Min * 1997 Ph.D. avhandling* som ett blogginlägg.
μ \mu μ M a ¨ r n e g a t i v t b o ¨ j d dim = n s t a ¨ n g t R i e m a n n m a ˚ n g f a l d m e d m e t r i s k g , m a ¨ t e t a l a n s l u t n i n g \nablaoch ( i c k e − n e g a t i v ) L a p l a c e − B e l t r a m i − o p e r a t o ¨ r Δ M . L a ˚ t k − t Δ / 2 ( x , y ) r e p r e s e n t e r a r v a ¨ r m e k a ¨ r n a n p a ˚ M M är negativt böjd \dim=n stängt Riemannmångfald med metrisk g, mätetalanslutning \nablaoch (icke-negativ) Laplace-Beltrami-operatör \Delta_M. Låt k_{-t\Delta/2}(x,y) representerar värmekärnan på M M a ¨ r n e g a t i v t b o ¨ j d dim = n s t a ¨ n g tR i e mannm a ˚ n g f a l d m e d m e t r i s k g , m a ¨ t e t a l an s l u t nin g \nablaoch ( i c k e − n e g a t i v ) L a pl a ce − B e lt r ami − o p er a t o ¨ r Δ M . L a ˚ t k − t Δ/2 ( x , y ) re p rese n t er a r v a ¨ r m e k a ¨ r nan p a ˚ M
Därför k − t Δ / 2 ( x , x ) = d D M ∗ μ / g d x k_{-t\Delta/2}(x,x) = dDM_*\mu/\sqrt{g}dx k − t Δ/2 ( x , x ) = d D M ∗ μ / g d x μ \mu μ Ω t ( M ) ∣ x \Omega_t(M)\vert_x Ω t ( M ) ∣ x D M DM D M Obs! D M − 1 Ω t ∣ x DM^{-1}\Omega_t\vert_x D M − 1 Ω t ∣ x
Ω t 0 a ¨ r u t r y m m e t f o ¨ r k o n t i n u e r l i g a k o n t r a k t i b l a s l i n g o r p a ˚ M \Omega_t^0 är utrymmet för kontinuerliga kontraktibla slingor på M Ω t 0 a ¨ r u t ry mm e t f o ¨ r k o n t in u er l i g ak o n t r ak t ib l a s l in g or p a ˚ M
Ω t [ γ ] a ¨ r u t r y m m e t f o ¨ r k o n t i n u e r l i g a s l i n g o r p a ˚ M h o m o t o p i s k t i l l s l u t e n g e o d e s i s k γ . L a ˚ t γ 0 \Omega_t[\gamma] är utrymmet för kontinuerliga slingor på M homotopisk till sluten geodesisk \gamma. Låt \gamma_0 Ω t [ γ ] a ¨ r u t ry mm e t f o ¨ r k o n t in u er l i g a s l in g or p a ˚ M h o m o t o p i s k t i ll s l u t e n g eo d es i s kγ . L a ˚ t γ 0
