Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Smooth Metric Measure Spaces and Ricci Introduction Solitons Comparison Geometry for Bakry-Emery Ricci Tensor Guofang Wei Applications to Ricci Solitons UCSB, Santa Barbara Hsinchu, Taiwan, 6/9/2012
Smooth Metric Measure Spaces Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei A smooth metric measure space is triple ( M n , g , e − f dvol g ), Introduction where ( M n , g ) is a Riemannian manifolds with metric g , Comparison Geometry for f is a smooth real valued function on M . Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Smooth Metric Measure Spaces Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei A smooth metric measure space is triple ( M n , g , e − f dvol g ), Introduction where ( M n , g ) is a Riemannian manifolds with metric g , Comparison Geometry for f is a smooth real valued function on M . Bakry-Emery Ricci Tensor Applications Namely a Riemannian manifold with a conformal change in the to Ricci Solitons measure
Motivation It occurs naturally as collapsed measured Gromov-Hausdorff Smooth Metric limit. Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Motivation It occurs naturally as collapsed measured Gromov-Hausdorff Smooth Metric limit. Measure Spaces and Ricci Solitons Let ( M n × F m , g ǫ ) be equipped with warped product metric Guofang Wei g ǫ = g M + ( ǫ e − f ) 2 g F . Then, as ǫ → 0, Introduction Comparison ( M n × F m , � dvol g ǫ ) mGH → ( M n , e − mf dvol g M ) . Geometry for − Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Motivation It occurs naturally as collapsed measured Gromov-Hausdorff Smooth Metric limit. Measure Spaces and Ricci Solitons Let ( M n × F m , g ǫ ) be equipped with warped product metric Guofang Wei g ǫ = g M + ( ǫ e − f ) 2 g F . Then, as ǫ → 0, Introduction Comparison ( M n × F m , � dvol g ǫ ) mGH → ( M n , e − mf dvol g M ) . Geometry for − Bakry-Emery Ricci Tensor Applications to Ricci Here � dvol g ǫ is a renormalized Riemannian measure. Solitons
Motivation It occurs naturally as collapsed measured Gromov-Hausdorff Smooth Metric limit. Measure Spaces and Ricci Solitons Let ( M n × F m , g ǫ ) be equipped with warped product metric Guofang Wei g ǫ = g M + ( ǫ e − f ) 2 g F . Then, as ǫ → 0, Introduction Comparison ( M n × F m , � dvol g ǫ ) mGH → ( M n , e − mf dvol g M ) . Geometry for − Bakry-Emery Ricci Tensor Applications to Ricci Here � dvol g ǫ is a renormalized Riemannian measure. Solitons Recall ( X i , µ i ) mGH − → ( X ∞ , µ ∞ ) (compact) if for all sequences of continuous functions f i : X i → R converging to f ∞ : X ∞ → R , we have � � f i d µ i → f ∞ d µ ∞ . X i X ∞
Motivation Smooth Metric Measure Spaces and We have, as ǫ → 0, Ricci Solitons Guofang Wei ( M n × F m , � dvol g ǫ ) mGH → ( M n , e − f dvol g M ) , − Introduction where g ǫ = g M + ( ǫ e − f Comparison m ) 2 g F . Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Motivation Smooth Metric Measure Spaces and We have, as ǫ → 0, Ricci Solitons Guofang Wei ( M n × F m , � dvol g ǫ ) mGH → ( M n , e − f dvol g M ) , − Introduction where g ǫ = g M + ( ǫ e − f Comparison m ) 2 g F . Geometry for Bakry-Emery Ricci Tensor By O’Neill’s formula, the Ricci curvature of the warped product Applications to Ricci metric g ǫ in the M direction is Solitons Ric M + Hess f − 1 mdf ⊗ df .
m -Bakry-Emery Ricci tensor Smooth Metric Measure Therefore for smooth metric measure spaces ( M n , g , e − f dvol g ), Spaces and Ricci Solitons the corresponding Ricci tensor is Guofang Wei f = Ric + Hess f − 1 Introduction Ric m mdf ⊗ df for 0 < m ≤ ∞ , Comparison Geometry for Bakry-Emery — the m -Bakry-Emery Ricci tensor. Ricci Tensor Applications to Ricci Solitons
m -Bakry-Emery Ricci tensor Smooth Metric Measure Therefore for smooth metric measure spaces ( M n , g , e − f dvol g ), Spaces and Ricci Solitons the corresponding Ricci tensor is Guofang Wei f = Ric + Hess f − 1 Introduction Ric m mdf ⊗ df for 0 < m ≤ ∞ , Comparison Geometry for Bakry-Emery — the m -Bakry-Emery Ricci tensor. Ricci Tensor Applications to Ricci When m = ∞ , denote Ric f = Ric ∞ = Ric + Hess f Solitons f
m -Bakry-Emery Ricci tensor Smooth Metric Measure Therefore for smooth metric measure spaces ( M n , g , e − f dvol g ), Spaces and Ricci Solitons the corresponding Ricci tensor is Guofang Wei f = Ric + Hess f − 1 Introduction Ric m mdf ⊗ df for 0 < m ≤ ∞ , Comparison Geometry for Bakry-Emery — the m -Bakry-Emery Ricci tensor. Ricci Tensor Applications to Ricci When m = ∞ , denote Ric f = Ric ∞ = Ric + Hess f Solitons f If m 1 ≥ m 2 , then Ric m 1 ≥ Ric m 2 f . f So Ric m f ≥ λ g implies Ric f ≥ λ g .
