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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
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Guofang Wei
UCSB, Santa Barbara
Hsinchu, Taiwan, 6/9/2012
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
A smooth metric measure space is triple (Mn, g, e−f dvolg), where (Mn, g) is a Riemannian manifolds with metric g, f is a smooth real valued function on M.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
A smooth metric measure space is triple (Mn, g, e−f dvolg), where (Mn, g) is a Riemannian manifolds with metric g, f is a smooth real valued function on M. Namely a Riemannian manifold with a conformal change in the measure
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
It occurs naturally as collapsed measured Gromov-Hausdorff limit.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
It occurs naturally as collapsed measured Gromov-Hausdorff limit. Let (Mn × F m, gǫ) be equipped with warped product metric gǫ = gM + (ǫe−f )2gF. Then, as ǫ → 0, (Mn × F m, dvolgǫ) mGH − → (Mn, e−mf dvolgM).
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
It occurs naturally as collapsed measured Gromov-Hausdorff limit. Let (Mn × F m, gǫ) be equipped with warped product metric gǫ = gM + (ǫe−f )2gF. Then, as ǫ → 0, (Mn × F m, dvolgǫ) mGH − → (Mn, e−mf dvolgM). Here dvolgǫ is a renormalized Riemannian measure.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
It occurs naturally as collapsed measured Gromov-Hausdorff limit. Let (Mn × F m, gǫ) be equipped with warped product metric gǫ = gM + (ǫe−f )2gF. Then, as ǫ → 0, (Mn × F m, dvolgǫ) mGH − → (Mn, e−mf dvolgM). Here dvolgǫ is a renormalized Riemannian measure. Recall (Xi, µi) mGH − → (X∞, µ∞) (compact) if for all sequences of continuous functions fi : Xi → R converging to f∞ : X∞ → R, we have
fidµi →
f∞dµ∞.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
We have, as ǫ → 0, (Mn × F m, dvolgǫ) mGH − → (Mn, e−f dvolgM), where gǫ = gM + (ǫe− f
m )2gF.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
We have, as ǫ → 0, (Mn × F m, dvolgǫ) mGH − → (Mn, e−f dvolgM), where gǫ = gM + (ǫe− f
m )2gF.
By O’Neill’s formula, the Ricci curvature of the warped product metric gǫ in the M direction is RicM + Hessf − 1 mdf ⊗ df .
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Therefore for smooth metric measure spaces (Mn, g, e−f dvolg), the corresponding Ricci tensor is Ricm
f = Ric + Hessf − 1
mdf ⊗ df for 0 < m ≤ ∞, — the m-Bakry-Emery Ricci tensor.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Therefore for smooth metric measure spaces (Mn, g, e−f dvolg), the corresponding Ricci tensor is Ricm
f = Ric + Hessf − 1
mdf ⊗ df for 0 < m ≤ ∞, — the m-Bakry-Emery Ricci tensor. When m = ∞, denote Ricf = Ric∞
f
= Ric + Hessf
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Therefore for smooth metric measure spaces (Mn, g, e−f dvolg), the corresponding Ricci tensor is Ricm
f = Ric + Hessf − 1
mdf ⊗ df for 0 < m ≤ ∞, — the m-Bakry-Emery Ricci tensor. When m = ∞, denote Ricf = Ric∞
f
= Ric + Hessf If m1 ≥ m2, then Ricm1
f
≥ Ricm2
f .
So Ricm
f ≥ λg implies Ricf ≥ λg.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricm
f = Ric when f is constant
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricm
f = Ric when f is constant
The quasi-Einstein equation Ricm
f = Ric + Hessf − 1
mdf ⊗ df = λg (1) has very nice geometric interpretations:
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricm
f = Ric when f is constant
The quasi-Einstein equation Ricm
f = Ric + Hessf − 1
mdf ⊗ df = λg (1) has very nice geometric interpretations: when m = ∞, (1) is exactly the gradient Ricci soliton equation.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricm
f = Ric when f is constant
The quasi-Einstein equation Ricm
f = Ric + Hessf − 1
mdf ⊗ df = λg (1) has very nice geometric interpretations: when m = ∞, (1) is exactly the gradient Ricci soliton equation. when m is a positive integer, (1) ⇔ the warped product metric M ×e− f
m F m is Einstein for some F m.
