BFK-gluing formula for zeta-determinants of Laplacians and a warped product metric

Transcription

1 BFK-gluing formula for zeta-determinants of Laplacians and a warped product metric Yoonweon Lee (Inha University, Korea) Geometric and Singular Analysis Potsdam University February 20-24, 2017 (Joint work with Klaus Kirsten) March 1, 2017

2 Zeta-determinants of Laplacians 1. Zeta-determinants of Laplacians (M, g M ) : a compact Riemannian manifold (possibly with boundary M) If M, impose a well-posed boundary condition D For example, Dirichlet, Neumann, absolute/relative boundary conditions. (or D ) : Laplacian, non-negative, self-adjoint elliptic D.O. of order 2 with σ L( ) (x, ξ) = ξ 2. (or D ) has a discrete spectrum 0 = λ 1 = = λ l0 < λ l0+1 λ l0+2 +

3 Zeta function If is invertible (i.e. l 0 = 0), we define ζ (s) = λ s j = 1 t s 1 Tr e t dt Γ(s) 0 j=l 0+1 If dim ker 1, we define the modified zeta function by ζ (s) = λ s j = 1 ( ) t s 1 Tr e t l 0 dt Γ(s) 0 j=l 0+1 ζ (s) (ζ dim M (s)) are analytic for Re s > 2, admits an analytic continuation to the whole complex plane, has a regular value at s = 0. Det := e ζ (0), or log Det := ζ (0), Det := e ζ (0), or log Det := ζ (0). Zeta-determinant plays a central role in the theory of the analytic torsion (Ray-Singer, 1971).

4 BFK-gluing formula 2. BFK-gluing formula (Burghelea, Friedlander and Kappeler) (1) Basic setting (M, g M ) : a compact closed Riemannian manifold with hypersurface Y Assume that M Y has two components, say, M 1 and M 2. M : Laplacian on M M = M 1 Y M 2 M1,D, M2,D : restriction of M to M 1 and M 2 with the Dirichlet boundary condition Goal : log Det M log Det M1,D log Det M2,D =???

5 (2) Dirichlet-to-Neumann operator R(λ) : C (Y ) C (Y ), 0 λ R Y : unit normal vector field to Y, points outward (inward) to M 1 (M 2 ) Assumption : Spec( Mi,D) (, 0] =, i = 1, 2, For f C (Y ), choose ψ 1 C (M 1 ), ψ 2 C (M 2 ) such that ( M1 + λ)ψ 1 = 0, ( M2 + λ)ψ 2 = 0, ψ 1 Y = ψ 2 Y = f. R(λ)f := ( Y ψ 1 ) Y ( Y ψ 2 ) Y. Remark : ψ i = f ( Mi,D + λ) 1 ( Mi + λ) f, where f is an arbitrary extension of f on a collar neighborhood of Y.

6 R(λ) : an elliptic ΨDO of order 1, non-negative, self-adjoint, invertible if λ > 0. When λ = 0, ker R(0) = {φ Y φ ker M } (= dim ker R(0) = dim ker M ) log Det R(λ) is well defined. R(λ) is defined on C (Y ) but R(λ) does depend on the interior part of M. (3) BFK-gluing formula ν = [ dim Y 2 ] + 1. d ν { log dλ ν Det( M + λ) log Det( M1,D + λ) log Det( M2,D + λ) } = d ν log Det R(λ) dλν d Remark : ν dλ log Det( ν M + λ) = ( 1) ν 1 (ν 1)! Tr ( M + λ) ν is a trace class operator.

7 = = log Det( M + λ) log Det( M1,D + λ) log Det( M2,D + λ) ν 1 c j λ j + log Det R(λ). j=0 As λ, log Det( M + λ) and log Det( Mi,D + λ) have asymptotic expansions, having zero constant terms. log Det R(λ) j= dim Y dim Y π j λ j 2 + j=0 q j λ j 2 log λ = c 0 = π 0 =: c(m, Y ). As λ 0 +, log Det( Mi,D + λ) log Det Mi,D Let dim ker M = dim ker R(0) = l 0. {φ 1,, φ l0 } : O. N. basis of ker M. a ij = φ i Y, φ j Y Y, A 0 = ( ) a ij = A 0 : l 0 l 0 positive definite symmetric matrix

8 lim λ 0 + {log Det( M + λ) log Det R(λ) } = log Det M log Det R (0) log det A 0. BFK-gluing formula log Det M log Det M1,D log Det M2,D = c(m, Y ) log det A 0 + log Det R(0). dim ker M = 1 (i.e. l 0 = 1) = det A 0 = vol(y ) vol(m). dim M is even, = c(m, Y ) = 0. c(m, Y ) is determined by the homogeneous symbol of R(λ) of order up to dim Y, is determined by some data on an arbitrarily small collar neighborhood of Y. g M : product metric on a collar nbd of Y so that M = 2 r + Y near Y, = c(m, Y ) = log 2 ( ζ Y (0) + dim ker Y ).

