course these can be \mixed" by integration by parts. Thus the precise assertion of the next two lemmas is that the sum of the formulas they give is the appropriate a 4 coecient. 6.1 Lemma [BG]. The interior terms of 60(4) m=2 a 4 (f; Y; D) or of 60(4) m=2 a 4 (f; Y; R) are = f 2jCj 2 2(m 2)(m 6)jV j 2 + (5m 16)(m 6)J 2 + 6(m 6)J f 2jCj 2 + 2(m 6)Q 2(m 4)(m 6)jV j 2 + 4(m 4)(m 6)J 2 + 4(m 6)J : n the last expression, we have used a highly linearly dependent list of local invariants; the terms that survive upon restriction to dimension m = 4 are linearly independent. 6.2 Lemma [BG]. The boundary terms of 60(4) m=2 a 4 (f; Y; D) are 6(2m 7) 10(m 4) f X 1 X 2 4X + 12X 4 4X m 1 m 1 21 X X X 8 15(m 4) + Y 1 (f) + 24Y 2 (f) + 24Y (f) 180 m 1 7 Y 4(f) Y 7(f) 0Y 8 (f) : (The invariants Y 5 (f) and Y 6 (f) appear with coecient 0.) Restricting to dimension m = 4 and changing the basis of local scalar invariants, we get: 6. Theorem [BG]. f = (4) 2 60 and similarly for i, j, and c k, then for the problem (Y; D) in dimension m = 4, 1 = 2; 2 = = 8; 4 = 4; 5 = 88 7 ; 1 = 20 7 ; 2 = 0; = 0; 1 = 20; 2 = 5 = 1 9 c = 8; c 4 = 152 6

16 16 BRANSON AND GLKEY For the problem (Y; R), 1 = 2; 2 = = 8; 4 = 4; 5 = 8; 1 = 4; 2 = 0; = 0; 1 = 20; 2 = 25 = 29 9 c = 8; c 4 = 8 : 9 When the background is the ball or shell, we are interested in the following combinations: ( c )(Y; D) = 48; ( c )(Y; R) = 72; ( )(Y; D) = 162; ( )(Y; R) = 18; ( c 4 )(Y; D) = 810=7; ( c 4 )(Y; R) = 78; (6 2 9c 4 )(Y; D) = 642=7; (6 2 9c 4 )(Y; R) = 42; ( c )(Y; D) = ( c )(Y; R) = 8: 6.4 Corollary. f g[0] is the standard B 4 metric and g[] the standard H 4 metric, then the conformal index is a 4 (Y; D) = a 4 (Y; R) = 1=180, and for = 1=180, log P (Y; D; g[]) P (Y; D; g[0]) log det(y D)[] det(y D )[0] 17 = (log 6 + )=60 > 0; = (4 log 2 + )=60 > 0; 21 log P (Y; R; g[]) P (Y; R; g[0]) = (log 6 1 )=60 > 0; log det(y R)[] det(y R )[0] = (4 log 2 1 )=60 > 0: f g[0] is the standard A 4 s metric and g[] the standard C4 h metric, s = eh, then the conformal index is a 4 (Y; D) = a 4 (Y; R) = 0, and log det(y D)[] det(y D )[0] = log det(y R)[] det(y R )[0] = h=60 < 0: Branson, Chang, rsted, and Yang [B, BCY] have shown that the scale invariant determinant functional for Y on the conformal class of the round metric g[0] on S 4 is minimized exactly at g[0], and at the metrics h g[0] gotten by pulling g[0] back under a conformal dieomorphism h of (S 4 ; g[0]). n light of this, Corollary 6.4 can be interpreted as saying that passage from H 4 to B 4 has improved (i.e. lowered) the scale-invariant determinant functionals for both (Y; D) and (Y; R). Roughly speaking, round is \best" in the boundariless case, but at is \better" when boundaries are allowed.

17 THE FUNCTONAL DETERMNANT Conjecture. For 1=180 0, the functionals P (Y; D; g[!]) and P (Y; R; g[!]) are minimized at the standard B 4 metric, i.e., at! = 0. References [B] T. Branson, Dierential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), 29{45. [BCY] T. Branson, S.-Y. A. Chang, and P. Yang, Estimates and extremals for zeta function determinants on four-manifolds, Commun. Math. Phys. 149 (1992), 241{262. [BG1] T. Branson and P. Gilkey, The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Dierential Equations 15 (1990), 245{272. [BG2] T. Branson and P. Gilkey, The functional determinant of a four-dimensional boundary value problem, Max-Planck-nstitut fur Mathematik (Bonn), preprint 92-90, [BG] T. Branson and P. Gilkey, The functional determinant of a four-dimensional boundary value problem, to appear, Trans. Amer. Math. Soc. [B1] T. Branson and B. rsted, Conformal indices of Riemannian manifolds, Compositio Math. 60 (1986), 261{29. [B2] T. Branson and B. rsted, Conformal geometry and global invariants, Di. Geom. Appl. 1 (1991), 279{08. [B] T. Branson and B. rsted, Explicit functional determinants in four dimensions, Proc. Amer. Math. Soc. 11 (1991), 669{682. [G] P. Gilkey, nvariance Theory, the Heat Equation, and the Atiyah-Singer ndex Theorem, Publish or Perish, Wilmington, Delaware, Thomas P. Branson, Department of Mathematics, University of owa, owa City A USA branson@math.uiowa.edu Peter B. Gilkey, Department of Mathematics, University of Oregon, Eugene OR 9740 USA gilkey@bright.math.uoregon.edu