Equality in the logarithmic Sobolev inequality
Shin-ichi OHTA∗,‡

Asuka TAKATSU†,‡

June 17, 2019
Dedicated to the memory of Kazumasa Kuwada
Abstract
We investigate the rigidity problem for the logarithmic Sobolev inequality on
weighted Riemannian manifolds satisfying Ric∞ ≥ K > 0. Assuming that equality
holds, we show that the 1-dimensional Gaussian space is necessarily split oﬀ, similarly to the rigidity results of Cheng–Zhou on the spectral gap as well as Morgan on
the isoperimetric inequality. The key ingredient of the proof is the needle decomposition method introduced on Riemannian manifolds by Klartag. We also present
several related open problems.

1

Introduction

The rigidity and stability properties of various inequalities are one of the most fundamental
problems in geometric analysis, and recently draw considerable attention from the geometric, analytic and probabilistic viewpoints. We refer to [CFMP, MN, El, BBJ] (resp.
[CMM]) for the quantitative isoperimetry on Gaussian spaces (resp. positively curved
spaces), and to [CZ, Ma1, GKKO] for rigidity results on the sharp spectral gap (equivalently, Poincaré inequality), to name a few.
In this article, we study the rigidity problem for the logarithmic Sobolev inequality
on positively curved weighted Riemannian manifolds. On a Riemannian manifold (M, g)equipped with a weighted probability measure ω = e−Ψ volM (Ψ ∈ C ∞ (M )) and satisfying
the weighted Ricci curvature bound Ric∞ := Ricg + Hessg Ψ ≥ K > 0, the logarithmic
Sobolev inequality
∫
∫
1
|∇ρ|2
dω
(1.1)
ρ log ρ dω ≤2K M ρ
M
holds for all nonnegative locally Lipschitz functions ρ : M −→ [0, ∞) with ∥ρ∥L1 (ω) = 1(see Section 2 for a further explanation).
∗

Department of Mathematics, Osaka University, Osaka 560-0043, Japan (s.ohta@math.sci.osaka-u.ac.jp)
Department of Mathematical Sciences, Tokyo Metropolitan University, Tokyo 192-0397, Japan(asuka@tmu.ac.jp)‡
RIKEN Center for Advanced Intelligence Project (AIP), 1-4-1 Nihonbashi, Tokyo 103-0027, Japan†1

We will characterize when equality holds in (1.1). We remark that, since the logarithmic Sobolev inequality implies the Poincaré inequality, equality in (1.1) is a weakerassumption than that for the spectral gap. In the Gaussian case (M = Rn , dω =2(K/2π)n/2 e−K|x| /2 dx), Carlen [Ca, Theorem 4] showed that equality holds if and only
if ρω is a translation of ω (see also [Le]). This was generalized to weighted Euclidean
spaces satisfying Ric∞ ≥ K in [AMTU, Theorem 3.11] (see also [OV, p. 390]). The manifold case was briefly discussed in [OV, p. 391], and our main theorem below gives a precise
answer to this question.
Theorem 1.1 (Main theorem) Let (M, g, ω) be a connected, complete, weighted C ∞ Riemannian manifold of dimension n ≥ 2 such that Ric∞ ≥ K > 0. Assume that there
is a nonconstant, nonnegative, locally Lipschitz function ρ : M −→ [0, ∞) such that∥ρ∥L1 (ω) = 1 and
∫
∫
1
|∇ρ|2
ρ log ρ dω =dω.
2K M ρ
M
Then we have the following.
(i) There exists a weighted Riemannian manifold (N, gN , ωN ) of dimension n − 1 satisfying Ric∞ ≥ K again such that (M, g, ω) is isometric to the product of (R, | · |, γK )and (N, gN , ωN ) as weighted Riemannian manifolds, where | · | denotes the Euclidean
metric and γK denotes the Gaussian measure with variance K −1 , namely√
K −Kx2 /2dγK :=edx.
2π(ii) In the product structure as in (i), ρ is a translation of the density of ω in the Rdirection. Precisely, there exists some t ∈ R \ {0} such that the isometry Tt :R × N −→ R × N defined by Tt (s, x) := (s + t, x) pushes ω forward to ρω.
The first assertion (i) has the same consequence as Cheng–Zhou’s result on the sharp
spectral gap [CZ, Theorem 2] as well as Morgan’s result on the sharp isoperimetric inequality [Mo, Theorem 18.7] under Ric∞ ≥ K. We will in fact show that equality in (1.1)
implies the sharp spectral gap and then apply Cheng–Zhou’s result (in the same spirit
as the argument in [Ma2] concerning equality of isoperimetric inequalities). The second
assertion (ii) shows a similar phenomenon to the characterizations in [Ca, AMTU]. The
key tool in the proof of Theorem 1.1 is the needle decomposition established by Klartag[Kl] (see Theorem 2.4), we refer to [CM1, CM2, CMM, Ma2] for other applications of thispowerful method.
Combining Theorem 1.1 with the results in [Mo, CZ], we complete the following chainof characterizations of the sharp geometric and functional inequalities.
Corollary 1.2 Let (M, g, ω) be a connected, complete, weighted C ∞ -Riemannian manifold of dimension n ≥ 2 such that Ric∞ ≥ K > 0. Then the following are equivalent.
(I) Equality holds in the logarithmic Sobolev inequality (1.1) on (M, g, ω).
2

