Ricci Flow for Warped Product Manifolds Kartik Prabhu (05PH2001) - PowerPoint PPT Presentation

Ricci Flow for Warped Product Manifolds Kartik Prabhu (05PH2001) Supervisor: Dr. Sayan Kar Dept. of Physics & Meteorology Indian Institute of Technology Kharagpur May 04, 2010 May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 1 / 24

Outline 1 Ricci flow: definition & motivation RF as a heat flow Example: Sphere 2 Warped manifolds Separable solution Scaling solution 3 RG flow Separable solution 4 Conclusion 5 Possible Further Work 6 References 7 Q & A May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 2 / 24

Ricci flow: definition & motivation Ricci Flow: definition • Ricci flow is a geometric flow defined on a manifold M with a metric g ij . It deforms the metric along a parameter λ according to the differential equation– ∂ g ij ∂λ = − 2 R ij (1) • To preserve the volume the Ricci flow can be normalized to give – ∂λ = − 2 R ij + 2 ∂ g ij n � R � g ij (2) May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 3 / 24

Ricci flow: definition & motivation Ricci Flow: definition • Ricci flow is a geometric flow defined on a manifold M with a metric g ij . It deforms the metric along a parameter λ according to the differential equation– ∂ g ij ∂λ = − 2 R ij (1) • To preserve the volume the Ricci flow can be normalized to give – ∂λ = − 2 R ij + 2 ∂ g ij n � R � g ij (2) May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 3 / 24

Ricci flow: definition & motivation Ricci Flow: motivation • RF is like the heat equation and tends to smooth out the irregularities in the metric. • Finding the best metric on a manifold to solve mathematical problems like the Poincare Conjecture. (Perelman [4] ) • Renormalization Group flow in non-linear σ -model of string theory leads to Ricci flow in the lowest order. May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 4 / 24

Ricci flow: definition & motivation Ricci Flow: motivation • RF is like the heat equation and tends to smooth out the irregularities in the metric. • Finding the best metric on a manifold to solve mathematical problems like the Poincare Conjecture. (Perelman [4] ) • Renormalization Group flow in non-linear σ -model of string theory leads to Ricci flow in the lowest order. May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 4 / 24

Ricci flow: definition & motivation Ricci Flow: motivation • RF is like the heat equation and tends to smooth out the irregularities in the metric. • Finding the best metric on a manifold to solve mathematical problems like the Poincare Conjecture. (Perelman [4] ) • Renormalization Group flow in non-linear σ -model of string theory leads to Ricci flow in the lowest order. May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 4 / 24

Ricci flow: definition & motivation RF as a heat flow RF as heat flow Conformally flat 2-d manifold – ds 2 = e 2 φ ( x , y ) � dx 2 + dy 2 � with Ricci curvature – ∂ 2 x φ + ∂ 2 � � R xx = R yy = − (3) y φ and the Ricci flow Eq.(1) becomes ∂φ ∂λ = △ φ (4) where △ = e − 2 φ � ∂ 2 x + ∂ 2 � is generalized Laplacian. y RF is like a generalized non-linear heat/diffusion equation. May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 5 / 24

Ricci flow: definition & motivation RF as a heat flow RF as heat flow Conformally flat 2-d manifold – ds 2 = e 2 φ ( x , y ) � dx 2 + dy 2 � with Ricci curvature – ∂ 2 x φ + ∂ 2 � � R xx = R yy = − (3) y φ and the Ricci flow Eq.(1) becomes ∂φ ∂λ = △ φ (4) where △ = e − 2 φ � ∂ 2 x + ∂ 2 � is generalized Laplacian. y RF is like a generalized non-linear heat/diffusion equation. May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 5 / 24

Ricci flow: definition & motivation RF as a heat flow RF as heat flow Conformally flat 2-d manifold – ds 2 = e 2 φ ( x , y ) � dx 2 + dy 2 � with Ricci curvature – ∂ 2 x φ + ∂ 2 � � R xx = R yy = − (3) y φ and the Ricci flow Eq.(1) becomes ∂φ ∂λ = △ φ (4) where △ = e − 2 φ � ∂ 2 x + ∂ 2 � is generalized Laplacian. y RF is like a generalized non-linear heat/diffusion equation. May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 5 / 24

Ricci flow: definition & motivation Example: Sphere Sphere • metric: ds 2 = r 2 ( λ ) d θ 2 + sin 2 θ d φ 2 � � • RF: r ∼ ( λ 0 − λ ) 1 / 2 NRF: r = constant m ( θ, φ ) then r 1 ∼ e − l ( l +1) λ • but if r = r 0 ( λ ) + r 1 with r 1 = Y l May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 6 / 24

Ricci flow: definition & motivation Example: Sphere Sphere • metric: ds 2 = r 2 ( λ ) d θ 2 + sin 2 θ d φ 2 � � • RF: r ∼ ( λ 0 − λ ) 1 / 2 NRF: r = constant m ( θ, φ ) then r 1 ∼ e − l ( l +1) λ • but if r = r 0 ( λ ) + r 1 with r 1 = Y l May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 6 / 24

