Sharp spectral gap and Li–Yau’s estimate on Alexandrov spaces

Abstract

In the previous work [35], the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen-Wang [9, 10] and Bakry-Qian [6], from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp) Li-Yau's estimate for positve solutions of heat equations on Alexandrov spaces.

Local Li–Yau’s estimates on RCD $$^*(K,N)$$ ∗ ( K , N ) metric measure spaces

“…There is a rich literature on extensions and improvements of the Li-Yau inequality, both the local version (1.1) and the global version (1.2), to diverse settings and evolution equations, 1 for example, in the setting of Riemannian manifolds with Ricci curvature bounded below [15,9,47,33,32], in the setting of weighted Riemannian manifolds with Bakry-Emery Ricci curvature bounded below [12,35,43,7] and some non-smooth setting [10,44], and so on.…”

Obata’s Rigidity Theorem for Metric Measure Spaces

“…proven by Qian, Zhang and Zhu [23]. They use a di erent notion of generalized lower Ricci curvature bound that implies a sharp curvature dimension condition and is inspired by Petrunin's second variation formula [22].…”

Gradient estimates for heat kernels and harmonic functions

“…These two gradient estimates are fundamental tools in geometric analysis and related fields, and there have been many efforts afterwards to generalise them to different settings, see for instance [34,38,39,40,44,52,67,73,82,90,91,95,113,114,115]. Let us review some of these generalisations.…”

“…Example 2. On an n-dimensional conical manifold with compact basis N without boundary, C(N) := R + × N, let λ 1 be the smallest nonzero eigenvalue of the Laplacian on the basis (see [29,90] for studies on the first eigenvalue). By a result of Li [81], the Riesz transform is bounded on L p (C(N)) for all p ∈ (1, p 0 ) and not bounded for p ≥ p 0 , where…”