(II) (M, g, ω) attains the sharp spectral gap λ1 = K or, equivalently, equality holds in
the Poincaré inequality (2.6).
(III) Equality holds in the Bakry–Ledoux isoperimetric inequality in (M, g, ω).
(IV) (M, g, ω) splits oﬀ the 1-dimensional Gaussian space of variance K −1 in the senseof Theorem 1.1(i).
The article is organized as follows. In the next section, we recall key definitions andtools necessary in our argument, including the curvature-dimension condition and needle
decompositions, from optimal transport theory. Section 3 is devoted to the proof of
Theorem 1.1. We finally discuss several further problems in Section 4.

2

Preliminaries for optimal transport theory

Throughout the article, let (M, g) be a connected, complete C ∞ -Riemannian manifold
without boundary of dimension n ≥ 2. We will denote by dM and volM the Riemannian
distance function and the volume measure on M , respectively. Let us also fix a probability
measure ω := e−Ψ volM on M with Ψ ∈ C ∞ (M ). Then (M, g, ω) is called a weighted
Riemannian manifold or a Riemannian manifold with density.
Optimal transport theory is quite useful in geometry and analysis of weighted manifolds. In this section, we recall some necessary notions and results. We refer to [Vi] forthe fundamentals of optimal transport theory as well as the curvature-dimension condition, and to [Kl, CM1] for the recent breakthrough called the needle decomposition or the
localization.

2.1

Wasserstein distanceLet P2 (M ) be the set of all Borel probability measures µ on M with finite second moments,
namely dM (o, ·) ∈ L2 (µ) for some (and hence any) o ∈ M . We denote by P2ac (M, ω) the
subset of P2 (M ) consisting of absolutely continuous measures with respect to ω.
Given µ0 , µ1 ∈ P2 (M ), we call π ∈ P(M × M ) a coupling of (µ0 , µ1 ) ifπ(A × M ) = µ0 (A),π(M × A) = µ1 (A)hold for any Borel set A ⊂ M . In other words, the push-forward measure of π to the first(resp. second) component M coincides with µ0 (resp. µ1 ). We denote by Π(µ0 , µ1 ) the setof all couplings of (µ0 , µ1 ). Then the L2 -Wasserstein distance between µ0 , µ1 ∈ P2 (M ) isdefined as
W2 (µ0 , µ1 ) :=inf
∥dM ∥L2 (π) .
π∈Π(µ0 ,µ1 )A coupling π ∈ Π(µ0 , µ1 ) is said to be optimal if it attains the infimum above. For
any µ0 , µ1 ∈ P2 (M ), W2 (µ0 , µ1 ) is finite and an optimal coupling always exists (see [Vi,Chapter 4]). Moreover, the pair (P2 (M ), W2 ) is a complete separable metric space (see[Vi, Chapter 6]).
3

