Li-Yau Multiplier Set and Optimal Li-Yau Gradient Estimate on Hyperbolic Spaces
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Li-Yau Multiplier Set and Optimal Li-Yau Gradient Estimate on Hyperbolic Spaces Chengjie Yu1 · Feifei Zhao2 Received: 6 November 2019 / Accepted: 15 October 2020 / © Springer Nature B.V. 2020
Abstract In this paper, motivated by finding sharp Li-Yau-type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature lower bound, we first introduce the notion of Li-Yau multiplier set and show that it can be computed by heat kernel of the manifold. Then, an optimal Li-Yau-type gradient estimate is obtained on hyperbolic spaces by using recurrence relations of heat kernels on hyperbolic spaces. Finally, as an application, we obtain sharp Harnack inequalities on hyperbolic spaces. Keywords Heat equation · Li-Yau-type gradient estimate · Heat kernel Mathematics Subject Classification (2010) Primary 35K05 · Secondary 53C44
1 Introduction The Li-Yau [18] gradient estimate: nα 2 k nα 2 + (1.1) 2t 2(α − 1) for positive solution u of the heat equation on complete Riemannian manifolds with Ric ≥ −k and k a nonnegative constant is of fundamental importance in geometric analysis. Here α is any constant greater than 1. On complete Riemannian manifolds with nonnegative Ricci curvature, by letting α → 1+ in Eq. 1.1, one has n ∇ log u2 − (log u)t ≤ . (1.2) 2t ∇ log u2 − α(log u)t ≤
Research partially supported by an NSF project of China with contract no. 11571215. Feifei Zhao
[email protected] Chengjie Yu [email protected] 1
Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China
2
School of Liberal Arts and Sciences, North China Institute of Aerospace Engineering, Langfang, 065000, China
C. Yu, F. Zhao
This estimate is sharp where the equality can be achieved by the fundamental solution of Rn . However, Eq. 1.1 is not sharp when k > 0. Finding sharp Li-Yau-type gradient estimate for k > 0 is still an unsolved problem (see [8, P. 393]). This is the motivation of this paper. We will assume that k > 0 without further indications in the rest of this paper. Li-Yau-type gradient estimates are important since they give Harnack inequalities immediately by taking integration on geodesics. Many authors have obtained Li-Yau-type gradient estimates in various forms or various settings. For example, in [14], Hamilton obtained a Li-Yau-type gradient estimate in matrix form, and in [15], Hamilton obtained a Li-Yau-type gradient estimate in matrix form for Ricci flow. Hamilton’s works were extended to the K¨ahler category by Cao-Ni [5] and Cao [4], and further extended to (p, p)forms on K¨ahler manifolds by Ni and Niu [19]. Recently, in [30], the authors extended the Li-Yau-type gradient estimate to metric measure spaces, and in [7, 28, 29], the authors obtained Li-Yau-type gradient estimates under integral curvature assumptions. Some other Li-Yau-type gradient estimates can be found in [1–3, 6, 9, 16, 20, 22–24]. Here, we only mention some of them that are more related to the topic of this paper and compare them. A slight improvement of Eq. 1
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