9 (4) Product case (Z, g Z ) : a compact closed Riemannian manifold Consider the Riemannian product M Z with M = M 1 Y M 2 M Z = (M 1 Z ) Y Z (M 2 Z ) (M 1 Z ) = (M 2 Z ) = Y Z Y Lift the unit normal vector field Y on Y to the unit normal vector field Y on Y Z Define the Dirichlet-to-Neumann operator R in the same way. R : C (Y Z ) C (Y Z ) For F C (Y Z ), choose Ψ 1 C (M 1 Z ), Ψ 2 C (M 2 Z ) such that M1 Z Ψ 1 = M2 Z Ψ 2 = 0, Ψ 1 Y Z = Ψ 2 Y Z = F. RF := ) ) ( Y Ψ 1 Y Z ( Y Ψ 2 Y Z

10 Let {φ 1,, φ l0 } : O. N. basis for ker M, {ψ 1,, ψ w } : O. N. basis for ker Z. {φ i ψ j 1 i l 0, 1 j w} : O. N. basis for ker M Z ( M Z = M Id + Id Z ). ã ij,rs = φ i ψ j Y Z, φ r ψ s Y Z Y Z = φ i Y, φ r Y Y ψ j, ψ s Z = δ js φ i Y, φ r Y Y. det ( ã ij,rs ) BFK-gluing formula for product case = det ( a ij ) w = det A w 0. log Det M Z log Det M1 Z,D log Det M2 Z,D = c(m Z, Y Z ) log det A w 0 + log Det R, where c(m Z, Y Z ) is given as follows. Theorem (Kirsten and L. 2015) c(m Z, Y Z ) = c(m, Y ) ( ν 1 ) ζ Z (0) + dim ker Z + c j ζ Z ( j). j=1

11 3. Warped product case f : [a, b] R + a smooth function, (called a warping function) [a, b] f Y : warped product manifold of [a, b] and Y. ( ) 1 0 with metric g(r, y) = 0 f (r) 2 h ij (y). [a,b] f Y = d 2 dr 2 m f (r) d f (r) dr + 1 f (r) 2 Y, m = dim Y. BFK-gluing for model case [a, b] f Y = ([a, c] f Y ) {c} Y ([c, b] f Y ). Assume that a collar nbd of Y is isometric to [a, b] f Y.

12 log Det ( ( ( ) [a,b] f Y,D) log Det [a,c] f Y,D) log Det [c,b] f Y,D = c + log Det R log Det M log Det M1,D log Det M2,D = c(m, Y ) log det A 0 + log Det R(0). c(m, Y ) and c are the constant parts in the asymptotic expansions of log Det R(λ) and log Det R (λ) as λ. R(λ) R (λ) is a smoothing operator = log Det R(λ) and log Det R (λ) have the same asymptotic expansion = c(m, Y ) = c. G. Fucci and K. Kirsten (2013) computed log Det ( [a,b] f Y,D) using WKB method. Using similar methods, we computed log Det R so that we computed c(m, Y ).

13 log Det ([a,b] f Y,D) b 1 = l 0 (m log f (a) + log a f (t) dt + log 2 m ) f (a) log m 2 f (b) ( b 1 m d(ν) log u 0 (b, ν) ν dt + log(2ν) ν 0 a f (t) 2 ( ) b 1 2(log 2 1) Res s= 1 ζ Y (s) FP s= 1 ζ Y (s) 2 2 a f (t) dt + 2 Res s= ζ (0) + 1 Y 2 ζ Y (0) (log f (a) + log f (b)) + [ m 1 2 ] 2i+2 ( ) FP s=i+ 1 ζ Y (s) G j,2i+1 (a, b) 2 i=0 j=0 log f (a) + m 1 log f (b) 2 ) m M i (0, a, b) i=1 b ζ Y (s) a [ m 1 2 ] 2i+2 ( ) j+i 2 Res s=i+ 1 ζ Y (s) (2 log 2 i=0 j=0 2 2k 1 ) G j,2i+1(a, b) I j,2i+1 (a, b) k=0 ν i log f (t) dt f (t) + [ m 2 ] 2i+1 {( FPs=i ζ Y (s) ) G j,2i (a, b) + Res s=i ζ Y (s) ( H i+j 1 G j,2i (a, b) + I j,2i (a, b) )}, i=1 j=0

