Conformal transformations and gluing formulas Klaus Kirsten Baylor - PowerPoint PPT Presentation

Conformal transformations and gluing formulas Klaus Kirsten Baylor University Microlocal and Global Analysis, Interactions with Geometry University of Potsdam, March 4, 2019 Joint work with Yoonweon Lee (Inha University, Korea) J. Math. Phys. 56 (2015) 123501 (19pp) J. Geom. Phys. 117 (2017) 197-213 Symmetry 10 (2018) 31 (16pp) J. Geom. Anal. 28 (2018) 3856-3891 The BFK-gluing formula and the curvature tensors on a two-dimensional compact manifold; submitted The gluing formula, conformal transformations, and geometry; in preparation Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 1 / 20

Analytical surgery M = M 1 ∪ N M 2 M 1 M 2 N Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 2 / 20

Analytical surgery M = M 1 ∪ N M 2 M 1 M 2 N Eigenvalue problem for the Laplacian: � − ∆ i ϕ ( i ) λ ( i ) ϕ ( i ) ϕ ( i ) = k , N = 0 , i = 1 , 2 , � k k k � Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 2 / 20

Analytical surgery M = M 1 ∪ N M 2 M 1 M 2 N Eigenvalue problem for the Laplacian: � − ∆ i ϕ ( i ) λ ( i ) ϕ ( i ) ϕ ( i ) = k , N = 0 , i = 1 , 2 , � k k k � − ∆ ϕ k = λ k ϕ k . Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 2 / 20

Outline 1 Introduction Heat kernel Zeta function BFK-gluing formula 2 Polynomial q ( λ ) and geometry 3 Applications: Casimir forces in pistons 4 Outlook Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 3 / 20

Introduction What are spectral functions? Eigenvalue problem for a suitable differential operator P def. on Y : Pu ℓ ( x ) = λ ℓ u ℓ ( x ) , B u ℓ | x ∈ ∂ Y = 0 0 < λ 1 ≤ λ 2 ..., λ ℓ → ∞ as ℓ → ∞ . Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 4 / 20

Introduction What are spectral functions? Eigenvalue problem for a suitable differential operator P def. on Y : Pu ℓ ( x ) = λ ℓ u ℓ ( x ) , B u ℓ | x ∈ ∂ Y = 0 0 < λ 1 ≤ λ 2 ..., λ ℓ → ∞ as ℓ → ∞ . Heat kernel ( m + 1 = dim( Y )): ∞ � e − t λ k K ( t ) = k =1 Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 4 / 20

Introduction What are spectral functions? Eigenvalue problem for a suitable differential operator P def. on Y : Pu ℓ ( x ) = λ ℓ u ℓ ( x ) , B u ℓ | x ∈ ∂ Y = 0 0 < λ 1 ≤ λ 2 ..., λ ℓ → ∞ as ℓ → ∞ . Heat kernel ( m + 1 = dim( Y )): ∞ ∞ � e − t λ k ∼ t − ( m +1) / 2 � c n t n K ( t ) = k =1 n =0 , 1 / 2 , 1 ,... Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 4 / 20

Introduction What are spectral functions? Eigenvalue problem for a suitable differential operator P def. on Y : Pu ℓ ( x ) = λ ℓ u ℓ ( x ) , B u ℓ | x ∈ ∂ Y = 0 0 < λ 1 ≤ λ 2 ..., λ ℓ → ∞ as ℓ → ∞ . Heat kernel ( m + 1 = dim( Y )): ∞ ∞ � e − t λ k ∼ t − ( m +1) / 2 � c n t n K ( t ) = k =1 n =0 , 1 / 2 , 1 ,... � � c n = dx a n ( x ) + dy b n ( y ) Y ∂ Y Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 4 / 20

Gluing formula for the heat kernel: K 1 ( t ) + K 2 ( t ) − K ( t ) Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 5 / 20

Gluing formula for the heat kernel: t → 0 K 1 ( t ) + K 2 ( t ) − K ( t ) ∼ ∞ � t − ( m +1) / 2 � dy ˜ t n b n ( y ) n =1 / 2 , 1 ,... N Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 5 / 20

Gluing formula for the heat kernel: t → 0 K 1 ( t ) + K 2 ( t ) − K ( t ) ∼ ∞ � t − ( m +1) / 2 � dy ˜ t n b n ( y ) n =1 / 2 , 1 ,... N The combination of the heat kernels has a small- t asymptotics only depending on a density on N . Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 5 / 20

Laplacian under conformal scaling: � � ∆ f = | g | − 1 / 2 ∂ i | g | 1 / 2 g ij ∂ j f ⇒ ∆ ℓ f = 1 k = 1 g ℓ = ℓ 2 g = ⇒ λ ℓ ℓ 2 ∆ f = ℓ 2 λ k Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 6 / 20

Laplacian under conformal scaling: � � ∆ f = | g | − 1 / 2 ∂ i | g | 1 / 2 g ij ∂ j f ⇒ ∆ ℓ f = 1 k = 1 g ℓ = ℓ 2 g = ⇒ λ ℓ ℓ 2 ∆ f = ℓ 2 λ k Heat kernel under conformal scaling: ∞ e − t λ ℓ t → 0 K ℓ � t − ( m +1) / 2 � c ℓ i , n t n i ( t ) = ∼ i , k k =1 n =0 , 1 / 2 , 1 ,... Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 6 / 20

