Geometric estimates on weighted p-fundamental tone and applications to the first eigenvalue of submanifolds with bounded mean curvature

Abstract

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Gradient Estimates for a Weighted Γ-nonlinear Parabolic Equation Coupled with a Super Perelman-Ricci Flow and Implications

“…For background and further discussion relating to the material covered here the reader is referred to the texts [10,18,27,30,34,36,42] and the references therein. For recent results and literature on gradient estimates see [3, 8, 13, 15, 19-21, 31, 35, 37, 39, 40] and for closely related topics see [1,2,7,11,14,16,23,25,26,28,32,38].…”

Weighted Cheeger constant and first eigenvalue lower bound estimates on smooth metric measure spaces

“…In some particular cases, it may be possible to show that the equality in the last bound is achieved, for instance, in the Euclidean ball setting. The reason for introducing h( ) (see [6]) is to obtain lower bounds for λ 1,p ( ) in terms of any lower bound for h( ). As for a lower bound on h, we note that if ∇φ ