{'fraction_non_alphanumeric': 0.055033126736482155, 'fraction_numerical': 0.01570848471895704, 'mean_word_length': 4.117003827227993, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The rigidity and stability of gradient estimates

In this note, we obtain the rigidity of the sharp Cheng-Yau gradient estimate for positive harmonic functions on surfaces with nonegative Gaussian curvature, the rigidity of the sharp Li-Yau gradient estimate for positive solutions to heat equations and the related estimates for Dirichlet Green's functions on Riemannian manifolds with nonnegative Ricci curvature. Moreover, we also obtain the corresponding stability results.


Introduction
The connection between harmonic functions and geometry can be traced back to the proof of the uniformization theorem of Riemann surfaces. For manifolds with non-negative Ricci curvature, one important result relying on harmonic functions, is Cheeger-Gromoll's splitting theorem [CG71]. Its proof reduces to the fact: the existence of a nontrivial harmonic function with constant norm of gradient results in the splitting of the manifold. This possibly can be viewed as the first geometric application of gradient estimates for harmonic functions. Yau's Liouville theorem [Yau75] inspired a lot of studies in harmonic functions on Riemannian manifolds with non-negative Ricci curvature. Especially, Cheng-Yau [CY75] proved the local gradient estimate of harmonic functions, which is not only important as a technical tool, but also provide a universal method to deal with similar analytical objects such as Green's function, heat kernel [LY86].
Recall in [Col12], for the global positive Green's function G on a complete Riemannian manifold with nonnegative Ricci curvature, let b = G 1 2−n , we have |∇b| ≤ 1, and M n is isometric to R n if the equality holds at some point in M n . This result also implies: If G and G are the global positive Green's functions on M and R n with n ≥ 3, then G(p, y) ≥ G(0,ȳ), where d(p, y) = d(0,ȳ). Moreover if the equality holds at some point, then M is isometric to R n . Hence, for the global positive Green's function, the sharp estimate and the related rigidity are known.
The philosophy of the above rigidity result is: If the sharp bounds of some estimates for certain kinds of functions (gradient estimates of positive harmonic functions, positive solutions to heat equations or Green's functions) are achieved on some manifold M n , then M n is isometric to the model space.
In the case of nonnegative curvature, the model space is usually R n .
Besides the isometric rigidity, due to Gromov [Gro99], we can discuss the stability of manifolds (in other words, the Gromov-Hausdorff distance between the manifolds and the model spaces) when the estimate is close to be sharp.
In this note, we consider rigidity and stability for the Cheng-Yau gradient estimate of positive harmonic functions on surfaces with nonnegative Gaussian curvature, Li-Yau gradient estimates for positive solutions to heat equations and some estimates of Dirichlet Green's functions on Riemannian manifolds with Rc ≥ 0. For the stability part, we only focus on local stability, more precisely, the stability of geodesic balls in the manifolds. Our results reveal that the geometry of the domain can be rediscovered by the properties of suitable functions defined on it.
The rest of the paper is organized as follows:We first review some preliminary results in Section 2, which will be used to establish the rigidity and stability in later sections. Although the Cheng-Yau gradient estimate for positive harmonic functions is sharp up to decay rate in global sense, it is not sharp pointwisely. The sharp Cheng-Yau gradient on geodesic balls is only known in 2-dim case (see [Xu20]). In this note, we obtain the related rigidity and stability for geodesic balls on surfaces in Section 3. It is well-known that Li-Yau's parabolic gradient estimate is sharp as a global inequality on the whole manifold when Ricci curvature is nonnegative. We will also establish the rigidity of Li-Yau's estimate in Section 3, which can be viewed as a parabolic analogue of Colding's rigidity result about global Green's functions. Finally, inspired by Colding's rigidity result of global Green's functions, in Section 4, we study the corresponding rigidity result for Dirichlet Green's functions in geodesic balls, and obtain the stability result for Dirichlet Green's functions.

Preliminary for local rigidity and stability
In this section, we recall some preliminary results that will be used in latter sections.
The following lemma is well-known in Riemannian geometry. We present it here for completeness.
for any x ∈ Ω \ {p} and Ω contains no cut-point of p, then Ω is isometric to a star-shaped domain Ω in R n .
The following theorem follows from [CC96], which will be used in the rest of the paper.
From [CC96, Theorem 6.33], we have the following existence of cut-off functions.
Finally, by reducing to Theorem 2.1, we have the following local stability for Riemannian surfaces in terms of ∆r.
Lemma 2.3. On a complete Riemannian surface (M 2 , g) with Rc ≥ 0, let Proof: By Hessian comparison, 2g − ∇ 2 r 2 is nonnegative. So Then, the conclusion follows from Theorem 2.1 directly.

