Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces

Monotonicity and rigidity of the W -entropy on RCD(0, N ) spaces

Kazumasa Kuwada

(Tokyo Institute of Technology)

joint work with X.-D. Li (Chinese Academy of Science)

Midlands Probability Seminar in University of Warwick (2 Nov. 2016)

1. Introduction

Perelman’s W -entropy

(M, g): m-dim. cpt. Riem. mfd, τ > 0, f ∈ C^∞(M),

Z

M

e^−f

(4πτ)^m/2 d vol = 1 W(g, f, τ)

:=

Z

M

τ(R+|∇f|²) +f − m e^−f

(4πτ)^m/2 d vol

F (g(t), f(t), τ(t)): ∂_tτ = −1,

∂_tg = −2 Ric, ∂_tf = −∆f + |∇f|²− R + m 2τ

⇒ d

dtW(g, f, τ) ≥ 0

Perelman’s W -entropy

(M, g): m-dim. cpt. Riem. mfd, τ > 0, f ∈ C^∞(M),

Z

M

e^−f

(4πτ)^m/2 d vol = 1 W(g, f, τ)

:=

Z

M

τ(R+|∇f|²) +f − m e^−f

(4πτ)^m/2 d vol F (g(t), f(t), τ(t)): ∂_tτ = −1,

∂_tg = −2 Ric, ∂_tf = −∆f + |∇f|²− R + m 2τ

⇒ d

dtW(g, f, τ) ≥ 0

3 / 32

Perelman’s W -entropy

(M, g): m-dim. cpt. Riem. mfd, τ > 0, f ∈ C^∞(M),

Z

M

e^−f

(4πτ)^m/2 d vol = 1 W(g, f, τ)

:=

Z

M

τ(R+|∇f|²) +f − m e^−f

(4πτ)^m/2 d vol F (g(t), f(t), τ(t)): ∂_tτ = −1,

∂_tg = −2 Ric, ∂_tf = −∆f + |∇f|²− R + m 2τ

⇒ d

W(g, f, τ) ≥ 0

Entropy formula

d

dtW(g, f, τ) ≥ 0

⇑ d

dtW = 2 Z

M

τ

Ric +∇²f − g 2τ

2 e^−f

(4πτ)^m/2 d vol

F d

dtW = 0 ⇒ Ric +∇²f − g

2τ = 0

(gradient shrinking Ricci soliton) u := e^−f

(4πτ)^m/2 ⇒ ∂tu = −∆u+Ru

∂_τ vol = Rvol ⇒ ∂_τ(uvol) = ∆(uvol)

4 / 32

Entropy formula

d

dtW = 2 Z

M

τ

Ric +∇²f − g 2τ

2 e^−f

(4πτ)^m/2 d vol F d

dtW = 0 ⇒ Ric +∇²f − g

2τ = 0

(gradient shrinking Ricci soliton)

u := e^−f

(4πτ)^m/2 ⇒ ∂_tu = −∆u+Ru

∂_τ vol = Rvol ⇒ ∂_τ(uvol) = ∆(uvol)

Entropy formula

d

dtW = 2 Z

M

τ

Ric +∇²f − g 2τ

2 e^−f

(4πτ)^m/2 d vol F d

dtW = 0 ⇒ Ric +∇²f − g

2τ = 0

(gradient shrinking Ricci soliton) u := e^−f

(4πτ)^m/2 ⇒ ∂_tu = −∆u+Ru

∂_τ vol = Rvol ⇒ ∂_τ(uvol) = ∆(uvol)

4 / 32

Entropy formula

d

dtW = 2 Z

M

τ

Ric +∇²f − g 2τ

2 e^−f

(4πτ)^m/2 d vol F d

dtW = 0 ⇒ Ric +∇²f − g

2τ = 0

(gradient shrinking Ricci soliton) u := e^−f

(4πτ)^m/2 ⇒ ∂_tu = −∆u+Ru

∂_τ vol = Rvol ⇒ ∂_τ(uvol) = ∆(uvol)

W -entropy on Riem. mfd

(M, g): m-dim. cpt. Riem. mfd, τ > 0, f ∈ C^∞(M),

Z

M

e^−f

(4πτ)^m/2 d vol = 1 W(g, f, τ)

:=

Z

M

τ(R+|∇f|²) +f − m e^−f

(4πτ)^m/2 d vol

W(f, τ) := Z

τ|∇f|² +f − m e^−f

(4πτ)^m/2 d vol F Ric ≥ 0,

∂τf = ∆f − |∇f|² − m

2τ (or ∂τu = ∆u)

⇒ d

dτW(f, τ) ≤ 0

5 / 32

W -entropy on Riem. mfd

(M, g): m-dim. cpt. Riem. mfd, τ > 0, f ∈ C^∞(M),

Z

M

e^−f

(4πτ)^m/2 d vol = 1 W(g, f, τ)

:=

Z

M

τ(R+|∇f|²) +f − m e^−f

(4πτ)^m/2 d vol

W(f, τ) :=

Z

τ|∇f|² +f − m e^−f

(4πτ)^m/2 d vol

F Ric ≥ 0,

∂τf = ∆f − |∇f|² − m

2τ (or ∂τu = ∆u)

⇒ d

dτW(f, τ) ≤ 0

W -entropy on Riem. mfd

(M, g): m-dim. cpt. Riem. mfd, τ > 0, f ∈ C^∞(M),

Z

M

e^−f

(4πτ)^m/2 d vol = 1 W(f, τ) :=

Z

τ|∇f|² +f − m e^−f

(4πτ)^m/2 d vol F Ric ≥ 0,

∂_τf = ∆f − |∇f|² − m

2τ (or ∂_τu = ∆u)