Laplacian under conformal scaling: � � ∆ f = | g | − 1 / 2 ∂ i | g | 1 / 2 g ij ∂ j f ⇒ ∆ ℓ f = 1 k = 1 g ℓ = ℓ 2 g = ⇒ λ ℓ ℓ 2 ∆ f = ℓ 2 λ k Heat kernel under conformal scaling: ∞ e − t λ ℓ t → 0 K ℓ � t − ( m +1) / 2 � c ℓ i , n t n i ( t ) = ∼ i , k k =1 n =0 , 1 / 2 , 1 ,... � t � t � − ( m +1) / 2 � n t → 0 � ∼ c i , n ℓ 2 ℓ 2 n =0 , 1 / 2 , 1 ,... Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 6 / 20

Laplacian under conformal scaling: � � ∆ f = | g | − 1 / 2 ∂ i | g | 1 / 2 g ij ∂ j f ⇒ ∆ ℓ f = 1 k = 1 g ℓ = ℓ 2 g = ⇒ λ ℓ ℓ 2 ∆ f = ℓ 2 λ k Heat kernel under conformal scaling: ∞ e − t λ ℓ t → 0 K ℓ � t − ( m +1) / 2 � c ℓ i , n t n i ( t ) = ∼ i , k k =1 n =0 , 1 / 2 , 1 ,... � t � t � − ( m +1) / 2 � n t → 0 � ∼ c i , n ℓ 2 ℓ 2 n =0 , 1 / 2 , 1 ,... ⇒ c ℓ ℓ m +1 − 2 n c i , n = = i , n Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 6 / 20

Heat kernel coefficients under conformal scaling: c ℓ ℓ m +1 − 2 n c i , n = i , n � � ( ℓ m +1 dx ) ℓ − 2 n a n ( x ) + ( ℓ m dy ) ℓ 1 − 2 n b n ( y ) = M i N Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 7 / 20

Heat kernel coefficients under conformal scaling: c ℓ ℓ m +1 − 2 n c i , n = i , n � � ( ℓ m +1 dx ) ℓ − 2 n a n ( x ) + ( ℓ m dy ) ℓ 1 − 2 n b n ( y ) = M i N Example: m = 2, n = 3 / 2: � b 1 R + b 2 R rr + b 3 K 2 + b 4 K ab K ab c ℓ � � i , 3 / 2 = c i , 3 / 2 = dy N Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 7 / 20

Zeta function: ∞ ℜ s > m + 1 � λ − s ζ ( s ) = ℓ , 2 ℓ =1 Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 8 / 20

Zeta function: ∞ ℜ s > m + 1 � λ − s ζ ( s ) = ℓ , 2 ℓ =1 Functional determinant: � ∞ ∞ ∞ ln λ ℓ = − d � � � � λ − s = − ζ ′ (0) ” ” ln det P = ln λ ℓ = � ℓ ds � � ℓ =1 ℓ =1 ℓ =1 s =0 D.B. Ray and I.M. Singer, Adv. Math. 7 (1971) 145-210 Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 8 / 20

Zeta function: ∞ ℜ s > m + 1 � λ − s ζ ( s ) = ℓ , 2 ℓ =1 Functional determinant: � ∞ ∞ ∞ ln λ ℓ = − d � � � � λ − s = − ζ ′ (0) ” ” ln det P = ln λ ℓ = � ℓ ds � � ℓ =1 ℓ =1 ℓ =1 s =0 D.B. Ray and I.M. Singer, Adv. Math. 7 (1971) 145-210 BFK-gluing formula: ln det ( − ∆ + λ ) − ln det ( − ∆ 1 + λ ) − ln det ( − ∆ 2 + λ ) = ln det R ( λ ) + q ( λ ) Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 8 / 20

Zeta function: ∞ ℜ s > m + 1 � λ − s ζ ( s ) = ℓ , 2 ℓ =1 Functional determinant: � ∞ ∞ ∞ ln λ ℓ = − d � � � � λ − s = − ζ ′ (0) ” ” ln det P = ln λ ℓ = � ℓ ds � � ℓ =1 ℓ =1 ℓ =1 s =0 D.B. Ray and I.M. Singer, Adv. Math. 7 (1971) 145-210 BFK-gluing formula: ln det ( − ∆ + λ ) − ln det ( − ∆ 1 + λ ) − ln det ( − ∆ 2 + λ ) = ln det R ( λ ) + q ( λ ) [ m / 2] q j λ j :integral of some local density over N . � q ( λ ) = j =0 D. Burghelea, L. Friedlander and T. Kappeler, J. Funct. Anal. 107 (1992) 34-65 G. Carron, Am. J. Math. 124 (2002) 307-352 Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 8 / 20

Let h ∈ C ∞ ( N ), φ i ∈ C ∞ ( M i ), such that ( − ∆ 1 + λ ) φ 1 = 0 , ( − ∆ 2 + λ ) φ 2 = 0 , φ 1 | N = φ 2 | N = h , Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 9 / 20

Let h ∈ C ∞ ( N ), φ i ∈ C ∞ ( M i ), such that ( − ∆ 1 + λ ) φ 1 = 0 , ( − ∆ 2 + λ ) φ 2 = 0 , φ 1 | N = φ 2 | N = h , then the Dirichlet-to-Neumann map is defined as follows: R ( λ ) : C ∞ ( N ) → C ∞ ( N ) , R ( λ )( h ) = ( ∂ r φ 1 − ∂ r φ 2 | N . Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 9 / 20

Polynomial q ( λ ) and geometry BFK-gluing formula: ln det ( − ∆ + λ ) − ln det ( − ∆ 1 + λ ) − ln det ( − ∆ 2 + λ ) = ln det R ( λ ) + q ( λ ) q : integral of some local density over N . D. Burghelea, L. Friedlander and T. Kappeler, J. Funct. Anal. 107 (1992) 34-65 G. Carron, Am. J. Math. 124 (2002) 307-352 M 1 M 2 N Klaus Kirsten (Baylor University) Gluing formula Potsdam, March 4, 2019 10 / 20