Rigidity and stability of Cheng-Yau and Li-Yau estimate
In this section, we prove the rigidity and stability result for Cheng-Yau gradient estimates on surfaces, and the rigidity of Li-Yau gradient estimate.
We first prove the rigidity of the sharp Cheng-Yau gradient for complete Riemannian surfaces with nonnegative Ricci curvature. In fact, we also provide a simpler proof of [Xu20, Theorem 3.4].
Theorem 3.1. Let (M 2 , g) be a complete Riemannian surface with Rc ≥ 0 and u be a positive harmonic function on B 1 (p). Then the follows hold: (1) . Let r(x) = d(p, x), Remark 3.1. The inequality (3.1) was proved in [Xu20] by a carefully chosen cut-off function. Here, we firstly give a more direct proof of (3.1) without using cut-off functions, and then use the strong maximum principle to get the rigidity result.
Proof: (1). It was shown in [Xu20] that for any positive harmonic function u on B 1 (p) in a complete Riemannian surface (M 2 , g) with nonnegative Ricci curvature, one has By Laplacian comparison, We claim that v ≤ v ǫ on B 1−ǫ (p). Then, by letting ǫ → 0 + , we will get (1). If the claim is not true, then there is . If x 0 is not a cut-point of p, then, by (3.3) and (3.4), which is a contradiction. When x 0 is a cut-point of p, the classical Calabi trick (see [Cal58]) will also give us a contradiction similarly. This complete the proof of (1). ( It is then clear that Ω is nonempty and closed in B 1 (p) by continuity. Moreover, for any in B δ (x 0 ). Since c(x) > 0, by the strong maximum principle (see [Cal58] or [ is smooth in B 1 (p) which implies that there is no cut-point of p in B 1 (p) (If there is a cut-point q of p in B 1 (p), then q is not a conjugate point of p since B 1 (p) is flat. So, there are at least two different minimal geodesics joining p to q which contradicts that ∇r is well defined at q since r is smooth at q ). Now, by Lemma 2.1, B 1 (p) is isometric to the ball B 1 (0) in R 2 . Finally by [Xu20], u is the Poisson kernel.
(3). It is straightforward to verify that So, by (3.3), Let φ be the cut-off function in Lemma 2.2, and note that 4 − ∆(r 2 ) ≥ 0 by Laplacian comparison and w − v ≥ 0 by (1). Then Hence, Now the conclusion follows from the above inequality and Lemma 2.3. Next, we come to prove the rigidity for the Li-Yau gradient estimate on complete Riemannian manifolds with nonnegative Ricci curvature. We first recall the following Li-Yau's theorem (see [LY86]).
The rigidity result of the Li-Yau gradient estimate on complete Riemannian manifolds with nonnegative Ricci curvature was obtained in [Ni04], we present another proof here. Proof: By Theorem 3.2, we get and Combining the above with Lemma 2.1, we get that M n is R n , and where p(t) ∈ R n . Substituting (3.9) into the equation of f : Hence d dt p(t) = 0 and d dt f (p(t), t) = n 2t . Thus p(t) = p is a constant point and f (x, t) = |x−p| 2 4t + n 2 ln t + C for some constant C ∈ R. Therefore u = e − f is a multiple of the heat kernel.
As a direct consequence of the rigidity for Li-Yau gradient estimate, we have the following rigidity for Li-Yau's sharp Harnack inequality of positive solutions to the heat equation on complete Riemannian manifolds with nonnegative Ricci curvature (See [LY86]). For Dirichlet Green's functions, we have the following rigidity result related to the gradient of the Dirichlet Green functions, which only applies to higher dimensional case (the dimension is n ≥ 3). |∇b(y)| ≥ nω n V (∂B 1 (p)) .
Proof: Note for y ∈ ∂B 1 (p), we have ∂b(y) ∂ n y = nω n ∂G(p, y) ∂ n y , where ∂ n y is unit inward normal vector at y. Recall that ∂G(p, y) ∂ n y ≥ nω n .
The first conclusion then follows. From the rigidity part of the Bishop-Gromov's volume comparison theorem, we get the second conclusion.
Note that we do not have the global rigidity result as [Col12] for the case n = 2 because of the following example.
(2). If the equality of (1) holds at some point, then G(p, y) = G(y) for y ∈ B 1 (p)\{p} by the strong maximum principle. Thus B 1 (p) is isometric to B 1 (0) ⊂ R n by Lemma 2.1.
So, 2n − ∆ r 2 = 2nω n r n ∆ G − G . Now let φ be the cut-off function as in Lemma 2.2. Then, by the Laplacian comparison, and noting that ∆G ≥ 0, Then, the conclusion follows from Lemma 2.3 and the above estimate.