Sharp gradient estimate, rigidity and almost rigidity of Greenfunctions on non-parabolic RCD(0, N ) spaces
Yuanlin Peng
June 24, 2024
My talk is based on a recent work [HP24] by Prof. Honda and myself. Our main results
include a sharp gradient estimate, a rigidity theorem and an almost rigidity theorem forGreen’s function on a special class of metric measure spaces, namely RCD spaces.

1

Green’s function in the smooth framework

The Green’s function (at the pole x) is the minimal positive solution to the heat equation∆Gx = −δx , where δx denotes the Dirac distribution centered at x. As a simplest example,
the Green’s function of N -dimensional Euclidean space (N > 2) is:NGRx (y) =
1

log |x − y|, N = 2 −2π|x − y|2−N,N (N − 2)ωN(1.1)

N >3

The notion of Green’s function is also valid for other spaces. Another well-known example
is that on Riemannian manifolds. For an N -dimensional complete Riemannian manifold(M N , g) (N > 2) with non-negative Ricci curvature, the existence of the (global) Greenfunction G is equivalent to the following non-parabolic assumption (See [V81] by Varapoulosfor the details):
Z ∞rdr < ∞, ∀x ∈ M N .
(1.2)
VolB(x)1r
Although for such manifolds it is in general not easy to determine the global Green’s
function explicitly, at least it is known that asymptotically it is close to the Euclidean case:

G(MxN ,g)=

1
log dx + o(− log dx ), N = 2
 −


2π
dx2−N


+ o(d2−N),xN (N − 2)ωN(1.3)

N >3

where dx denotes the distance function from x. This indicates that the function
1bx := Gx2−N(1.4)

can be regarded as a counterpart of the distance function dx up to a dimensional constantwhen N > 3. The following result by Colding supports our observation:
Theorem 1.1 (Theorem 3.1 of [C12]). Let (M N , g) be an N -dimensional (N > 3), complete
and non-parabolic Riemannian manifold with non-negative Ricci curvature and let x ∈ M N .
Then we have the following.

1. (Sharp gradient estimate) We have1|∇bx |(z) 6 (N (N − 2)ωN ) N −2(1.5)

for any z ∈ M N \ {x}.
2. (Rigidity) (M N , g) is isometric to the N -dimensional Euclidean space RN with
1bx = (N (N − 2)ωN ) N −2 dx if the equality of (1.5) holds for some z ∈ M N \ {x}.
Our work extends the above result to metric measure spaces with non-negative Ricci
curvature and a dimension upper bound N > 2 (where N is not necessarily an integer) insome synthetic sense, namely RCD(0, N ) spaces.

2RCD spaces and Green’s function thereon

Here we only give a rough definition of RCD spaces. We say a triple (X, d, m) is anRCD(K, N ) space for K ∈ R and N ∈ [1, ∞] if1) (X, d) is a complete and separable metric space, and m is a locally finite measure on(X, d) with full support.
2) The volume growth is exponentially bounded
2m(Br (x)) 6 C1 eC2 r , ∀x ∈ X.

(2.1)

3) The Sobolev space W 1,2 (X, d, m) is a Hilbert space so that first and second order
calculus is well-posed.
4) The Sobolev-to-Lipschitz property is satisfied, namely if there is a function f ∈W 1,2 (X) with |∇f | 6 1, then f admits a 1-Lipschitz representative in m-a.e. sense.
5) The (K, N )-Bochner’s inequality holds for all nice functions f and ϕ such that bothsides make sense12(∆f )2ϕ∆ϕ|∇f |2 dm 6+ h∇f, ∇∆f i + K|∇f |2NX
XZZ!

dm.

(2.2)

To seek for the more precise statement or other equivalent definitions of RCD spaces,
we refer to, for instance, [EKS15] by Erbar-Kuwada-Sturm. It is easily checked that all
Riemannian manifolds with lower Ricci curvature bound K and upper dimension boundN (equipped with volume measure induced by the Riemannian metric) are RCD(K, N )spaces.
Moreover, the RCD framework also covers weighted Riemannian manifolds. That is,consider a Riemannian manifold (M n , g) and f ∈ C ∞ (M N ). For any N > n, if
Ricg,N := Ricg + Hessf −df ⊗ df> Kg,N −n(2.3)

then the metric measure space (M N , dg , e−f Volg ) satisfies RCD(K, N ) conditions. Besides,
some other well-studied classes of metric spaces, for instance, Alexandrov spaces with lower
curvature bound and Ricci limit spaces, are also examples of RCD spaces.

Similarly as in the Riemannian manifold case, the global Green’s function can also be
defined on an RCD(0, N ) space (X, d, m) if it satisfies the non-parabolic assumption
Z ∞0r
dr < ∞, ∀x ∈ X.
m(Br (x))(2.4)

See [BS19] by Bruè-Semola for the proof. This makes it possible to consider the generalization of Theorem 1.1 to non-parabolic RCD(0, N ) setting. However, there are still somedifficulties• Functions on RCD spaces are defined only in the almost-everywhere sense a priori
due to the non-smooth nature of the space. Thus the pointwise estimate like (1.5)
does not make sense.
• For the rigidity part, counterexamples are known where bx is proportional to thedistance function, but the space itself is not Euclidean. For instance, consider theweighted half line
(X, d, m) := ([0, ∞), dEucl , rN −1 dr),
(2.5)
thenG0 (r) =r2−N.
N −2(2.6)

In our work, we successfully overcome these difficulties, and establish the sharp gradientestimate and rigidity result for Green’s function on non-parabolic RCD(0, N ) spaces.
Besides, given that the class of RCD(K, N ) spaces enjoys the sequential compactness in
Gromov-Hausdorff topology, we also showed a quantitative version of the rigidity result.