In the case where µ0 ∈ P2ac (M, ω), an optimal coupling is given by the push-forward
measure of µ0 as follows (see [Br, Mc, FG, Vi]).
Theorem 2.1 (Optimal transports) Given any µ0 ∈ P2ac (M, ω) and µ1 ∈ P2 (M ),there exists a locally semi-convex function ϕ : Ω → R on an open set Ω ⊂ M withµ0 (Ω) = 1 such that the map Tt (x) := expx (t∇ϕ(x)), t ∈ [0, 1], provides a unique minimal
geodesic from µ0 to µ1 with respect to W2 . Precisely, (T0 × T1 )♯ µ0 is a unique optimal
coupling of (µ0 , µ1 ), and µt := (Tt )♯ µ0 gives the unique W2 -minimal geodesic from µ0 toµ1 . Moreover, µt ∈ P2ac (M, ω) holds for all t ∈ [0, 1).
Denoted by (Tt )♯ µ0 is the push-forward measure of µ0 by Tt . A locally semi-convex
function is locally Lipschitz and twice diﬀerentiable almost everywhere by the Alexandrov–
Bangert theorem (see [Ba, Vi]). Hence Tt is diﬀerentiable almost everywhere on Ω, andµt = (Tt )♯ µ0 implies the following Monge–Ampère equation (or the Jacobian equation)()()ρt Tt (x) · eΨ(x)−Ψ(Tt (x)) detg dTt (x) = ρ0 (x)(2.1)
for all t ∈ [0, 1) at µ0 -almost every x ∈ Ω, where µt = ρt ω and detg (dTt ) is the Jacobian
determinant of Tt with respect to g. If µ1 ∈ P2ac (M, ω), then (2.1) holds also at t = 1.

2.2

Curvature-dimension condition

The deep relation between optimal transport theory and Riemannian geometry was revealed in the curvature-dimension condition, the convexity of the relative entropy characterizing the lower weighted Ricci curvature bounds.
For µ = ρω ∈ P2ac (M, ω), we define the relative entropy with respect to ω as∫Hω (µ) :=ρ log ρ dω.
M

We need to modify the Ricci curvature of (M, g), taking the eﬀect of the weight function Ψ into account, to the weighted Ricci curvature (also called the Bakry–Émery–Ricci
curvature)
Ric∞ (v) := Ricg (v, v) + Hessg Ψ(v, v)
for v ∈ T M , where Ricg is the Ricci curvature tensor and Hessg Ψ is the Hessian of Ψ withrespect to g. We will say that Ric∞ ≥ K holds for some K ∈ R if we have Ric∞ (v) ≥ K|v|2
for all v ∈ T M . The next theorem was established in [CMS1, CMS2, vRS, St1].
Theorem 2.2 (Curvature-dimension condition) For a weighted Riemannian manifold (M, g, ω) and K ∈ R, we have Ric∞ ≥ K if and only if Hω is K-convex in the sense
that, for any pair µ0 , µ1 ∈ P2ac (M, ω),Hω (µt ) ≤ (1 − t)Hω (µ0 ) + tHω (µ1 ) −Kt(1 − t)W2 (µ0 , µ1 )2
2

(2.2)

holds for all t ∈ (0, 1), where (µt )t∈[0,1] ⊂ P2ac (M, ω) is the unique minimal geodesic fromµ0 to µ1 as in Theorem 2.1.
4

The above K-convexity of Hω with respect to W2 is called the curvature-dimension
condition CD(K, ∞). We refer to [St2, St3, LV, Vi] and recent [Am] for the rapidly
developing theory of metric measure spaces satisfying the curvature-dimension condition.
Going back to the pioneering work [OV], it is known that the curvature-dimension condition intimately connects the curvature bound Ric∞ ≥ K with some functional inequalities. For µ = ρω ∈ P2ac (M, ω) with locally Lipschitz density ρ, the Fisher information
with respect to ω is defined as∫|∇ρ|2Iω (µ) :=dω,ρM
where we set |∇ρ|2 /ρ := 0 on ρ−1 (0). The Fisher information appears as the derivativeof Hω along W2 -geodesics, and then the K-convexity (2.2) shows the following (see [OV],[LV, §6]).
Proposition 2.3 (Talagrand, HWI inequalities) If (M, g, ω) satisfies Ric∞ ≥ K >0, then we have2W2 (µ, ω)2 ≤ Hω (µ)(2.3)
K
for any µ ∈ P2ac (M, ω), and√KHω (µ) ≤ W2 (µ, ω) Iω (µ) − W2 (µ, ω)2
2

(2.4)

for any µ ∈ P2ac (M, ω) whose density function is locally Lipschitz.
The first inequality (2.3) is called the Talagrand inequality and (2.4) is called the HWI
inequality. Completing the square in (2.4), we obtain the logarithmic Sobolev inequality(1.1) as
(
)2
K 1√
1
1
Iω (µ) −Iω (µ) − W2 (µ, ω) ≤Iω (µ).
(2.5)
Hω (µ) ≤2K
2 K
2K
Moreover, we can deduce from the logarithmic Sobolev inequality the Poincaré inequality
)2
(∫
∫
∫
1
2
f dω −
f dω ≤|∇f |2 dω
(2.6)
K M
M
M
for any locally Lipschitz function f ∈ L2 (ω) (see [OV], [LV, §6]).

2.3

Needle decompositions

Let us close the section by recalling the needle decomposition (also called the localization). This is a powerful method, with its root in convex geometry, for reducing a highdimensional problem to its 1-dimensional counterpart. We refer to [Kl] for the backgroundand the generalization to weighted Riemannian manifolds, and to [CM1] for a further generalization to metric measure spaces satisfying the curvature-dimension condition. We will
use the following version in [Kl, Theorems 1.2, 1.5].
5

Theorem 2.4 (Needle decompositions) Assume that (M, g, ω) satisfies Ric∞ ≥ Kand let f ∈ L1 (ω) satisfy∫
f dω = 0
M

and f · dM (o, ·) ∈ L1 (ω) for some (and hence all) o ∈ M . Then there exists a 1-Lipschitz
function u, a partition L of M , a probability measure ν on L and a family of probability
measures {ω ℓ }ℓ∈L on M satisfying the following conditions.
(i) For any measurable set A ⊂ M , we have∫
ω ℓ (A) ν(dℓ).
ω(A) =
L(ii) For ν-almost all ℓ ∈ L, we have supp(ω ℓ ) ⊂ ℓ and ℓ is a transport ray associatedwith u. Moreover, if ℓ is not a singleton, then (ℓ, | · |, ω ℓ ) satisfies Ric∞ ≥ K in the
weak sense.
(iii) For ν-almost all ℓ ∈ L, we have∫f dω ℓ = 0.
ℓ

The 1-Lipschitz function u is called the guiding function. In (ii), ℓ ⊂ M being atransport ray means that |u(x) − u(y)| = dM (x, y) holds for any x, y ∈ ℓ and ℓ cannot
be extended keeping this property. A transport ray is necessarily the image of a minimal
geodesic, and | · | denotes the distance structure of dM restricted on it. Also in (ii),
Ric∞ ≥ K in the weak sense means that, if we write dω ℓ = e−Ψℓ dx (by identifying ℓ withan interval in R), then Ψℓ is K-convex in the same way as (2.2). The construction of
the needle decomposition is closely related to the L1 -optimal transport theory. In such acontext, the function f is regarded as the diﬀerence between two probability measures,the guiding function u is a Kantorovich potential, and the L1 -optimal transport is done
along needles ℓ.

3

Proof of Theorem 1.1

Let (M, g, ω) be a weighted Riemannian manifold as in Theorem 1.1, namely it satisfies
Ric∞ ≥ K > 0 and there is a nonconstant, nonnegative, locally Lipschitz function ρ with∥ρ∥L1 (ω) = 1 attaining equality in the logarithmic Sobolev inequality (1.1). Notice that
µ := ρω ∈ P2 (M ) by the Talagrand inequality (2.3) (precisely, we take the truncation
ρi := min{ρ, i} and apply (2.3) to the normalization of ρi , then we have (2.3) for µ as
i → ∞ thanks to the lower semi-continuity of W2 found in [Vi, Lemma 4.3] for instance).
Recalling (2.5), then we haveK1Iω (µ) = W2 (µ, ω)2 .
(3.1)
2K
2
We shall show that log ρ attains equality in the Poincaré inequality (2.6). Then therigidity result in [CZ] applies.
Hω (µ) =6

3.1

Decomposition into needles

We first recall how to derive the logarithmic Sobolev inequality∫ on M from those on needles(see [CM2]). Put f := ρ − 1 and notice that f ∈ L1 (ω) and M f dω = 0. Moreover, sinceµ ∈ P2 (M ), we have for any o ∈ M∫∫()|f (x)|dM (o, x) ω(dx) ≤ρ(x) + 1 dM (o, x) ω(dx)
M