D M − 1 Ω t 0 [ γ ] a ¨ r p r e i m a g e a v k o n t i n u e r l i g a k o n t r a k t i b l a s l i n g o r p a ˚ M s k r i v s s o m f o ¨ r s k j u t n i n g a r h o m o t o p i c t i l l γ ( s ) = D M ( s ℓ ( γ ) t e ⃗ 1 ) , 0 ≤ s ≤ t . T a ¨ n k H o r o c y k l i s k a K o o r d i n a t e r — v a r j e f i b e r s o m d e n g e o m e t r i s k a g r a ¨ n s e n f o ¨ r p e r i o d i s k a g e o d e t i s k a s f a ¨ r e r S γ 0 ( s ) n − 1 ( k ℓ ( γ 0 ) ) , 0 ≤ s ≤ t , k → ∞ , v e k t o r i s e r a d i N o r m a l B u n d l e o ¨ v e r γ 0 . V a ˚ r a k u r v a t u r b e g r a ¨ n s n i n g a r i n n e b a ¨ r H o r o c y k l i s k a k o o r d i n a t e r f o ¨ r v a r j e γ 0 s o m e n j a ¨ m n , D M − k o m p a t i b e l k o o r d i n a t m a p p n i n g f o ¨ r Ω t 0 [ γ ] DM^{-1}\Omega_t^0[\gamma] är preimage av kontinuerliga kontraktibla slingor på M skrivs som förskjutningar homotopic till \gamma(s) = DM(\frac{s\ell(\gamma)}{t}\vec{e}^1), 0\leq s \leq t. Tänk Horocykliska Koordinater — varje fiber som den geometriska gränsen för periodiska geodetiska sfärer S_{\gamma_0(s)}^{n-1}(k\ell(\gamma_0)), 0\leq s \leq t, k\rightarrow\infty, vektoriserad i Normal Bundle över \gamma_0. Våra kurvaturbegränsningar innebär Horocykliska koordinater för varje \gamma_0 som en jämn, DM-kompatibel koordinatmappning för \Omega_t^0[\gamma] D M − 1 Ω t 0 [ γ ] a ¨ r p re ima g e a v k o n t in u er l i g ak o n t r ak t ib l a s l in g or p a ˚ M s k r i v sso m f o ¨ rs kj u t nin g a r h o m o t o p i c t i ll γ ( s ) = D M ( t s ℓ ( γ ) e 1 ) , 0 ≤ s ≤ t . T a ¨ nk Horocy k l i s ka Koor d ina t er — v a r j e f ib erso m d e n g eo m e t r i s ka g r a ¨ n se n f o ¨ r p er i o d i s ka g eo d e t i s ka s f a ¨ rer S γ 0 ( s ) n − 1 ( k ℓ ( γ 0 )) , 0 ≤ s ≤ t , k → ∞ , v e k t or i ser a d i N or ma lB u n d l e o ¨ v er γ 0 . V a ˚ r ak u r v a t u r b e g r a ¨ n s nin g a r inn e b a ¨ rHorocy k l i s kak oor d ina t er f o ¨ r v a r j e γ 0 so m e nj a ¨ mn , D M − k o m p a t ib e l k oor d ina t ma pp nin g f o ¨ r Ω t 0 [ γ ]
Nu x ⃗ ( τ ) + ℓ ( γ ) e ⃗ 1 \vec{x}(\tau)+\ell(\gamma)\vec{e}^1 x ( τ ) + ℓ ( γ ) e 1 outvecklad för “offset” kinked geodesic homotopic till γ : D M ( x ⃗ ( τ ) + s ℓ ( γ ) t e ⃗ 1 ) , 0 ≤ s ≤ t \gamma: DM(\vec{x}(\tau) + \frac{s\ell(\gamma)}{t}\vec{e}^1), 0\leq s \leq t γ : D M ( x ( τ ) + t s ℓ ( γ ) e 1 ) , 0 ≤ s ≤ t ℓ ( γ 0 ) \ell(\gamma_0) ℓ ( γ 0 ) D M ( x ⃗ ( τ ) ) DM(\vec{x}(\tau)) D M ( x ( τ )) t t t J = lim s ↑ t D M ′ ∣ D M ( x ⃗ ( τ ) + ℓ ( γ ) s t e ⃗ 1 ) J=\lim_{s\uparrow t}DM^\prime\vert_{DM(\vec{x}(\tau) + \frac{\ell(\gamma)s}{t}\vec{e}^1)} J = lim s ↑ t D M ′ ∣ D M ( x ( τ ) + t ℓ ( γ ) s e 1 ) T D M ( x ⃗ ( τ ) ) M T_{DM(\vec{x}(\tau))}M T D M ( x ( τ )) M J D M ( x ⃗ ( τ ) + ℓ ( γ ) e ⃗ 1 ) J_{DM(\vec{x}(\tau)+\ell(\gamma)\vec{e}^1)} J D M ( x ( τ ) + ℓ ( γ ) e 1 ) Jacobi Fields , eftersom D M DM D M R n \Reals^n R n 1 / 2 ∫ 0 t < d X | d X > s 1/2 \int_0^t \bra{dX}\ket{dX}_s 1/2 ∫ 0 t ⟨ d X ∣ d X ⟩ s
X t = X 0 + ∫ 0 t J X t d B t \begin{aligned}
X_t &= X_0 + \int_0^t \sqrt{J}_{X_t} dB_t \\
\end{aligned} X t = X 0 + ∫ 0 t J X t d B t
Z − Δ / 2 ( t ) : = ∫ M k − t Δ / 2 ( x , x ) g d x = ∑ j = 0 ∞ e − λ i t / 2 Z_{-\Delta/2}(t) := \int_M k_{-t\Delta/2}(x,x) \sqrt{g}dx = \sum_{j=0}^\infty e^{-\lambda_i t/2} Z − Δ/2 ( t ) := ∫ M k − t Δ/2 ( x , x ) g d x = ∑ j = 0 ∞ e − λ i t /2
Låt oss slutligen definiera följande från deras Radon-Nicodym derivat:
D M ∗ μ ( Ω t ) : = ∫ M D M ∗ μ ( Ω t ∣ x g d x ) D M ∗ μ ( Ω t 0 ) : = ∫ M D M ∗ μ ( Ω t 0 ∣ x g d x ) D M ∗ μ ( Ω t [ γ ] ) : = ∫ M D M ∗ μ ( Ω t [ γ ] ∣ x g d x ) \begin{aligned}
DM_*\mu(\Omega_t) &:= \int_M DM_*\mu(\Omega_t\vert_x \sqrt{g}dx)\\
DM_*\mu(\Omega^0_t) &:= \int_M DM_*\mu(\Omega^0_t\vert_x \sqrt{g}dx)\\
DM_*\mu(\Omega_t[\gamma]) &:= \int_M DM_*\mu(\Omega_t[\gamma]\vert_x \sqrt{g}dx) \\
\end{aligned} D M ∗ μ ( Ω t ) D M ∗ μ ( Ω t 0 ) D M ∗ μ ( Ω t [ γ ]) := ∫ M D M ∗ μ ( Ω t ∣ x g d x ) := ∫ M D M ∗ μ ( Ω t 0 ∣ x g d x ) := ∫ M D M ∗ μ ( Ω t [ γ ] ∣ x g d x )
Z − Δ / 2 ( t ) = D M ∗ μ ( Ω t ) = D M ∗ μ ( Ω t 0 ) + ∑ { γ } D M ∗ μ ( Ω t [ γ ] ) D M ∗ μ ( Ω t 0 ) ≈ t → 0 ( 2 π t ) − n / 2 ( v o l ( M ) + t / 6 ∫ M K ( x ) g d x + O ( t 2 ) ) by McKean-Singer D M ∗ μ ( Ω t [ γ ] ) = e − ℓ ( γ ) 2 / 2 t ∫ M D M ∗ μ ( e t < J B B t | B t > Ω t 0 [ γ ] ∣ x g d x ) by Cameron-Martin = e − ℓ ( γ ) 2 / 2 t ∫ T γ 0 