3

Main results and outline of the proof

Consider a non-parabolic RCD(0, N ) space (X, d, m) and a base point x ∈ X. We firstshow that the pointwise supremum of |∇bx | is well-defined by finding a good representativeof it,
Theorem 3.1. There exists a representative of |∇bx |, say |∇bx |∗ , with|∇bx |∗ (y) = lim sup |∇bx |∗ (z) < ∞, ∀y ∈ X.

(3.1)

z→y

This can done by the potential theory for metric measure spaces. Indeed, by thecalculus rules and the Bochner’s inequality, we can show that |∇bx |2 Gx is subharmonic,
which allows us to prove that|∇bx |∗ (y) := lim ess sup |∇bx |(z)r↓0 z∈Br (y)(3.2)

is a representative of |∇bx | satisfying the desired property. Again thanks to the subharmonicity of |∇bx |2 Gx , we can prove the sharp gradient estimate:
Theorem 3.2 (Sharp gradient estimate). If x ∈ X satisfies
m(Br (x))< ∞,r↓0rN

νx := limthen(3.3)

1|∇bx |∗ 6 CN νxN −2 ,1where CN := (N (N − 2)) N −2 is the sharp constant.

(3.4)

Note that here νx is in place of ωN in (1.5). Actually, νx somehow reflects the shapeof tangent spaces. For a Riemannian manifold, since every point admits a unique N dimensional Euclidean tangent space, νx ≡ ωN . But the case is more complicated for RCDspaces. This also gives hint to the second difficulty we mentioned above. That is, weshould consider the rigidity in comparison with the tangent spaces at x instead of merely
the Euclidean spaces. In view of this, we have
Theorem 3.3 (Rigidity). If1|∇bx |∗ (z) = CN νxN −2(3.5)

for some z ∈ X \ {x}, then there exists a compact RCD(N − 2, N − 1) space (Y, dY , mY )such that (X, d, m) is isomorphic to the metric measure cone (C(Y ), dC(Y ) , mC(Y ) ) built
over Y , whereC(Y ) := Y × [0, ∞)/Y × {0},(3.6)
dC(Y ) (y1 , r1 ), (y2 , r2 ) :=q

r12 + r22 − 2r1 r2 cos dY (y1 , y2 ) ,dmC(Y ) (y, r) := rN −1 dmY (y) ⊗ dL1 (r).

(3.7)
(3.8)

1

In particular, |∇bx |∗ ≡ CN νxN −2 .
This result can be proved as a corollary of Gigli-Violo’s functional rigidity theorem([GV23, Theorem 5.1]) combining with the subharmonicity of |∇bx |2 Gx and the sharp
gradient estimate.
Combining the results above with the compactness of non-parabolic RCD(0, N ) spaces
with respect to the pointed-measured-Gromov-Hausdorff (pmGH) topology, we obtain thefollowing almost rigidity result.
Theorem 3.4 (Almost rigidity). For all N > 2, 0 < ε < 1, v > 0, 0 < r < R < ∞and ϕ ∈ L1 ([0, ∞), L1 ) there exists δ := δ(N, ε, v, r, R, ϕ) > 0 such that, if a pointed
non-parabolic RCD(0, N ) space (X, d, m, x) satisfies νx 6 v,s6 ϕ(s),m(Bs (x))
andfor L1 -a.e. s ∈ [1, ∞),(3.9)

1CN νxN −2 − |∇bx |∗ (z) 6 δ(3.10)

holds for some z ∈ BR (x) \ Br (x), then (X, d, m, x) is ε-pmGH close to the N -metricmeasure cone over an RCD(N − 2, N − 1) space.
It is worth mentioning that this result is even new restricted to the Riemannian manifold
setting:
Corollary 3.5. For all N > 3, 0 < ε < 1, 0 < r < R < ∞ and ϕ ∈ L1 ([0, ∞), L1 ) there
exists δ := δ(N, ε, r, R, ϕ) > 0 such that if a pointed non-parabolic Riemannian manifold(M N , g, x) with non-negative Ricci curvature satisfiess
6 ϕ(s),Volg (Bs (x))
andfor L1 -a.e. s ∈ [1, ∞)(3.11)

1

CN ωNN −2 − |∇bx |∗ (z) 6 δ(3.12)

holds for some z ∈ BR (x) \ Br (x), then (M N , g, x) ε-pGH is close to the N -dimensional
Euclidean space.

References[BS19]

E. Bruè, D. Semola: Constancy of dimension for RCD∗ (K, N ) spaces via
regularity of Lagrangian flows. Comm. Pure and Appl. Math. 73 (2019), 11411204[C12]

T. H. Colding New monotonicity formulas for Ricci curvature and applications.
I., Acta. Math. (2) 209 (2012), 229-263.

[EKS15] M. Erbar, K. Kuwada, K.-T. Sturm: On the equivalence of the entropic
curvature-dimension condition and Bochner’s inequality on metric measure spaces.
Invent. Math. 201 (2015), 993–1071.
[GV23]

N. Gigli, I. Y. Violo: Monotonicity formulas for harmonic functions inRCD(0, N ) spaces, J. Geom. Anal. 33 (2023), https://doi.org/10.1007/s12220022-01131-7[HP24]

S. Honda, Y. Peng Sharp gradient estimate, rigidity and almost rigidity ofGreen functions on non-parabolic RCD(0, N ) spaces, Proceedings of the Royal
Society of Edinburgh Section A: Mathematics (2024), 1–54.

[V81]

N. Varopoulos: The Poinsson Kernel on positively curved manifolds, J. Funct.
Anal. 44 (1981), 359–380.