(∫

≤

M

)1/2 ( ∫)1/2 ∫2dM (o, x) ω(dx)ρ(x) ω(dx)ρ(x)dM (o, x) ω(dx)
+

M

M

M

< ∞.
In the last∫ inequality, we used the fact that ω has the Gaussian decay ([St2, Theorem 4.26])
to see M dM (o, x) ω(dx) < ∞. Hence we can apply Theorem 2.4 to f and obtain a 1Lipschitz function u, a partition L of M along with a family of probability measures{ω ℓ }ℓ∈L , and a probability measure
ν on L.
∫∫ℓ
Theorem 2.4(iii) shows ℓ f dω = 0 ν-almost every ℓ ∈ L, which yields ℓ ρ dω ℓ = 1.
Thus, since (ℓ, |·|, ω ℓ ) enjoys Ric∞ ≥ K by Theorem 2.4(ii), the 1-dimensional logarithmic
Sobolev inequality on the needle (ℓ, | · |, ω ℓ ) yields∫∫
1|∇ℓ ρ|2 ℓℓ
ρ log ρ dω ≤
dω ,
(3.2)
2K ℓ ρ
ℓwhere |∇ℓ ρ| denotes the slope of ρ taken along ℓ. Note that |∇ℓ ρ| ≤ |∇ρ|, thereby the
integration of (3.2) in ν provides the logarithmic Sobolev inequality on M (by Theorem 2.4(i)). Under our assumption (3.1), we have equality in (3.2) and∫∫∫1|∇ℓ ρ|2 ℓ|∇ρ|2 ℓ K
1ℓ
ρ log ρ dω =dω =dω = W2 (µℓ , ω ℓ )2
(3.3)
2K
ρ
2K
ρ
2
ℓ
ℓ
ℓ
ν-almost every ℓ ∈ L, where µℓ := ρω ℓ ∈ P2ac (ℓ, ω ℓ ).

3.2

Analysis on needles

Next, in order to consider the outcome of the equality (3.3), let us fix a needle (ℓ, | · |, ω ℓ )on which ρ enjoys (3.3). Let()
T (x) = expx ∇ϕ(x) = x + ϕ′ (x)be the unique optimal transport map from ω ℓ to µℓ by the 1-dimensional version of Theorem 2.1, thus ϕ : ℓ −→ R is locally Lipschitz and twice diﬀerentiable almost everywhere.
For t ∈ [0, 1], set()
Tt (x) := expx t∇ϕ(x) = x + tϕ′ (x),µℓt := (Tt )♯ ω ℓ = ρℓt ω ℓ .
Note that W2 (µℓs , µℓt ) = |s − t|W2 (µℓ0 , µℓ1 ) for s, t ∈ [0, 1].
7

On the one hand, we deduce from the Talagrand inequality (2.3) and the CD(K, ∞)inequality (2.2) that, for any t ∈ [0, 1],KKW2 (µℓ0 , µℓt )2 ≤ Hωℓ (µℓt ) ≤ (1 − t)Hωℓ (µℓ0 ) + tHωℓ (µℓ1 ) − t(1 − t)W2 (µℓ0 , µℓ1 )2 .
2
2

(3.4)

On the other hand, Hωℓ (µℓ0 ) = Hωℓ (ω ℓ ) = 0 and the hypothesis (3.3) yield that the RHSof (3.4) coincides withK 2
Kt W2 (µℓ0 , µℓ1 )2 = W2 (µℓ0 , µℓt )2 .
2
2Therefore all the inequalities in (3.4) become equality.
Equality in the CD(K, ∞)-inequality implies rigidity-type results (see [BK] for a related work). To this end, we have a closer look on the derivation of the CD(K, ∞)inequality (2.2) from Ψ′′ℓ ≥ K in Theorem 2.4(ii) (recall dω ℓ = e−Ψℓ dx). Let us consider()
Jt (x) := Tt′ (x)eΨℓ (x)−Ψℓ (Tt (x)) = 1 + tϕ′′ (x) eΨℓ (x)−Ψℓ (Tt (x))
existing almost everywhere. Recall from (2.1) that()ρℓt Tt (x) Jt (x) = ρℓ0 (x)(3.5)

holds almost everywhere. We deduce from Ψ′′ℓ ≥ K and |ϕ′ (x)| = dM (x, T (x)) that
(
) ′ 2
(
)2
d2 [log Jt (x)]ϕ′′ (x)2
′′
=
−Ψ
T
(x)
ϕ
(x)
−
≤ −KdM x, T (x)
t
ℓ
2
′′
2
dt
(1 + tϕ (x))(3.6)