⇒ d

dτW(f, τ) ≤ 0

5 / 32

W -entropy on Riem. mfd

(M, g): m-dim. cpt. Riem. mfd, τ > 0, f ∈ C^∞(M),

Z

M

e^−f

(4πτ)^m/2 d vol = 1 W(f, τ) :=

Z

τ|∇f|² +f − m e^−f

(4πτ)^m/2 d vol F Ric ≥ 0,

∂_τf = ∆f − |∇f|² − m

2τ (or ∂_τu = ∆u)

⇒ d

dτW(f, τ) ≤ 0

W -entropy on Riem. mfd

(M, g): m-dim. cpt. Riem. mfd, τ > 0, f ∈ C^∞(M),

Z

M

e^−f

(4πτ)^m/2 d vol = 1 W(f, τ) :=

Z

τ|∇f|² +f − m e^−f

(4πτ)^m/2 d vol F Ric ≥ 0,

∂_τf = ∆f − |∇f|² − m

2τ (or ∂_τu = ∆u)

⇒ d

dτW(f, τ) ≤ 0

5 / 32

Entropy formula and rigidity

d

dτW(f, τ) ≤ 0

⇑

d

dτW =−2 Z

M

τ

∇²f − g 2τ

2

+ Ric(∇f,∇f)

!

ud vol

[L. Ni ’04]

F Extension to M: cpl. Riem. mfd with “bdd geom.” Ric ≥ 0, u: heat kernel & d

dτW = 0

⇒ M ' RRR^m [Ni ’04]

Extension to weighted Riem. mfds [X.-D. Li ’12]

Entropy formula and rigidity

d

dτW =−2 Z

M

τ

∇²f − g 2τ

2

+ Ric(∇f,∇f)

!

ud vol

[L. Ni ’04]

F Extension to M: cpl. Riem. mfd with “bdd geom.”

Ric ≥ 0, u: heat kernel & d

dτW = 0

⇒ M ' RRR^m [Ni ’04]

Extension to weighted Riem. mfds [X.-D. Li ’12]

6 / 32

Entropy formula and rigidity

d

dτW =−2 Z

M

τ

∇²f − g 2τ

2

+ Ric(∇f,∇f)

!

ud vol

[L. Ni ’04]

F Extension to M: cpl. Riem. mfd with “bdd geom.”

Ric ≥ 0, u: heat kernel & d

dτW = 0

⇒ M ' RRR^m [Ni ’04]

Extension to weighted Riem. mfds [X.-D. Li ’12]

Entropy formula and rigidity

d

dτW =−2 Z

M

τ

∇²f − g 2τ

2

+ Ric(∇f,∇f)

!

ud vol

[L. Ni ’04]

F Extension to M: cpl. Riem. mfd with “bdd geom.”

Ric ≥ 0, u: heat kernel & d

dτW = 0

⇒ M ' RRR^m [Ni ’04]

Extension to weighted Riem. mfds [X.-D. Li ’12]

6 / 32

Purpose

Q.

Can one extend the monotonicity/rigidity of W on metric measure spaces with “Ric ≥ 0 & dim ≤ N” (RCD(0, N) spaces)?

A. Yes!

Weaken ass’n(s) even on (weighted) Riem. mfds Without the entropy formula

optimal transport approach

Singular sp.’s other than RRR^m appear in rigidity

Purpose

Q.

Can one extend the monotonicity/rigidity of W on metric measure spaces with “Ric ≥ 0 & dim ≤ N” (RCD(0, N) spaces)?

A.

Yes!

Weaken ass’n(s) even on (weighted) Riem. mfds Without the entropy formula

optimal transport approach

Singular sp.’s other than RRR^m appear in rigidity

7 / 32

Purpose

Q.

Can one extend the monotonicity/rigidity of W on metric measure spaces with “Ric ≥ 0 & dim ≤ N” (RCD(0, N) spaces)?

A.

Yes!

Weaken ass’n(s) even on (weighted) Riem. mfds

Without the entropy formula

optimal transport approach

Singular sp.’s other than RRR^m appear in rigidity

Purpose

Q.

Can one extend the monotonicity/rigidity of W on metric measure spaces with “Ric ≥ 0 & dim ≤ N” (RCD(0, N) spaces)?

A.

Yes!

Weaken ass’n(s) even on (weighted) Riem. mfds Without the entropy formula

optimal transport approach

Singular sp.’s other than RRR^m appear in rigidity

7 / 32

Purpose

Q.

Can one extend the monotonicity/rigidity of W on metric measure spaces with “Ric ≥ 0 & dim ≤ N” (RCD(0, N) spaces)?

A.

Yes!

Weaken ass’n(s) even on (weighted) Riem. mfds Without the entropy formula

optimal transport approach

Singular sp.’s other than RRR^m appear in rigidity

Purpose

Q.

Can one extend the monotonicity/rigidity of W on metric measure spaces with “Ric ≥ 0 & dim ≤ N” (RCD(0, N) spaces)?

A.

Yes!

Weaken ass’n(s) even on (weighted) Riem. mfds Without the entropy formula

optimal transport approach

Singular sp.’s other than RRR^m appear in rigidity

7 / 32

Outline of the talk 1. Introduction

2. Framework: RCD spaces 3. Main results

4. Proof

4.1 Monotonicity 4.2 Rigidity

4.3 Additional remarks

Outline of the talk

1. Introduction

2. Framework: RCD spaces 3. Main results

4. Proof

4.1 Monotonicity 4.2 Rigidity

4.3 Additional remarks

Met. meas. sp. & heat flow on it

(X, d,m): Polish geod. met. meas. sp.