M E ( e t J B ∣ Ω t 0 [ γ ] ∣ x ( τ ) ) d x 1 ( τ ) … d x n ( τ ) d τ d D M ∗ μ ( e − ℓ ( γ ) x 1 ( t ) Ω t 0 [ γ ] ) d x 1 ( τ ) … d x n ( τ ) d τ ∣ y ( τ ) ⃗ ≈ t → 0 e − < ∣ I − J D M ( x ⃗ ( τ ) , y ⃗ ( τ ) ) x ⃗ ( τ ) | x ⃗ ( τ ) > / 2 t ( 2 π t ) ( n + 1 ) / 2 ( 1 + O ( t 2 ) ) semi-classical limit Horocyclic coordinates : z ( τ ) − x ( τ ) = x + ℓ ( γ ) e ⃗ 1 ⟹ ∫ M / S 1 ⊕ S 1 k t ( x , z ) d x = lim j → ∞ e − ℓ ( γ ) 2 / 2 t 2 π t E ( e < J X t j x ⃗ | x ⃗ > ) = lim j → ∞ e − ℓ ( γ ) 2 / 2 t 2 π t ∫ M j / S 1 ⊕ S 1 1 2 π t j n det ∣ I − J X j ∣ e − ℓ ( X j ) 2 / 2 t X j \begin{aligned}
Z_{-\Delta/2}(t) = DM_*\mu(\Omega_t) &= DM_*\mu(\Omega^0_t) + \sum_{\set{\gamma}} DM_*\mu(\Omega_t[\gamma]) \\
DM_*\mu(\Omega_t^0) &\approx_{t\rightarrow 0} (2\pi t)^{-n/2}(vol(M) + t/6\int_M K(x)\sqrt{g} dx + O(t^2))\space \small\text{by McKean-Singer}\\
DM_*\mu(\Omega_t[\gamma]) &= e^{-\ell(\gamma)^2/2t}\int_M DM_*\mu(e^{\bra{J_BB_t}\ket{B_t}} _t \Omega_t^0[\gamma]\vert_x\sqrt{g}dx)\space\small \text{ by Cameron-Martin}\\
&= e^{-\ell(\gamma)^2/2t}\int_{T_{\gamma_0}M} E(e^{J_B}_{t} | \Omega_t^0[\gamma]\vert_{x(\tau)})dx^1(\tau)\dots dx^n(\tau) d\tau\\
\frac{dDM_*\mu(e^{-\ell(\gamma)x^1(t)}\Omega^0_t[\gamma])}{dx^1(\tau)\dots dx^n(\tau)d\tau}\vert_{\vec{y(\tau)}}&\approx_{t\rightarrow 0} \frac{e^{-\bra{|I-J_{DM(\vec{x}(\tau),\vec{y}(\tau))}\vec{x}(\tau)}\ket{\vec{x}(\tau)}/2t}}{(2 \pi t)^{(n+1)/2}}(1+O(t^2))\small \text{ semi-classical limit}\\
\text{Horocyclic coordinates}: z(\tau) - x(\tau) &= x + \ell(\gamma)\vec{e}^1\implies\\
\int_{M/S^1\oplus S^1}k_t(x,z) dx &=\lim_{j\rightarrow\infty}\frac{e^{-\ell(\gamma)^2/2t}}{\sqrt{2\pi t}}E(e^{\bra{J_{X^j_t}\vec{x}}\ket{\vec{x}}})\\
&=\lim_{j\rightarrow\infty}\frac{e^{-\ell(\gamma)^2/2t}}{\sqrt{2\pi t}}\int_{M^j/S^1\oplus S^1}\frac{1}{\sqrt{2\pi t}^{jn}\det|I-J_{X^j}|}e^{-\ell(X^j)^2/2t}X^{j}\\