in the weak sense. Since
∫
∫
∫
ℓ
ℓ
ℓ
ℓ
ℓ
Hωℓ (µt ) = log ρt dµt = log ρt (Tt ) dω = (log ρℓ0 − log Jt ) dω ℓℓℓℓ

by µℓt = (Tt )♯ ω ℓ and (3.5), we obtain from (3.6) the CD(K, ∞)-inequality
∫
(
)2
K
ℓ
ℓ
ℓ
Hωℓ (µt ) ≤ (1 − t)Hωℓ (µ0 ) + tHωℓ (µ1 ) − t(1 − t) dM x, T (x) ω ℓ (dx).
2ℓNow we find from (3.6) that the equality (3.3) shows ϕ′′ (x) = 0, hence ϕ is an aﬃnefunction on ℓ, as well as Ψ′′ℓ (Tt (x))dM (x, T (x))2 = KdM (x, T (x))2 for all t.
If ϕ′ ̸= 0, then T is a translation and hence ℓ is isometric to the real line R. Moreover,by Ψ′′ℓ ≡ K, ω ℓ is a Gaussian measure on ℓ with variance K −1 . Notice also that, since therestriction of ρ on ℓ is the ratio between two Gaussian densities with the same variance,log ρ is also an aﬃne function on ℓ.
If ϕ′ ≡ 0, then T is identity and µℓ = ω ℓ , thereby ρ ≡ 1 on ℓ. In both cases ρ > 0
holds, therefore we have supp µ = M .

8

3.3

Integration in needles

In the previous step we showed that h := log ρ on each ℓ is either an aﬃne function on the
1-dimensional Gaussian space or constant. In both cases h yields equality of the Poincaréinequality (2.6) on ℓ, namely
(∫

∫

)2

h dω −
2

ℓ

h dω

ℓ

ℓ

ℓ

1
=
K

∫
|∇ℓ h|2 dω ℓ .
ℓRecall from (3.3) that |∇ℓ ρ| = |∇ρ| holds almost everywhere. Therefore, by Theorem 2.4(i) and the Cauchy–Schwarz inequality on (L, ν),(∫∫h dω −
2

M

)2

(∫ ∫

)2
h dω ν(dℓ) −h dω ν(dℓ)
=
L ℓ
L ℓ
)2
∫ ∫
∫ (∫
ℓ
2
ℓ
h dω ν(dℓ) −
h dω
ν(dℓ)
≥
L
ℓ
L ℓ
∫ ∫
1
=
|∇ℓ h|2 dω ℓ ν(dℓ)
K L ℓ
∫
1
=
|∇h|2 dω.
K M
∫ ∫

2

h dω
M

ℓ

ℓ

Hence h gives equality of the Poincaré inequality on M . Thanks to [CZ, Theorem 2],(M, g, ω) splits oﬀ the 1-dimensional Gaussian space as described in Theorem 1.1(i).
Moreover, by the construction in [CZ], every ℓ is Gaussian and |∇h| is constant on M .
Since µ ̸= ω ensures |∇h| ≠ 0, µ is a translation of ω along the R-direction. This
completes the proof of Theorem 1.1(ii).

4

Further problems

Theorem 1.1 is concerned with one of the simplest situations of the quantitative study ofgeometric or functional inequalities, there are many related open problems.
(A) It is natural to consider the analogue of Theorem 1.1 for N ∈ [n, ∞). In this case,however, there is no extremal function. Precisely, for instance, tracing the proof ofthe dimensional logarithmic Sobolev inequality under RicN ≥ K > 0,
Hω (ρω) ≤N −1Iω (ρω),2KN

in [BGL, §5.7], we see that equality forces ρ to be constant. Therefore we can only
assume the existence of a sequence of functions ρi such thatHω (ρi ω)
N −1=.
i→∞ Iω (ρi ω)
2KN
lim