(m: loc.-finite, suppm = X) P_t = e^t∆ ↔ Cheeger’s L²-energy

loc. Lip. const.

2Ch(f) := inf

lim

n

Z

X

|Df|²dm

f_n : Lip. f_n → f in L²

= Z

X

|Df|²_w dm Definition 1

(X, d,m): infinitesimally Hilbertian

⇔def Ch: quadratic form (⇔ P_t: linear ⇔ ∆: linear)

⇒ ^∃∃∃hD·, D·i_w bilinear s.t. hDf, Dfi_w = |Df|²_w

Met. meas. sp. & heat flow on it

(X, d,m): Polish geod. met. meas. sp.

(m: loc.-finite, suppm = X) P_t = e^t∆ ↔ Cheeger’s L²-energy

loc. Lip. const.

2Ch(f) := inf

lim

n

Z

X

|Df|²dm

f_n : Lip. f_n → f in L²

= Z

X

|Df|²_w dm Definition 1

(X, d,m): infinitesimally Hilbertian

⇔def Ch: quadratic form (⇔ P_t: linear ⇔ ∆: linear)

⇒ ^∃∃∃hD·, D·i_w bilinear s.t. hDf, Dfi_w = |Df|²_w

9 / 32

Met. meas. sp. & heat flow on it

(X, d,m): Polish geod. met. meas. sp.

(m: loc.-finite, suppm = X) P_t = e^t∆ ↔ Cheeger’s L²-energy

loc. Lip. const.

2Ch(f) := inf

lim

n

Z

X

lip(f_n)²dm

f_n : Lip.

f_n → f in L²

= Z

X

|Df|²_w dm Definition 1

(X, d,m): infinitesimally Hilbertian

⇔def Ch: quadratic form (⇔ P_t: linear ⇔ ∆: linear)

⇒ ^∃∃∃hD·, D·i_w bilinear s.t. hDf, Dfi_w = |Df|²_w

Met. meas. sp. & heat flow on it

(X, d,m): Polish geod. met. meas. sp.

(m: loc.-finite, suppm = X) P_t = e^t∆ ↔ Cheeger’s L²-energy

loc. Lip. const.

2Ch(f) := inf

lim

n

Z

X

lip(f_n)²dm

f_n : Lip.

f_n → f in L²

= Z

X

|Df|²_w dm Definition 1

(X, d,m): infinitesimally Hilbertian

⇔def Ch: quadratic form (⇔ P_t: linear ⇔ ∆: linear)

⇒ ^∃∃∃hD·, D·i_w bilinear s.t. hDf, Dfi_w = |Df|²_w

9 / 32

Met. meas. sp. & heat flow on it

(X, d,m): Polish geod. met. meas. sp.

(m: loc.-finite, suppm = X) P_t = e^t∆ ↔ Cheeger’s L²-energy

loc. Lip. const.

2Ch(f) := inf

lim

n

Z

X

lip(f_n)²dm

f_n : Lip.

f_n → f in L²

= Z

X

|Df|²_w dm Definition 1

(X, d,m): infinitesimally Hilbertian

⇔def Ch: quadratic form (⇔ P_t: linear ⇔ ∆: linear)

⇒ ^∃∃∃hD·, D·i_w bilinear s.t. hDf, Dfi_w = |Df|²_w

Met. meas. sp. & heat flow on it

(X, d,m): Polish geod. met. meas. sp.

(m: loc.-finite, suppm = X) P_t = e^t∆ ↔ Cheeger’s L²-energy

loc. Lip. const.

2Ch(f) := inf

lim

n

Z

X

lip(f_n)²dm

f_n : Lip.

f_n → f in L²

= Z

X

∃∃∃|Df|²_wdm

Definition 1

(X, d,m): infinitesimally Hilbertian

⇔def Ch: quadratic form (⇔ P_t: linear ⇔ ∆: linear)

⇒ ^∃∃∃hD·, D·i_w bilinear s.t. hDf, Dfi_w = |Df|²_w

9 / 32

Met. meas. sp. & heat flow on it

(X, d,m): Polish geod. met. meas. sp.

(m: loc.-finite, suppm = X) P_t = e^t∆ ↔ Cheeger’s L²-energy

2Ch(f) := inf

lim

n

Z

X

lip(f_n)²dm

f_n : Lip.

f_n → f in L²

= Z

X

|Df|²_w dm Definition 1

(X, d,m): infinitesimally Hilbertian

⇔def Ch: quadratic form (⇔ P_t: linear ⇔ ∆: linear)

⇒ ^∃∃∃hD·, D·i_w bilinear s.t. hDf, Dfi_w = |Df|²_w

Met. meas. sp. & heat flow on it

(X, d,m): Polish geod. met. meas. sp.

P_t = e^t∆ ↔ Cheeger’s L²-energy 2Ch(f) := inf

lim

n

Z

X

lip(fn)²dm

fn : Lip.

f_n → f in L²

= Z

X

|Df|²_w dm Definition 1

(X, d,m): infinitesimally Hilbertian

⇔def Ch: quadratic form (⇔ P_t: linear ⇔ ∆: linear)

⇒ ^∃∃∃hD·, D·i_w bilinear s.t. hDf, Dfi_w = |Df|²_w

9 / 32

RCD spaces

Definition 2 (RCD(0, N) (N ∈ (0,∞))) (X, d,m): infin. Hilb.

