Global Lower Bounds on the First Eigenvalue for a Diffusion Operator
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Global Lower Bounds on the First Eigenvalue for a Diffusion Operator Liangdi Zhang1 Received: 19 August 2019 / Revised: 3 January 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract We derive global lower bounds for the first eigenvalue of a symmetric diffusion X := − ∇ X on Riemannian manifolds with the Bakry–Émery–Ricci curvature bounded from below. Keywords First eigenvalue · Diffusion operator · Bakry–Émery–Ricci curvature Mathematics Subject Classification 35P15 · 53C21
1 Introduction Let (M n , g) be an n-dimensional Riemannian manifold and X be a smooth vector field on M n . The diffusion operator X := + ∇ X
(1.1)
is an important generalization of the Laplacian operator , in particular, the Witten– Laplacian f := − ∇∇ f
(1.2)
is a special case of (1.1) by taking X = −∇ f for some f ∈ C ∞ (M n ).
Communicated by Young Jin Suh.
B 1
Liangdi Zhang [email protected] Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
123
L. Zhang
As in [1,3,9], the m-dimensional Bakry–Émery–Ricci curvature of the diffusion operator X is defined as Ricm X := Ric −
1 X⊗X LXg − 2 m−n
(1.3)
for any number m ∈ (n, ∞), where L X stands for the Lie derivative along the direction X . In particular, the m-dimensional Bakry–Émery–Ricci curvature of the Witten– Laplacian operator f is defined as Ricmf := Ric + H ess f −
df ⊗df , m−n
(1.4)
where H ess f is the Hessian of f . For m-dimensional Bakry–Émery–Ricci curvatures, we can allow m to be infinite: 1 L X g, 2
(1.5)
Ric∞ f := Ric + H ess f ,
(1.6)
Ric∞ X := Ric − and
which are called the ∞-dimensional Bakry–Émery–Ricci curvature of the diffusion operator X (of the Witten–Laplacian operator), respectively. We refer the readers to [5,6,11,12,15,16] for applications of Bakry–Émery–Ricci curvatures. There are several well-known results on lower bound estimates for the first eigenvalue of Laplacian operator on closed Riemannian manifolds (see Section 5 of [7] for a summary): Lichnerowicz [10] (see also [14]) showed that the first nonzero eigenvalue of the Laplacian on a closed manifold must satisfy λ1 ≥ m K if the Ricci curvature is bounded from below by (m − 1)K . When the Ricci curvature is nonnegative, Li–Yau π2 [8] proved that λ1 ≥ (1+a)d 2 , where 0 ≤ a < 1 is a constant and d is the diameter of the underlying closed manifold. More generally, they [8] also derived a lower bound estimate that depends on the lower bound of the Ricci curvature, the upper bound of the diameter, and the dimension of the manifold alone. In this note, we will prove that these results are still valid for the first eigenvalue of the diffusion operator X on the closed manifold M n under the condition of the Barky–Émery–Ricci curvature bounded from below. Wu [17,18] established upper bounded first nontrivial eigenvalue for the Witten– Laplacian under the condition that the m-dimensional (∞-dimensional) Barky– Émery–Ricci curvature bounded below, respectively. We will consider global lower bound
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