\end{aligned} Z − Δ/2 ( t ) = D M ∗ μ ( Ω t ) D M ∗ μ ( Ω t 0 ) D M ∗ μ ( Ω t [ γ ]) d x 1 ( τ ) … d x n ( τ ) d τ d D M ∗ μ ( e − ℓ ( γ ) x 1 ( t ) Ω t 0 [ γ ]) ∣ y ( τ ) Horocyclic coordinates : z ( τ ) − x ( τ ) ∫ M / S 1 ⊕ S 1 k t ( x , z ) d x = D M ∗ μ ( Ω t 0 ) + { γ } ∑ D M ∗ μ ( Ω t [ γ ]) ≈ t → 0 ( 2 π t ) − n /2 ( v o l ( M ) + t /6 ∫ M K ( x ) g d x + O ( t 2 )) by McKean-Singer = e − ℓ ( γ ) 2 /2 t ∫ M D M ∗ μ ( e t ⟨ J B B t ∣ B t ⟩ Ω t 0 [ γ ] ∣ x g d x ) by Cameron-Martin = e − ℓ ( γ ) 2 /2 t ∫ T γ 0 M E ( e t J B ∣ Ω t 0 [ γ ] ∣ x ( τ ) ) d x 1 ( τ ) … d x n ( τ ) d τ ≈ t → 0 ( 2 π t ) ( n + 1 ) /2 e − ⟨ ∣ I − J D M ( x ( τ ) , y ( τ )) x ( τ ) ∣ x ( τ ) ⟩ /2 t ( 1 + O ( t 2 )) semi-classical limit = x + ℓ ( γ ) e 1 ⟹ = j → ∞ lim 2 π t e − ℓ ( γ ) 2 /2 t E ( e ⟨ J X t j x x ⟩ ) = j → ∞ lim 2 π t e − ℓ ( γ ) 2 /2 t ∫ M j / S 1 ⊕ S 1 2 π t jn det ∣ I − J X j ∣ 1 e − ℓ ( X j ) 2 /2 t X j
I dim = 2 \dim = 2 dim = 2 − κ 2 -\kappa^2 − κ 2
J x ⃗ , y ⃗ d R B = ( e κ d ( x ⃗ , y ⃗ ) / 2 0 0 e − κ d ( x ⃗ , y ⃗ ) / 2 ) ⟹ < J d R B | J d R B > = e κ ℓ ( B ) d R B 1 2 − e − κ ℓ ( B ) d R B 2 2 ∫ 0 t < J d B | J d B > = e κ ℓ ( γ ) − e − κ ℓ ( γ ) det I − J γ = ( e κ ℓ ( γ ) / 2 − e − κ ℓ ( γ ) / 2 ) 2 \begin{aligned}
\sqrt{J_{\vec{x}, \vec{y}}}dRB&=
\begin{pmatrix}
e^{\kappa d(\vec{x},\vec{y})/2} && 0\\
0 && e^{-\kappa d(\vec{x},\vec{y})/2}\\
\end{pmatrix}
\implies&\\
\bra{\sqrt{J}dRB}\ket{\sqrt{J}dRB} &= e^{\kappa \ell(B)}dRB_1^2 - e^{-\kappa \ell(B)}dRB_2^2\\
\int_0^t \bra{\sqrt{ J}dB}\ket{\sqrt{ J}dB} &= e^{\kappa\ell(\gamma)} - e^{-\kappa\ell(\gamma)}\\
\det I-J_{\gamma} &= (e^{\kappa\ell(\gamma)/2}- e^{-\kappa\ell(\gamma)/2})^2
\end{aligned} J x , y d RB ⟨ J d RB J d RB ⟩ ∫ 0 t ⟨ J d B J d B ⟩ det I − J γ = ( e κ d ( x , y ) /2 0 0 e − κ d ( x , y ) /2 ) ⟹ = e κ ℓ ( B ) d R B 1 2 − e − κ ℓ ( B ) d R B 2 2 = e κ ℓ ( γ ) − e − κ ℓ ( γ ) = ( e κ ℓ ( γ ) /2 − e − κ ℓ ( γ ) /2 ) 2
som är konstant över ( x ⃗ , τ ) (\vec{x},\tau) ( x , τ ) ≈ t → 0 \approx_{t\rightarrow 0} ≈ t → 0
D M ∗ μ ( Ω t [ γ ] ) = e − ℓ ( γ ) 2 / 2 t ℓ ( γ 0 ) 2 π t ( e κ ℓ ( γ ) / 2 − e − κ ℓ ( γ ) / 2 ) γ ( t ) = γ 0 ( k t ) ⟹ = e − k 2 ℓ ( γ 0 ) 2 / 2 t ℓ ( γ 0 ) 2 2 π t sinh k κ ℓ ( γ 0 ) / 2 \begin{aligned}
DM_*\mu(\Omega_t[\gamma]) &= \frac{e^{-\ell(\gamma)^2/2t}\ell(\gamma_0)}{\sqrt{2 \pi t}(e^{\kappa\ell(\gamma)/2} -e^{-\kappa\ell(\gamma)/2})}\\