Z

X

exp

−^∃∃∃cd(^∃∃∃x₀, x)²

m(dx) < ∞

(⇒ kP_tfk_L¹_(m) = kfk_L¹_(m) for f ≥ 0)

∀∀∀f ∈ D(Ch), |Df|_w ≤ 1 ⇒ f: Lip., lip(f) ≤ 1

W2(Psf m, Ptgm)²

≤ W₂(f m, gm)² + 2N(√

t− √ s)² (^∀∀∀f, g: prob. density)

F RCD^∗(K, N) (K 6= 0) can be defined similarly

RCD spaces

Definition 2 (RCD(0, N) (N ∈ (0,∞))) (X, d,m): infin. Hilb.

Z

X

exp

−^∃∃∃cd(^∃∃∃x₀, x)²

m(dx) < ∞

(⇒ kP_tfk_L¹_(m) = kfk_L¹_(m) for f ≥ 0)

∀∀∀f ∈ D(Ch), |Df|_w ≤ 1 ⇒ f: Lip., lip(f) ≤ 1

W2(Psf m, Ptgm)²

≤ W₂(f m, gm)² + 2N(√

t− √ s)² (^∀∀∀f, g: prob. density)

F RCD^∗(K, N) (K 6= 0) can be defined similarly

10 / 32

RCD spaces

Definition 2 (RCD(0, N) (N ∈ (0,∞))) (X, d,m): infin. Hilb.

Z

X

exp

−^∃∃∃cd(^∃∃∃x₀, x)²

m(dx) < ∞

(⇒ kP_tfk_L¹_(m) = kfk_L¹_(m) for f ≥ 0)

∀∀∀f ∈ D(Ch), |Df|_w ≤ 1 ⇒ f: Lip., lip(f) ≤ 1 W2(Psf m, Ptgm)²

≤ W₂(f m, gm)² + 2N(√

t− √ s)² (^∀∀∀f, g: prob. density)

F RCD^∗(K, N) (K 6= 0) can be defined similarly

RCD spaces

Definition 2 (RCD(0, N) (N ∈ (0,∞))) (X, d,m): infin. Hilb.

Z

X

exp

−^∃∃∃cd(^∃∃∃x₀, x)²

m(dx) < ∞

(⇒ kP_tfk_L¹_(m) = kfk_L¹_(m) for f ≥ 0)

∀∀∀f ∈ D(Ch), |Df|_w ≤ 1 ⇒ f: Lip., lip(f) ≤ 1 W2(Psf m, Ptgm)²

≤ W₂(f m, gm)² + 2N(√

t− √ s)² (^∀∀∀f, g: prob. density)

F RCD^∗(K, N) (K 6= 0) can be defined similarly

10 / 32

RCD spaces

Definition 2 (RCD(0, N) (N ∈ (0,∞))) (X, d,m): infin. Hilb.

Z

X

exp

−^∃∃∃cd(^∃∃∃x₀, x)²

m(dx) < ∞

(⇒ kP_tfk_L¹_(m) = kfk_L¹_(m) for f ≥ 0)

∀∀∀f ∈ D(Ch), |Df|_w ≤ 1 ⇒ f: Lip., lip(f) ≤ 1 W2(Psf m, Ptgm)²

≤ W₂(f m, gm)² + 2N(√

t− √ s)² (^∀∀∀f, g: prob. density)

F RCD^∗(K, N) (K 6= 0) can be defined similarly

Examples

(X, g): m-dim. cpl. Riem. mfd., ∂X = ∅, d: Riem. dist., m = e^−Vvol_g (V : X →RRR)

(Weighted Riem. mfd)

⇓

RCD^∗(K, N) ⇔ Ric +∇²V − ∇V^⊗2

N − m ≥ K (Pointed) measured GH lim. of RCD^∗(K, N) sp.’s

[Gigli, Mondino & Savar´e ’15]

m-dim. Alexandrov sp. of curv. ≥ k

⇒ RCD^∗((m − 1)k, m) sp.

[Petrunin ’09]

11 / 32

Examples

(X, g): m-dim. cpl. Riem. mfd., ∂X = ∅, d: Riem. dist., m = e^−Vvol_g (V : X →RRR)

(Weighted Riem. mfd)

⇓

RCD^∗(K, N) ⇔ Ric +∇²V − ∇V^⊗2

N − m ≥ K (Pointed) measured GH lim. of RCD^∗(K, N) sp.’s

[Gigli, Mondino & Savar´e ’15]

m-dim. Alexandrov sp. of curv. ≥ k

⇒ RCD^∗((m − 1)k, m) sp.

[Petrunin ’09]

Examples

(X, g): m-dim. cpl. Riem. mfd., ∂X = ∅, d: Riem. dist., m = e^−Vvol_g (V : X →RRR)

(Weighted Riem. mfd)

⇓

RCD^∗(K, N) ⇔ Ric +∇²V − ∇V^⊗2

N − m ≥ K (Pointed) measured GH lim. of RCD^∗(K, N) sp.’s

[Gigli, Mondino & Savar´e ’15]

m-dim. Alexandrov sp. of curv. ≥ k

⇒ RCD^∗((m − 1)k, m) sp.

[Petrunin ’09]

11 / 32

Characterizations of RCD cond.

RCD(0, N): Some regularity ass’ns &

W₂(P_sf m, P_tgm)²

≤ W₂(f m, gm)² + 2N(√

t − √ s)² F Equiv. cond’ns to RCD(0, N) (up to reg. assn’s)

“1

2∆|Df|²_w − hDf, D∆fi_w ≥ 1

N|∆f|²” (Bakry-´Emery’s curv.-dim. cond.) On (P₂(X), W₂), ^∀∀∀µ₀, ^∃∃∃(µ_t)_t≥0 sol. to

(0, N)-evolution variational inequality of Ent (a (metric) formulation of “µ˙_t = −∇Ent(µ_t)”) [Erbar, K. & Sturm ’15]

Characterizations of RCD cond.