(Case-Shu-Wei using D.S.Kim-Y.S. Kim’s work)
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricm
f = Ric when f is constant
The quasi-Einstein equation Ricm
f = Ric + Hessf − 1
mdf ⊗ df = λg (1) has very nice geometric interpretations: when m = ∞, (1) is exactly the gradient Ricci soliton equation. when m is a positive integer, (1) ⇔ the warped product metric M ×e− f
m F m is Einstein for some F m.
(Case-Shu-Wei using D.S.Kim-Y.S. Kim’s work) Corresponding versions for non-smooth metric measure spaces (Lott-Villani, Sturm)
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricm
f = Ric when f is constant
The quasi-Einstein equation Ricm
f = Ric + Hessf − 1
mdf ⊗ df = λg (1) has very nice geometric interpretations: when m = ∞, (1) is exactly the gradient Ricci soliton equation. when m is a positive integer, (1) ⇔ the warped product metric M ×e− f
m F m is Einstein for some F m.
(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
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Question What geometric and topological results for the Ricci tensor extend to the Bakry-Emery Ricci tensor?
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Question What geometric and topological results for the Ricci tensor extend to the Bakry-Emery Ricci tensor? When 0 < m < ∞, many geometry and topology results for Ricci curvature lower bound extend directly to Ricm
f .
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
What about m = ∞? Example Hn the hyperbolic space. Fixed any p ∈ Hn, let f (x) = (n − 1)d2(p, x), then Ricf ≥ (n − 1). Myers’ theorem and Cheeger-Gromoll’s isometric splitting theorem do not hold for Ricf .
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
What about m = ∞? Example Hn the hyperbolic space. Fixed any p ∈ Hn, let f (x) = (n − 1)d2(p, x), then Ricf ≥ (n − 1). Myers’ theorem and Cheeger-Gromoll’s isometric splitting theorem do not hold for Ricf . Example Rn with Euclidean metric, f (x1, · · · , xn) = x1. Ricf = Ric = 0. volf (B(0, r)) =
Bishop-Gromov’s volume comparison doesn’t extend.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Let ∆f u = ∆u − ∇u, ∇f . Theorem (Wei-Wylie2009, JDG) If Ricf ≥ 0 then ∆f (r) ≤ n − 1 r + 2 r2 r (f (t) − f (r))dt. In particular, when |f | ≤ k then ∆f (r) ≤ n + 4k − 1 r .
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Let volf (B(p, r)) =
Theorem (Wei-Wylie2009) Fix p ∈ (Mn, g, e−f dvolg). Assume Ricf ≥ 0, if |f (x)| ≤ k then for R ≥ r > 0, volf (B(p, R)) volf (B(p, r)) ≤ R r n+4k . Namely M has polynomial f -volume growth.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Let volf (B(p, r)) =
Theorem (Wei-Wylie2009) Fix p ∈ (Mn, g, e−f dvolg). Assume Ricf ≥ 0, if |f (x)| ≤ k then for R ≥ r > 0, volf (B(p, R)) volf (B(p, r)) ≤ R r n+4k . Namely M has polynomial f -volume growth. In fact, Ning Yang, 2009 observed that the degree is ≤ n.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
(Mn, g, f ) is a gradient Ricci soliton if Ricf = Ric + Hessf = λg. (2)
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
(Mn, g, f ) is a gradient Ricci soliton if Ricf = Ric + Hessf = λg. (2) λ > 0, shrinking λ = 0, steady λ < 0, expanding
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
(Mn, g, f ) is a gradient Ricci soliton if Ricf = Ric + Hessf = λg. (2) λ > 0, shrinking λ = 0, steady λ < 0, expanding Self-similar solutions of Ricci flow, singularity models
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricci flat manifolds
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricci flat manifolds (Rn, g0, f (x) = x1)
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricci flat manifolds (Rn, g0, f (x) = x1) Cigar Soliton (Hamilton, 1988): (R2, g = dx2+dy2
1+x2+y2 , f = − ln(1 + x2 + y2))
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricci flat manifolds (Rn, g0, f (x) = x1) Cigar Soliton (Hamilton, 1988): (R2, g = dx2+dy2
1+x2+y2 , f = − ln(1 + x2 + y2))
Bryant Solitons: Rn(n ≥ 3), SO(n) symmetry
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricci flat manifolds (Rn, g0, f (x) = x1) Cigar Soliton (Hamilton, 1988): (R2, g = dx2+dy2