Warped manifolds Warped manifolds • Extra-dimensional brane world metric ds 2 = e 2 f ( σ,λ ) η µν dx µ dx ν + r 2 c ( σ, λ ) d σ 2 (5) • The RF is now a system of PDEs – f ′′ + 4 f ′ 2 − f ′ r ′ f = 1 � � ˙ c (6) r 2 r c c f ′′ + f ′ 2 − f ′ r ′ r c = 4 � � c ˙ (7) r c r c May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 7 / 24

Warped manifolds Warped manifolds • Extra-dimensional brane world metric ds 2 = e 2 f ( σ,λ ) η µν dx µ dx ν + r 2 c ( σ, λ ) d σ 2 (5) • The RF is now a system of PDEs – f ′′ + 4 f ′ 2 − f ′ r ′ f = 1 � � ˙ c (6) r 2 r c c f ′′ + f ′ 2 − f ′ r ′ r c = 4 � � c ˙ (7) r c r c May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 7 / 24

Warped manifolds Separable solution Separable solution • Assume separable functions – r c ( σ, λ ) = r c ( λ ) f ( σ, λ ) = f σ ( σ ) + f λ ( λ ) • The equations become separable and the solution becomes – � � � � � η µν dx µ dx ν + d σ 2 � ds 2 = 1 + λ σ exp ± √ 2 λ c λ c • Curvature scalar R = − 5 / 2 λ + λ c . Flow becomes singular at λ = − λ c . May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 8 / 24

Warped manifolds Separable solution Separable solution • Assume separable functions – r c ( σ, λ ) = r c ( λ ) f ( σ, λ ) = f σ ( σ ) + f λ ( λ ) • The equations become separable and the solution becomes – � � � � � η µν dx µ dx ν + d σ 2 � ds 2 = 1 + λ σ exp ± √ 2 λ c λ c • Curvature scalar R = − 5 / 2 λ + λ c . Flow becomes singular at λ = − λ c . May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 8 / 24

Warped manifolds Separable solution Separable solution • Assume separable functions – r c ( σ, λ ) = r c ( λ ) f ( σ, λ ) = f σ ( σ ) + f λ ( λ ) • The equations become separable and the solution becomes – � � � � � η µν dx µ dx ν + d σ 2 � ds 2 = 1 + λ σ exp ± √ 2 λ c λ c • Curvature scalar R = − 5 / 2 λ + λ c . Flow becomes singular at λ = − λ c . May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 8 / 24

Warped manifolds Scaling solution Scaling solution • Invariance under σ → ασ and λ → α 2 λ . So use variable x = 1 2 ln λ σ 2 and convert to ODE. • Also use B = e x r c df dx = A (8) dA A 2 B 2 − A − 24 A 3 B 2 dx = (9) dB dx = − 4 AB + B + 24 A 2 B 3 (10) May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 9 / 24

Warped manifolds Scaling solution Scaling solution • Invariance under σ → ασ and λ → α 2 λ . So use variable x = 1 2 ln λ σ 2 and convert to ODE. • Also use B = e x r c df dx = A (8) dA A 2 B 2 − A − 24 A 3 B 2 dx = (9) dB dx = − 4 AB + B + 24 A 2 B 3 (10) May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 9 / 24

Warped manifolds Scaling solution Scaling solution: plot � f � 0 � � 1, A � 0 � � 1, B � 0 � � 1 � � f � 0 � � 1, A � 0 � � 1, B � 0 � � 1 � 1.0 1.04 0.8 1.03 0.6 A f 1.02 0.4 1.01 0.2 1.00 0.0 0 1 2 3 4 5 0 1 2 3 4 5 x x (a) f v/s x (b) A v/s x May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 10 / 24

Warped manifolds Scaling solution Scaling solution: plot � f � 0 � � 1, A � 0 � � 1, B � 0 � � 1 � � f � 0 � � 1, A � 0 � � 1, B � 0 � � 1 � 8 1.0 0.8 6 0.6 Σ 2 R 4 r 0.4 2 0.2 0.0 0 0 1 2 3 4 5 0 1 2 3 4 5 x x (d) σ 2 R v/s x (c) r c v/s x May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 11 / 24

Warped manifolds Scaling solution Scaling solution: singularities 5 4 3 r c � 1 � 2 1 0 0 1 2 3 4 5 A � 1 � Figure: phase diagram showing non-singular and singular flows in space of ( A (0), r c (0)) May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 12 / 24

RG flow RG flow • Behaviour of non-linear σ -models under renormalization given by – ∂ g ij ∂λ = − β ij (11) • Perturbative expansion of β in terms of α ′ the inverse string tension � α ′ 5 � β ij = α ′ β (1) + α ′ 2 β (2) + α ′ 3 β (3) + α ′ 4 β (4) + O (12) ij ij ij ij May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 13 / 24

RG flow RG flow • Behaviour of non-linear σ -models under renormalization given by – ∂ g ij ∂λ = − β ij (11) • Perturbative expansion of β in terms of α ′ the inverse string tension � α ′ 5 � β ij = α ′ β (1) + α ′ 2 β (2) + α ′ 3 β (3) + α ′ 4 β (4) + O (12) ij ij ij ij May 04, 2010 Kartik Prabhu (IIT-Kgp) Ricci Flow 13 / 24