RCD(0, N): Some regularity ass’ns &

W₂(P_sf m, P_tgm)²

≤ W₂(f m, gm)² + 2N(√

t − √ s)² F Equiv. cond’ns to RCD(0, N) (up to reg. assn’s)

“1

2∆|Df|²_w − hDf, D∆fi_w ≥ 1

N|∆f|²” (Bakry-´Emery’s curv.-dim. cond.) On (P₂(X), W₂), ^∀∀∀µ₀, ^∃∃∃(µ_t)_t≥0 sol. to

(0, N)-evolution variational inequality of Ent (a (metric) formulation of “µ˙_t = −∇Ent(µ_t)”) [Erbar, K. & Sturm ’15]

12 / 32

Characterizations of RCD cond.

RCD(0, N): Some regularity ass’ns &

W₂(P_sf m, P_tgm)²

≤ W₂(f m, gm)² + 2N(√

t − √ s)² F Equiv. cond’ns to RCD(0, N) (up to reg. assn’s)

“1

2∆|Df|²_w − hDf, D∆fi_w ≥ 1

N|∆f|²” (Bakry-´Emery’s curv.-dim. cond.) On (P₂(X), W₂), ^∀∀∀µ₀, ^∃∃∃(µ_t)_t≥0 sol. to

(0, N)-evolution variational inequality of Ent (a (metric) formulation of “µ˙_t = −∇Ent(µ_t)”) [Erbar, K. & Sturm ’15]

Heat flow

Properties of the heat semigr. P_t under RCD^∗(K, N) P_t : L²(m) → L²(m) can be extended

to P_t : P₂(X) → P₂(X)

Pt admits a continuous kernel (heat kernel) pt

µt = Ptµ(= ρtm) ∈ P(X) satisfies

||µ˙_t||² := lim

δ↓0

W₂(µ_t, µ_t+δ)² δ²

= − d

dt Ent(µt) =

Z |Dρ_t|²_w

ρ_t dm =: I(µt) (Fisher information)

13 / 32

Heat flow

Properties of the heat semigr. P_t under RCD^∗(K, N) P_t : L²(m) → L²(m) can be extended

to P_t : P₂(X) → P₂(X)

Pt admits a continuous kernel (heat kernel) pt

µt = Ptµ(= ρtm) ∈ P(X) satisfies

||µ˙_t||² := lim

δ↓0

W₂(µ_t, µ_t+δ)² δ²

= − d

dt Ent(µt) =

Z |Dρ_t|²_w

ρ_t dm =: I(µt) (Fisher information)

Heat flow

Properties of the heat semigr. P_t under RCD^∗(K, N) P_t : L²(m) → L²(m) can be extended

to P_t : P₂(X) → P₂(X)

Pt admits a continuous kernel (heat kernel) pt

µt = Ptµ(= ρtm) ∈ P(X) satisfies

||µ˙_t||² := lim

δ↓0

W₂(µ_t, µ_t+δ)² δ²

= − d

dt Ent(µt) =

Z |Dρ_t|²_w

ρ_t dm =: I(µt) (Fisher information)

13 / 32

Heat flow

Properties of the heat semigr. P_t under RCD^∗(K, N) P_t : L²(m) → L²(m) can be extended

to P_t : P₂(X) → P₂(X)

Pt admits a continuous kernel (heat kernel) pt

µt = Ptµ(= ρtm) ∈ P(X) satisfies

||µ˙_t||² := lim

δ↓0

W₂(µ_t, µ_t+δ)² δ²

= − d

dt Ent(µt) =

Z |Dρ_t|²_w

ρ_t dm =: I(µt) (Fisher information)

1. Introduction

2. Framework: RCD spaces 3. Main results

4. Proof

4.1 Monotonicity 4.2 Rigidity

4.3 Additional remarks

W -entropy

µ = ρm ∈ P(X), ρ =: e^−f

(4πt)^N/2 (τ t) W(µ, t) :=

Z

X

t|Df|²_w +f −N

ρdm

= tI(µ) −Ent(µ) − N

2 logt+ c1

I(µ) :=

Z |Dρ|²_w ρ dm Ent(µ) :=

Z

X

ρlogρdm

W -entropy

µ = ρm ∈ P(X), ρ =: e^−f

(4πt)^{N /2} (τ t) W(µ, t) :=

Z

X

t|Df|²_w +f −N

ρdm

= tI(µ) −Ent(µ) − N

2 logt+ c1

I(µ) :=

Z |Dρ|²_w ρ dm Ent(µ) :=

Z

X

ρlogρdm

15 / 32

W -entropy

µ = ρm ∈ P(X), ρ =: e^−f

(4πt)^{N /2} (τ t) W(µ, t) :=

Z

X

t|Df|²_w +f −N

ρdm

= tI(µ) −Ent(µ) − N

2 logt+ c1

I(µ) :=

Z |Dρ|²_w ρ dm Ent(µ) :=

Z

X

ρlogρdm

W -entropy

µ = ρm ∈ P(X), ρ =: e^−f

(4πt)^{N /2} (τ t) W(µ, t) :=

Z

X

t|Df|²_w +f −N

ρdm

= tI(µ) −Ent(µ) − N

2 logt+ c1

I(µ) :=

Z |Dρ|²_w ρ dm Ent(µ) :=

Z

X

ρlogρdm

15 / 32

W -entropy

µ = ρm ∈ P(X), ρ =: e^−f

(4πt)^{N /2} (τ t) W(µ, t) :=

Z

X

t|Df|²_w +f −N

ρdm

= tI(µ) −Ent(µ) − N

2 logt+ c1

I(µ) :=

Z |Dρ|²_w ρ dm Ent(µ) :=

Z

X

ρlogρdm

W -entropy

µ = ρm ∈ P(X), ρ =: e^−f

(4πt)^{N /2} (τ t) W(µ, t) :=

Z

X

t|Df|²_w +f −N

ρdm

= tI(µ) −Ent(µ) − N

2 logt+ c1

I(µ) :=

Z |Dρ|²_w ρ dm Ent(µ) :=

Z

X

ρlogρdm

15 / 32

Main thm

Theorem 3 ([X.-D. Li & K.])