1+x2+y2 , f = − ln(1 + x2 + y2))
Bryant Solitons: Rn(n ≥ 3), SO(n) symmetry H.D. Cao (1996), Cn, U(n) symmetry, Kahler
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricci flat manifolds (Rn, g0, f (x) = x1) Cigar Soliton (Hamilton, 1988): (R2, g = dx2+dy2
1+x2+y2 , f = − ln(1 + x2 + y2))
Bryant Solitons: Rn(n ≥ 3), SO(n) symmetry H.D. Cao (1996), Cn, U(n) symmetry, Kahler Ivey(1994), Dancer-Wang (2011), double and multiple warped product
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Ricci flat manifolds (Rn, g0, f (x) = x1) Cigar Soliton (Hamilton, 1988): (R2, g = dx2+dy2
1+x2+y2 , f = − ln(1 + x2 + y2))
Bryant Solitons: Rn(n ≥ 3), SO(n) symmetry H.D. Cao (1996), Cn, U(n) symmetry, Kahler Ivey(1994), Dancer-Wang (2011), double and multiple warped product Any Riemannian product of steady solitons
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
f has no local minimum unless f is constant In particular, all compact steady Ricci solitons are Einstein
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
f has no local minimum unless f is constant In particular, all compact steady Ricci solitons are Einstein Cao-Chen (2009) Non-compact, locally conformally flat ⇒ flat or Bryant soliton
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
f has no local minimum unless f is constant In particular, all compact steady Ricci solitons are Einstein Cao-Chen (2009) Non-compact, locally conformally flat ⇒ flat or Bryant soliton
infinity, then it’s the Bryant soliton. In particular, all noncollapsing, not flat 3-dim ones are Bryant solitons.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Hamilton (1995) showed that R + |∇f |2 = Λ, where Λ is a constant, and R is the scalar curvature.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Hamilton (1995) showed that R + |∇f |2 = Λ, where Λ is a constant, and R is the scalar curvature. Combine with trace soliton equation R + ∆f = 0, we have ∆f f = ∆f − |∇f |2= −Λ.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Hamilton (1995) showed that R + |∇f |2 = Λ, where Λ is a constant, and R is the scalar curvature. Combine with trace soliton equation R + ∆f = 0, we have ∆f f = ∆f − |∇f |2= −Λ.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Hamilton (1995) showed that R + |∇f |2 = Λ, where Λ is a constant, and R is the scalar curvature. Combine with trace soliton equation R + ∆f = 0, we have ∆f f = ∆f − |∇f |2= −Λ.
⇒ |∇f |2 ≤ Λ and f decays at most linearly.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Question Does f always decays linearly? For shrinking Ricci soliotn, Cao-Zhou (2010) showed λ 2 (r − c1)2 ≤ f (r) ≤ λ 2 (r + c2)2 volB(x, r) ≤ crn.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Question Does f always decays linearly? For shrinking Ricci soliotn, Cao-Zhou (2010) showed λ 2 (r − c1)2 ≤ f (r) ≤ λ 2 (r + c2)2 volB(x, r) ≤ crn. Cao-Chen (2009) If Ric > 0 and R attains its maximum at some point, then − √ Λr ≤ f (r) ≤ −c1r + c2, 0 < c1 ≤ √ Λ.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
In general, the answer is “No”. R× steady soliton is a counterexample!
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
In general, the answer is “No”. R× steady soliton is a counterexample! Unlike shrinking one, f could behave very differently in different direction for steady ones. But the infimum does always decay linearly.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Theorem (Munteanu-Sesum, P. Wu) For a gradient Ricci soliton with R + |∇f |2 = 1, −r ≤ inf
y∈∂Br(x) f (y) − f (x) ≤ −r +
√ 2n(√r + 1), r ≫ 1. In particular, if f is sublinear, then f is constant, and liminfy→∞R(y) = 0. Also observed by Fernandez-Lopez and Garcia-Rio, Chow-Lu
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Among all known examples, the infimum of f is like −r + O(ln r).
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Among all known examples, the infimum of f is like −r + O(ln r). Fix x ∈ Mn, write f in polar coordinate, f (r, θ) = −r + φ(r, θ).