14 Theorem (Kirsten and L. 2015) ( ) ( ) ( ) c(m, Y ) = log Det [a,b] f Y,D log Det [a,c] f Y,D log Det [c,b] f Y,D log Det R = log 2(ζ Y (0) + l 0 ) [ m 2 ] 2k + 2 log 2 D2k 1 (0, c) + H k+j 1 Ω j,k (c)f (c) 2j+2k Ress=k ζ Y (s) + k=1 j=0 [ m 2 ] k 1 D 2j 1 (0, c)d 2k 2j 1 (0, c) Ress=k ζ Y (s). k=2 j=1 dim Y is odd = Res s=k ζ Y (s) = 0 for all integer k Z, and ζ Y (0) + l 0 = 0 = c(m, Y ) = 0. f 1 (a constant function) = all Ω j,k (c) = 0 and D 2j 1 (0, c) = 0 = c(m, Y ) = log 2(ζ Y (0) + l 0 ). dim Y = 2 = c(m, Y ) = log 2(ζ Y (0) + l 0) { ( 1 + log 2 2 f (c)f (c) + 1 ) ( 3 4 f (c) 2 16 f (c) )} 4 f (c)f (c) Res s=k ζ Y (s).

15 4. Relative zeta-determinant on a manifold with cusp M : a compact manifold with boundary Y X := M Y (Y [a ɛ, )) g X : a Riemannian metric on X such that g X Y [a ɛ, ) = dr 2 + e 2r g Y. (Y [a ɛ, ) is a warped product manifold with [a ɛ, ) and Y, and f (r) = e r.) (X, g X ) : a manifold with cusp X : C (X ) L 2 (X ) C (X ) L 2 (X ), essentially self-adjoint ( ) X := M Y [a ɛ, a] (Y [a, )) =: M {a} Y Z Z = d 2 dr 2 + m d dr + e2r Y, m = dim Y Spec( X ) = σ pp( X ) σ ac( X ), σ ac( X ) = [ m2 4, )

16 Relative heat kernel asymptotics ) Tr (e t X e t Z,D ) Tr (e t X e t Z,D j=0 c j t m+1 j 2 for t 0 +, dim ker X + O(e ct ) for t. Relative zeta function = + ζ(s; X, Z,D ) 1 1 { } t s 1 Tr(e t X e t Z,D ) dim ker X dt Γ(s) 0 1 Γ(s) 1 (Re s > dim X ) 2 t s 1 { Tr(e t X e t Z,D ) dim ker X } dt (entire function). Relative zeta-determinant Det ( X, Z,D ) := e ζ (0; X, Z,D).

17 Dirichlet-to-Neumann operator : R a : C (Y a) C (Y a) For f C (Y a), choose ψ 1 C (M), ψ 2 C (Z ) L 2 (Z ) such that M ψ 1 = Z ψ 2 = 0, ψ 1 Ya = ψ 2 Ya = f. = R a(f ) := ( Ya ψ 1 ) Ya ( Ya ψ 2 ) Ya. BFK-type gluing formula log Det ( ) X, Z,D log Det M,Z = c(x, Y a) log det A 0 + log Det R a, where c(x, Y a) is computed as follows. Let f (r) = e r. [a ɛ, a + ɛ] f Y = ([a ɛ, a] f Y ) {a} Y ([a, a + ɛ] f Y ) c(x, Y a) = log Det [a ɛ,a+ɛ] f Y,D log Det [a ɛ,a] f Y,D log Det [a,a+ɛ] f Y,D log Det R a. If dim ker X = 1, then det A 0 = vol(ya) vol(x ).

18 + + c(x, Y a) = log 2(ζ Y (0) + l 0 ) [ m 2 ] 2k 2 log 2 D2k 1 (0, c) + H k+j 1 Ω j,k (c)f (c) 2j+2k Ress=k ζ Y (s) k=1 j=0 [ m 2 ] k 1 D 2j 1 (0, c)d 2k 2j 1 (0, c) Ress=k ζ Y (s). k=2 j=1 c(x, Y a) = 0 if dim Y a is odd. If dim Y a = 2 and a = 0, then c(x, Y a) = log 2 ( ζ Y (0) + dim ker Y a ) + ( 3 4 log ) Res s=1 ζ Ya (s)

19 Thank you for attention!!

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