9

According to the argument in [OV] (see also [LV, §6]), we may expect that hi := ρi −1
provides the sharp spectral gap, namely
∫
KN
1
lim
|∇hi |2 dω =.
i→∞ ∥hi ∥2 2
N −1
L (ω) M
To this end, we need to analyze the asymptotic behavior of hi . We remark that it isunclear if the needle decomposition is useful in this strategy.
(B) Cheng–Zhou’s characterization in [CZ] of the sharp spectral gap was extended to
RCD(K, ∞)-spaces in [GKKO] (RCD(K, ∞) stands for the Riemannian curvaturedimension condition corresponding to Ric∞ ≥ K, for which we refer to [AGS, EKS]).
Hence it is natural to expect the analogue to Theorem 1.1 for RCD(K, ∞)-spaces.
Currently, however, the needle decomposition is known only for RCD(K, N )-spaces
with N ∈ [1, ∞) (or, more generally, essentially non-branching CD(K, N )-spaces),see [CM1]. In order to generalize Theorem 1.1, we may generalize the needle decomposition to RCD(K, ∞)-spaces, or modify the discussion in this article to directly
show that log ρ attains the sharp spectral gap (then [GKKO] applies).
(C) It is also possible to consider a generalization of Theorem 1.1 for Finsler manifolds,
where the curvature-dimension condition and the needle decomposition are available([Oh1, Oh3]). In this case, however, the splitting phenomenon is not as simple as the
Riemannian world, see [Oh2] for the case of Cheeger–Gromoll splitting theorem.
(D) One of the most challenging problems is a quantitative version of the logarithmic
Sobolev inequality, namely analysis of the situation where we have
∫
∫
1
|∇ρ|2
ρ log ρ dω ≥dω2K ′ M ρM
for some K ′ > K close to K. Quantitative properties of geometric or functional
inequalities on Riemannian manifolds are less investigated than the Euclidean orGaussian cases. The recent important work [CMM] on a quantitative isoperimetricinequality on CD(K, N )-spaces with K > 0 and N ∈ (1, ∞) made use of the needle
decomposition. On Euclidean spaces equipped with Gaussian measures, quantitative
versions are studied for the isoperimetric inequality in [CFMP, MN, El, BBJ] andfor the logarithmic Sobolev inequality in [BGRS, FIL, CF].
(E) Another interesting problem is the rigidity of gradient estimates, in this case K canbe nonpositive as well. See [ABS] for a recent result in this direction.
Acknowledgements. SO was supported in part by JSPS Grant-in-Aid for Scientific
Research (KAKENHI) 15K04844, 17H02846. AT was supported in part by JSPS Grantin-Aid for Scientific Research (KAKENHI) 15K17536, 16KT0132.

10References[Am]

L. Ambrosio, Calculus, heat flow and curvature-dimension bounds in metric measurespaces. Proceedings of the International Congress of Mathematician, 2018.

[ABS]

L. Ambrosio, E. Brué and D. Semola, Rigidity of the 1-Bakry–Émery inequality andsets of finite perimeter in RCD spaces. Preprint (2018). Available at arXiv:1812.07890[AGS]

L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci
curvature bounded from below. Duke Math. J. 163 (2014), 1405–1490.

[AMTU] A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On logarithmic Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations.
Comm. Partial Diﬀerential Equations 26 (2001), 43–100.
[BGL]

D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diﬀusion operators. Springer, Cham, 2014.

[BK]

Z. M. Balogh and A. Kristály, Equality in Borell–Brascamp–Lieb inequalities on curved
spaces. Adv. Math. 339 (2018), 453–494.

[Ba]

V. Bangert, Analytische Eigenschaften konvexer Funktionen auf Riemannschen Mannigfaltigkeiten (German). J. Reine Angew. Math. 307/308 (1979), 309–324.

[BBJ]

M. Barchiesi, A. Brancolini and V. Julin, Sharp dimension free quantitative estimates
for the Gaussian isoperimetric inequality. Ann. Probab. 45 (2017), 668–697.

[BGRS] S. G. Bobkov, N. Gozlan, C. Roberto and P.-M. Samson, Bounds on the deficit in thelogarithmic Sobolev inequality. J. Funct. Anal. 267 (2014), 4110–4138.
[Br]

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 (1991), 375–417.

[Ca]

E. Carlen, Superadditivity of Fisher’s information and logarithmic Sobolev inequalities. J. Funct. Anal. 101 (1991), 194–211.

[CMM]

F. Cavalletti, F. Maggi and A. Mondino, Quantitative isoperimetry à la Levy–Gromov.
Comm. Pure Appl. Math. (to appear). Available at arXiv:1707.04326[CM1]

F. Cavalletti and A. Mondino, Sharp and rigid isoperimetric inequalities in metricmeasure spaces with lower Ricci curvature bounds. Invent. Math. 208 (2017), 803–849.