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Among all known examples, the infimum of f is like −r + O(ln r). Fix x ∈ Mn, write f in polar coordinate, f (r, θ) = −r + φ(r, θ). φ(r) ≥ 0 and φ(r, θ) is nondecreasing in r for any fixed θ.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Theorem (Wei-Wu) If there exists θ0 ∈ Sn−1, and constant C ≥ 0 such that r (φ(r, θ0) − φ(t, θ0)) dt ≤ C min
θ∈Sn−1
r (φ(r, θ) − φ(t, θ)) dt + Cr for sufficiently large r, then for r large, −r ≤ inf
y∈∂Br(x) f (y) − f (x) ≤ −r + C ′ ln r.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Theorem (Wei-Wu) If there exists θ0 ∈ Sn−1, and constant C ≥ 0 such that r (φ(r, θ0) − φ(t, θ0)) dt ≤ C min
θ∈Sn−1
r (φ(r, θ) − φ(t, θ)) dt + Cr for sufficiently large r, then for r large, −r ≤ inf
y∈∂Br(x) f (y) − f (x) ≤ −r + C ′ ln r.
In particular, the result is true when the minimum direction is the same for all r large. All know examples satisfy this.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
When φ(r, θ) is some what uniform in the sphere direction for all r large (as in the shrinking case), we also get a volume estimate. Theorem (Wei-Wu) If there exists a constant C such that for all r large, max
θ∈Sn−1
r φ(r, θ) − φ(t, θ)dt ≤ C min
θ∈Sn−1
r φ(r, θ) − φ(t, θ)dt + Cr, then Vol(Br(x)) ≤ C ′rn.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
When φ(r, θ) is some what uniform in the sphere direction for all r large (as in the shrinking case), we also get a volume estimate. Theorem (Wei-Wu) If there exists a constant C such that for all r large, max
θ∈Sn−1
r φ(r, θ) − φ(t, θ)dt ≤ C min
θ∈Sn−1
r φ(r, θ) − φ(t, θ)dt + Cr, then Vol(Br(x)) ≤ C ′rn. In particular, the condition holds when f is the distance function to a compact set. All known nonproduct examples satisfy this.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Use equation ∆f f = −1 and f -volume comparison to get min
y∈∂B(x,r)
r (φ(y) − φ(t))dt ≤ n 2(r + √r) + o(1 r ).
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Use equation ∆f f = −1 and f -volume comparison to get min
y∈∂B(x,r)
r (φ(y) − φ(t))dt ≤ n 2(r + √r) + o(1 r ). Use the conditions and an ODE
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
By integrating ∆f f = −1 with respect to e−f dvol and using |∇f | ≤ 1, we get volf (∂B(x, r)) volf (B(x, r)) ≥ 1.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
By integrating ∆f f = −1 with respect to e−f dvol and using |∇f | ≤ 1, we get volf (∂B(x, r)) volf (B(x, r)) ≥ 1. By adapting the f -volume comparison in Wei-Wylie to soliton, we get, for 0 < r1 ≤ r2, volf (∂B(x, r2)) volf (A(x, r1, r2)) ≤
n r2 + 1 − a
1 − (r1
r2 )n+(1−a)r2 ,
where a = min
y∈A(x,r1,r2)
2 r(y)2 r(y) (φ(y) − φ(t))dt.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Let r1 = r, r2 = r + √r, we get min
y∈∂B(x,r)
r (φ(y) − φ(t))dt ≤ n 2(r + √r) + o(1 r ).
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Let r1 = r, r2 = r + √r, we get min
y∈∂B(x,r)
r (φ(y) − φ(t))dt ≤ n 2(r + √r) + o(1 r ). Let Φ(r) = r
0 φ(t)dt. By assumption, in the direction of θ0,
Φ′(r) − 1 r Φ(r) ≤ C ′. and we get φ(r) ≤ C ′ ln r.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Write dvol = J(r, θ)drdθ. From the Laplacian comparison, (ln J)′ ≤ n − 1 r + 1 − 2 r φ(r) + 2 r2 r φ(t)dt + ∇f , ∇r.
Smooth Metric Measure Spaces and Ricci Solitons Guofang Wei Introduction Comparison Geometry for Bakry-Emery Ricci Tensor Applications to Ricci Solitons
Write dvol = J(r, θ)drdθ. From the Laplacian comparison, (ln J)′ ≤ n − 1 r + 1 − 2 r φ(r) + 2 r2 r φ(t)dt + ∇f , ∇r. ln J(r) J(1) ≤ (n−1) ln r−φ(r)+2
r r φ(t)dt
1 φ(t)dt Therefore Vol(B(x, r)) ≤ C ′rn.