More Motivations Smooth Ric m f = Ric when f is constant Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
More Motivations Smooth Ric m f = Ric when f is constant Metric Measure The quasi-Einstein equation Spaces and Ricci Solitons f = Ric + Hess f − 1 Ric m mdf ⊗ df = λ g (1) Guofang Wei Introduction has very nice geometric interpretations: Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
More Motivations Smooth Ric m f = Ric when f is constant Metric Measure The quasi-Einstein equation Spaces and Ricci Solitons f = Ric + Hess f − 1 Ric m mdf ⊗ df = λ g (1) Guofang Wei Introduction has very nice geometric interpretations: Comparison when m = ∞ , (1) is exactly the gradient Ricci soliton Geometry for Bakry-Emery equation. Ricci Tensor Applications to Ricci Solitons
More Motivations Smooth Ric m f = Ric when f is constant Metric Measure The quasi-Einstein equation Spaces and Ricci Solitons f = Ric + Hess f − 1 Ric m mdf ⊗ df = λ g (1) Guofang Wei Introduction has very nice geometric interpretations: Comparison when m = ∞ , (1) is exactly the gradient Ricci soliton Geometry for Bakry-Emery equation. Ricci Tensor when m is a positive integer, (1) ⇔ the warped product Applications to Ricci m F m is Einstein for some F m . metric M × e − f Solitons (Case-Shu-Wei using D.S.Kim-Y.S. Kim’s work)
More Motivations Smooth Ric m f = Ric when f is constant Metric Measure The quasi-Einstein equation Spaces and Ricci Solitons f = Ric + Hess f − 1 Ric m mdf ⊗ df = λ g (1) Guofang Wei Introduction has very nice geometric interpretations: Comparison when m = ∞ , (1) is exactly the gradient Ricci soliton Geometry for Bakry-Emery equation. Ricci Tensor when m is a positive integer, (1) ⇔ the warped product Applications to Ricci m F m is Einstein for some F m . metric M × e − f Solitons (Case-Shu-Wei using D.S.Kim-Y.S. Kim’s work) Corresponding versions for non-smooth metric measure spaces (Lott-Villani, Sturm)
More Motivations Smooth Ric m f = Ric when f is constant Metric Measure The quasi-Einstein equation Spaces and Ricci Solitons f = Ric + Hess f − 1 Ric m mdf ⊗ df = λ g (1) Guofang Wei Introduction has very nice geometric interpretations: Comparison when m = ∞ , (1) is exactly the gradient Ricci soliton Geometry for Bakry-Emery equation. Ricci Tensor when m is a positive integer, (1) ⇔ the warped product Applications to Ricci m F m is Einstein for some F m . metric M × e − f Solitons (Case-Shu-Wei using D.S.Kim-Y.S. Kim’s work) Corresponding versions for non-smooth metric measure spaces (Lott-Villani, Sturm) diffusion processes Sobolev inequality conformal geometry, Chang- Gursky-Yang 2006
Question Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Question Introduction Comparison What geometric and topological results for the Ricci tensor Geometry for Bakry-Emery extend to the Bakry-Emery Ricci tensor? Ricci Tensor Applications to Ricci Solitons
Question Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Question Introduction Comparison What geometric and topological results for the Ricci tensor Geometry for Bakry-Emery extend to the Bakry-Emery Ricci tensor? Ricci Tensor Applications When 0 < m < ∞ , many geometry and topology results for to Ricci Solitons Ricci curvature lower bound extend directly to Ric m f .
Examples Smooth Metric What about m = ∞ ? Measure Spaces and Ricci Solitons Example Guofang Wei H n the hyperbolic space. Fixed any p ∈ H n , let Introduction f ( x ) = ( n − 1) d 2 ( p , x ) , then Ric f ≥ ( n − 1) . Comparison Geometry for Myers’ theorem and Cheeger-Gromoll’s isometric splitting Bakry-Emery Ricci Tensor theorem do not hold for Ric f . Applications to Ricci Solitons
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