(X, d,m): RCD(0, N), N ≥ 2, µt := Ptµ (1) W(µ_t, t) & in t ∈ (0,∞)

(2) Suppose ^∃∃∃t_∗ > 0 s.t.

lim

t↓t∗

W(µ_t, t) − W(µ_t∗, t_∗) t− t_∗ = 0

⇒

∃∃∃x₀ ∈ X s.t. µ = δ_x0,

X ' (0, N − 1)-cone of

an RCD^∗(N − 2, N − 1) sp.

& t 7→ W(µ_t, t): const.

Main thm

Theorem 3 ([X.-D. Li & K.])

(X, d,m): RCD(0, N), N ≥ 2, µt := Ptµ (1) W(µ_t, t) & in t ∈ (0,∞)

(2) Suppose ^∃∃∃t_∗ > 0 s.t.

lim

t↓t∗

W(µ_t, t) − W(µ_t∗, t_∗) t− t_∗ = 0

⇒

∃∃∃x₀ ∈ X s.t. µ = δ_x0, X ' (0, N − 1)-cone of

an RCD^∗(N − 2, N − 1) sp.

& t 7→ W(µ_t, t): const.

16 / 32

Main thm

Theorem 3 ([X.-D. Li & K.])

(X, d,m): RCD(0, N), N ≥ 2, µt := Ptµ (1) W(µ_t, t) & in t ∈ (0,∞)

(2) Suppose ^∃∃∃t_∗ > 0 s.t.

lim

t↓t∗

W(µ_t, t) − W(µ_t∗, t_∗) t− t_∗ = 0

⇒ ^∃∃∃x₀ ∈ X s.t. µ = δ_x0, X ' (0, N − 1)-cone of

an RCD^∗(N −2, N − 1) sp.

& t 7→ W(µ_t, t): const.

Main thm

Theorem 3 ([X.-D. Li & K.])

(X, d,m): RCD(0, N), N ≥ 2, µt := Ptµ (1) W(µ_t, t) & in t ∈ (0,∞)

(2) ^∃∃∃t_∗ > 0 s.t.

lim

t↓t∗

W(µ_t, t) − W(µ_t∗, t_∗) t− t_∗ = 0

⇔ ^∃∃∃x₀ ∈ X s.t. µ = δ_x0, X ' (0, N − 1)-cone of

an RCD^∗(N −2, N − 1) sp.

& t 7→ W(µ_t, t): const.

16 / 32

Cone

Definition 4 ((0, N)-cone)

(X, d,m): (0, N)-cone of (Y, d_Y,m_Y)

⇔def

X = [0,∞) × Y /{0} × Y, d((r, x),(s, y))²

:= r²+ s² − 2rscos(d_Y(x, y) ∧ π) m(drdx) := r^Ndrm_Y(dx)

Cone

Definition 4 ((0, N)-cone)

(X, d,m): (0, N)-cone of (Y, d_Y,m_Y)

⇔def

X = [0,∞) × Y /{0} × Y, d((r, x),(s, y))²

:= r²+ s² − 2rscos(d_Y(x, y) ∧ π) m(drdx) := r^Ndrm_Y(dx)

17 / 32

Cone

Definition 4 ((0, N)-cone)

(X, d,m): (0, N)-cone of (Y, d_Y,m_Y)

⇔def

X = [0,∞) × Y /{0} × Y, d((r, x),(s, y))²

:= r²+ s² − 2rscos(d_Y(x, y) ∧ π) m(drdx) := r^Ndrm_Y(dx)

Remarks

Theorem 1 (1) is known when X: cpt.

[Jiang & Zhang ’16]

In previous results, µ = δ_x0 (intial data) is assumed It is a conclusion in Theorem 1

Considering the right upper derivative of W(µ_t, t) Requires no differentiability

(0, N)-cone of Y is a (smooth) Riem. mfd

⇔ Y ' SSS^N−1(1)

(⇒ dimX = N ∈ NNN)

Theorem 1 covers previous results

for weighted Riem. mfds Theorem 1 does not rely on the “entropy formula”

18 / 32

Remarks

Theorem 1 (1) is known when X: cpt.

[Jiang & Zhang ’16]

In previous results, µ = δ_x0 (intial data) is assumed It is a conclusion in Theorem 1

Considering the right upper derivative of W(µ_t, t) Requires no differentiability

(0, N)-cone of Y is a (smooth) Riem. mfd

⇔ Y ' SSS^N−1(1)

(⇒ dimX = N ∈ NNN)

Theorem 1 covers previous results

for weighted Riem. mfds Theorem 1 does not rely on the “entropy formula”

Remarks

Theorem 1 (1) is known when X: cpt.