[CM2]

F. Cavalletti and A. Mondino, Sharp geometric and functional inequalities in metricmeasure spaces with lower Ricci curvature bounds. Geom. Topol. 21 (2017), 603–645.

[CZ]

X. Cheng and D. Zhou, Eigenvalues of the drifted Laplacian on complete metric measure spaces. Commun. Contemp. Math. 19 (2017), 1650001, 17 pp.

[CFMP] A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, On the isoperimetric deficit in Gauss
space. Amer. J. Math. 133 (2011), 131–186.

11

[CMS1] D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146 (2001),
219–257.
[CMS2] D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, Prékopa–Leindler
type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport. Ann.
Fac. Sci. Toulouse Math. (6) 15 (2006), 613–635.
[CF]

T. A. Courtade and M. Fathi, Stability of the Bakry–Émery theorem on Rn . Preprint(2018). Available at arXiv:1807.09845[El]

R. Eldan, A two-sided estimate for the Gaussian noise stability deficit. Invent. Math.
201 (2015), 561–624.

[EKS]

M Erbar, K. Kuwada and K.-T. Sturm, On the equivalence of the entropic curvaturedimension condition and Bochner’s inequality on metric measure spaces. Invent. Math.
201 (2015), 993–1071.

[FIL]

M. Fathi, E. Indrei and M. Ledoux, Quantitative logarithmic Sobolev inequalities andstability estimates. Discrete Contin. Dyn. Syst. 36 (2016), 6835–6853.

[FG]

A. Figalli and N. Gigli, Local semiconvexity of Kantorovich potentials on non-compact
manifolds. ESAIM Control Optim. Calc. Var. 17 (2011), 648–653.

[GKKO] N. Gigli, C. Ketterer, K. Kuwada and S. Ohta, Rigidity for the spectral gap on
RCD(K, ∞)-spaces. Amer. J. Math. (to appear). Available at arXiv:1709.04017[Kl]

B. Klartag, Needle decompositions in Riemannian geometry. Mem. Amer. Math. Soc.
249 (2017), no. 1180.

[Le]

M. Ledoux, On an integral criterion and hypercontractivity of diﬀusion semigroups
and extremal functions. J. Funct. Anal. 105 (1992), 444–465.

[LV]

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport.
Ann. of Math. 169 (2009), 903–991.

[Ma1]

C. H. Mai, On Riemannian manifolds with positive weighted Ricci curvature of negative eﬀective dimension. Kyushu J. Math. (to appear). Available at arXiv:1704.06091[Ma2]

C. H. Mai, Rigidity for the isoperimetric inequality of negative eﬀective dimension on weighted Riemannian manifolds. Geom. Dedicata (to appear). Available atarXiv:1712.06904[Mc]

R. J. McCann, Polar factorization of maps on Riemannian manifolds. Geom. Funct.
Anal. 11 (2001), 589–608.

[Mo]

F. Morgan, Geometric Measure Theory. A beginner’s guide. Fifth edition. Elsevier/Academic Press, Amsterdam, 2016.

[MN]

E. Mossel and J. Neeman, Robust dimension free isoperimetry in Gaussian Space.
Ann. Probab. 43 (2015), 971–991.

12

[Oh1]

S. Ohta, Finsler interpolation inequalities. Calc. Var. Partial Diﬀerential Equations 36(2009), 211–249.

[Oh2]

S. Ohta, Splitting theorems for Finsler manifolds of nonnegative Ricci curvature. J.
Reine Angew. Math. 700 (2015), 155–174.

[Oh3]

S. Ohta, Needle decompositions and isoperimetric inequalities in Finsler geometry. J.
Math. Soc. Japan, 70 (2018), 651–693.

[OV]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links withthe logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000), 361–400.

[vRS]

M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy and Ricci curvature. Comm. Pure Appl. Math. 58 (2005), 923–940.

[St1]

K.-T. Sturm, Convex functionals of probability measures and nonlinear diﬀusions on
manifolds. J. Math. Pures Appl. 84 (2005), 149–168.

[St2]

K.-T. Sturm, On the geometry of metric measure spaces. I. Acta Math. 196 (2006),
65–131.

[St3]

K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006),
133–177.

[Vi]

C. Villani, Optimal transport, old and new. Springer-Verlag, Berlin, 2009.

13