\gamma(t) = \gamma_0(kt)\implies \\
&=\frac{e^{-k^2\ell(\gamma_0)^2/2t}\ell(\gamma_0)}{2\sqrt{2\pi t}\sinh k\kappa\ell(\gamma_0)/2}\\
\end{aligned} D M ∗ μ ( Ω t [ γ ]) γ ( t ) = γ 0 ( k t ) ⟹ = 2 π t ( e κ ℓ ( γ ) /2 − e − κ ℓ ( γ ) /2 ) e − ℓ ( γ ) 2 /2 t ℓ ( γ 0 ) = 2 2 π t sinh kκ ℓ ( γ 0 ) /2 e − k 2 ℓ ( γ 0 ) 2 /2 t ℓ ( γ 0 )
I dim = 3 \dim=3 dim = 3 ( z , z ˉ ) (z,\bar{z}) ( z , z ˉ )
J D M ( x ⃗ + ( τ + ℓ ( γ ) ) e ⃗ 1 ) = ( e κ ℓ ( γ ) 0 0 0 e − κ ℓ ( γ ) + i θ ( γ ) 0 0 0 e − κ ℓ ( γ ) − i θ ( γ ) ) ⟹ det I − ⊥ γ 0 k = ∣ 1 − e − k ( κ ℓ ( γ 0 ) − i θ ( γ 0 ) ) ∣ 2 \begin{aligned}
J_{DM(\vec{x}+(\tau+\ell(\gamma))\vec{e}^1)} &=
\begin{pmatrix}
e^{\kappa\ell(\gamma)} && 0 && 0\\
0 && e^{-\kappa\ell(\gamma)+i\theta(\gamma)} && 0 \\
0 && 0 && e^{-\kappa\ell(\gamma)-i\theta(\gamma)} \\
\end{pmatrix}\\
\implies& \\
\det I-{\perp_{\gamma_0}}^k &= |1-e^{-k(\kappa\ell(\gamma_0)-i\theta(\gamma_0))}|^2
\end{aligned} J D M ( x + ( τ + ℓ ( γ )) e 1 ) ⟹ det I − ⊥ γ 0 k = e κ ℓ ( γ ) 0 0 0 e − κ ℓ ( γ ) + i θ ( γ ) 0 0 0 e − κ ℓ ( γ ) − i θ ( γ ) = ∣1 − e − k ( κ ℓ ( γ 0 ) − i θ ( γ 0 )) ∣ 2
och sedan z = x 2 + i x 3 ⟹ d z ˉ ∧ d z = ( d x 2 − i d x 3 ) ∧ ( d x 2 + i d x 3 ) = 2 i d x 2 ∧ d x 3 z=x^2+ix^3 \implies d\bar{z}\wedge dz= (dx^2-idx^3)\wedge(dx^2+idx^3) = 2idx^2\wedge dx^3 z = x 2 + i x 3 ⟹ d z ˉ ∧ d z = ( d x 2 − i d x 3 ) ∧ ( d x 2 + i d x 3 ) = 2 i d x 2 ∧ d x 3
κ = 1 ⟹ D M ∗ μ ( Ω t [ γ ] ) = e − k 2 ℓ ( γ 0 ) 2 / 2 t ℓ ( γ 0 ) 2 2 π t ( 1 − e − k ℓ ( γ 0 ) ) ∣ e k ℓ ( γ 0 ) / 2 − e − k ( ℓ ( γ 0 ) / 2 − i θ ( γ 0 ) ) ∣ \begin{aligned}
\kappa &= 1 \implies \\
DM_*\mu(\Omega_t[\gamma])
&=\frac{e^{-k^2\ell(\gamma_0)^2/2t}\ell(\gamma_0)}{2\sqrt{2\pi t (1-e^{-k\ell(\gamma_0)})}|e^{k\ell(\gamma_0)/2}-e^{-k(\ell(\gamma_0)/2-i\theta(\gamma_0))}|}\\
\end{aligned} κ D M ∗ μ ( Ω t [ γ ]) = 1 ⟹ = 2 2 π t ( 1 − e − k ℓ ( γ 0 ) ) ∣ e k ℓ ( γ 0 ) /2 − e − k ( ℓ ( γ 0 ) /2 − i θ ( γ 0 )) ∣ e − k 2 ℓ ( γ 0 ) 2 /2 t ℓ ( γ 0 )
Joe Schaefer
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