[Jiang & Zhang ’16]

In previous results, µ = δ_x0 (intial data) is assumed It is a conclusion in Theorem 1

Considering the right upper derivative of W(µ_t, t) Requires no differentiability

(0, N)-cone of Y is a (smooth) Riem. mfd

⇔ Y ' SSS^N−1(1)

(⇒ dimX = N ∈ NNN)

Theorem 1 covers previous results

for weighted Riem. mfds Theorem 1 does not rely on the “entropy formula”

18 / 32

Remarks

Theorem 1 (1) is known when X: cpt.

[Jiang & Zhang ’16]

In previous results, µ = δ_x0 (intial data) is assumed It is a conclusion in Theorem 1

Considering the right upper derivative of W(µ_t, t) Requires no differentiability

(0, N)-cone of Y is a (smooth) Riem. mfd

⇔ Y ' SSS^N−1(1)

(⇒ dimX = N ∈ NNN)

Theorem 1 covers previous results

for weighted Riem. mfds Theorem 1 does not rely on the “entropy formula”

Remarks

Theorem 1 (1) is known when X: cpt.

[Jiang & Zhang ’16]

In previous results, µ = δ_x0 (intial data) is assumed It is a conclusion in Theorem 1

Considering the right upper derivative of W(µ_t, t) Requires no differentiability

(0, N)-cone of Y is a (smooth) Riem. mfd

⇔ Y ' SSS^N−1(1)

(⇒ dimX = N ∈ NNN)

Theorem 1 covers previous results

for weighted Riem. mfds Theorem 1 does not rely on the “entropy formula”

18 / 32

Remarks

Theorem 1 (1) is known when X: cpt.

[Jiang & Zhang ’16]

In previous results, µ = δ_x0 (intial data) is assumed It is a conclusion in Theorem 1

Considering the right upper derivative of W(µ_t, t) Requires no differentiability

(0, N)-cone of Y is a (smooth) Riem. mfd

⇔ Y ' SSS^N−1(1)

(⇒ dimX = N ∈ NNN)

Theorem 1 covers previous results

for weighted Riem. mfds Theorem 1 does not rely on the “entropy formula”

Remarks

Theorem 1 (1) is known when X: cpt.

[Jiang & Zhang ’16]

In previous results, µ = δ_x0 (intial data) is assumed It is a conclusion in Theorem 1

Considering the right upper derivative of W(µ_t, t) Requires no differentiability

(0, N)-cone of Y is a (smooth) Riem. mfd

⇔ Y ' SSS^N−1(1)

(⇒ dimX = N ∈ NNN)

Theorem 1 covers previous results

for weighted Riem. mfds Theorem 1 does not rely on the “entropy formula”

18 / 32

Remarks

Theorem 1 (1) is known when X: cpt.

[Jiang & Zhang ’16]

In previous results, µ = δ_x0 (intial data) is assumed It is a conclusion in Theorem 1

Considering the right upper derivative of W(µ_t, t) Requires no differentiability

(0, N)-cone of Y is a (smooth) Riem. mfd

⇔ Y ' SSS^N−1(1) (⇒ dimX = N ∈ NNN) Theorem 1 covers previous results

for weighted Riem. mfds Theorem 1 does not rely on the “entropy formula”

Remarks

Theorem 1 (1) is known when X: cpt.

[Jiang & Zhang ’16]

In previous results, µ = δ_x0 (intial data) is assumed It is a conclusion in Theorem 1

Considering the right upper derivative of W(µ_t, t) Requires no differentiability

(0, N)-cone of Y is a (smooth) Riem. mfd

⇔ Y ' SSS^N−1(1) (⇒ dimX = N ∈ NNN) Theorem 1 covers previous results

for weighted Riem. mfds Theorem 1 does not rely on the “entropy formula”

18 / 32

1. Introduction

2. Framework: RCD spaces 3. Main results

4. Proof

4.1 Monotonicity 4.2 Rigidity

4.3 Additional remarks

4.1. Monotonicity

Optimal transport approach on Ricci flow

∂τgτ = 2 Ric, µτ: ∂τµτ = ∆τµτ

L^t_s(x, y) := inf

γs=x, γt=y

Z t

s

√r(|γ˙_r|²_r + R(γ_r)) dr

T_L^t

s(µ, ν) := inf

π

Z

X×X

L^t_sdπ: L-opt. trans. cost

Ξ^τ_τ¹

0(t) := 2(√

τ₁t− √

τ₀t)T_L^τ1t

τ0t(µ_τ0t, µ_τ1t)

(τ₀ < τ₁) −2m(√

τ₁t−√ τ₀t)²

⇒ Ξ^τ_τ¹

0(t) & in t [Topping ’09 / K. & Philipowski ’11]

⇒ lim

τ↓1

Ξ^τ₁(t)

(τ − 1)² & ⇒ W & [Topping ’09]

Optimal transport approach on Ricci flow

∂τgτ = 2 Ric, µτ: ∂τµτ = ∆τµτ

L^t_s(x, y) := inf

γs=x, γt=y

Z t

s

√r(|γ˙_r|²_r + R(γ_r)) dr

T_L^t

s(µ, ν) := inf

π

Z

X×X

L^t_sdπ: L-opt. trans. cost Ξ^τ_τ¹

0(t) := 2(√

τ₁t− √

τ₀t)T_L^τ1t

τ0t(µ_τ0t, µ_τ1t)

(τ₀ < τ₁) −2m(√

τ1t−√ τ0t)²

⇒ Ξ^τ_τ¹

0(t) & in t [Topping ’09 / K. & Philipowski ’11]

⇒ lim

τ↓1

Ξ^τ₁(t)

(τ − 1)² & ⇒ W & [Topping ’09]

21 / 32

Optimal transport approach on Ricci flow

∂τgτ = 2 Ric, µτ: ∂τµτ = ∆τµτ

L^t_s(x, y) := inf

γs=x, γt=y

Z t

s

√r(|γ˙_r|²_r + R(γ_r)) dr

T_L^t

s(µ, ν) := inf

π

Z

X×X

L^t_sdπ: L-opt. trans. cost Ξ^τ_τ¹

0(t) := 2(√

τ₁t− √

τ₀t)T_L^τ1t

τ0t(µ_τ0t, µ_τ1t)

(τ₀ < τ₁) −2m(√

τ1t−√ τ0t)²

⇒ Ξ^τ_τ¹

0(t) & in t [Topping ’09 / K. & Philipowski ’11]

⇒ lim

τ↓1

Ξ^τ₁(t)

(τ − 1)² & ⇒ W & [Topping ’09]

Optimal transport approach on Ricci flow

∂τgτ = 2 Ric, µτ: ∂τµτ = ∆τµτ

L^t_s(x, y) := inf

γs=x, γt=y

Z t

s

√r(|γ˙_r|²_r + R(γ_r)) dr

T_L^t

s(µ, ν) := inf

π

Z

X×X

L^t_sdπ: L-opt. trans. cost Ξ^τ_τ¹

0(t) := 2(√

τ₁t− √

τ₀t)T_L^τ1t

τ0t(µ_τ0t, µ_τ1t)

(τ₀ < τ₁) −2m(√

τ1t−√ τ0t)²

⇒ Ξ^τ_τ¹

0(t) & in t [Topping ’09 / K. & Philipowski ’11]

⇒ lim

τ↓1

Ξ^τ₁(t) (τ − 1)² &

⇒ W & [Topping ’09]

21 / 32

Optimal transport approach on Ricci flow

∂τgτ = 2 Ric, µτ: ∂τµτ = ∆τµτ

L^t_s(x, y) := inf

γs=x, γt=y

Z t

s

√r(|γ˙_r|²_r + R(γ_r)) dr

T_L^t

s(µ, ν) := inf

π

Z

X×X

L^t_sdπ: L-opt. trans. cost Ξ^τ_τ¹

0(t) := 2(√

τ₁t− √

τ₀t)T_L^τ1t

τ0t(µ_τ0t, µ_τ1t)

(τ₀ < τ₁) −2m(√

τ1t−√ τ0t)²

⇒ Ξ^τ_τ¹

0(t) & in t [Topping ’09 / K. & Philipowski ’11]

⇒ lim

τ↓1

Ξ^τ₁(t)

(τ − 1)² & ⇒ W & [Topping ’09]

Toward the time-inhomogeneous case

L^t_s(x, y) := inf

γ_s=x, γ_t=y

Z t

s

√r(|γ˙_r|²_r + R(γ_r)) dr

L^t_s(x, y) := inf

γs=x, γt=y

Z t

s

√r|γ˙_r|²dr

F γ_r^∗ := γ_ξ(r), ξ(r) := ((1 −r)√

s + r√ t)²

⇒ 2(√

t −√ s)

Z t

s

√r|γ˙_r|²dr = Z 1

0

|γ˙_u^∗|²du

⇒ 2(√

t −√

s)L^t_s(x, y) = d(x, y)², 2(√

t −√

s)T_L^t

s(µ, ν) = W₂(µ, ν)² Ξ^τ_τ¹

0(t) := 2(√

τ₁t− √

τ₀t)T_L^τ1t

τ0t(µ_τ0t, µ_τ1t)

−2N(√

τ₁t −√ τ₀t)²

= W₂(µ_τ0t, µ_τ1t)² −2N(√

τ₁t− √ τ₀t)²

⇓ Ξ^τ₁(t) &

m

W₂(µ_t, µ_{τ t})² ≤ W₂(µ_s, µ_{τ s})² +2N(p

τ(t− s)− √

t− s)²

22 / 32

Toward the time-inhomogeneous case

L^t_s(x, y) := inf

γ_s=x, γ_t=y

Z t

s

√r(|γ˙_r|²_r + R(γ_r)) dr

L^t_s(x, y) := inf

γs=x, γt=y

Z t

s

√r|γ˙_r|²dr

F γ_r^∗ := γ_ξ(r), ξ(r) := ((1− r)√

s + r√ t)²

⇒ 2(√

t− √ s)

Z t

s

√r|γ˙_r|²dr = Z 1

0

|γ˙_u^∗|²du

⇒ 2(√

t −√

s)L^t_s(x, y) = d(x, y)², 2(√

t −√

s)T_L^t

s(µ, ν) = W₂(µ, ν)² Ξ^τ_τ¹

0(t) := 2(√

τ₁t− √

τ₀t)T_L^τ1t

τ0t(µ_τ0t, µ_τ1t)

−2N(√

τ₁t −√ τ₀t)²

= W₂(µ_τ0t, µ_τ1t)² −2N(√

τ₁t− √ τ₀t)²

⇓ Ξ^τ₁(t) &

m

W₂(µ_t, µ_{τ t})² ≤ W₂(µ_s, µ_{τ s})² +2N(p

τ(t− s)− √

t− s)²

Toward the time-inhomogeneous case

L^t_s(x, y) := inf

γ_s=x, γ_t=y

Z t

s

√r|γ˙_r|²dr

F γ_r^∗ := γ_ξ(r), ξ(r) := ((1− r)√

s + r√ t)²

⇒ 2(√

t− √ s)

Z t

s

√r|γ˙_r|²dr = Z 1

0

|γ˙_u^∗|²du

⇒ 2(√

t− √

s)L^t_s(x, y) = d(x, y)², 2(√

t− √

s)T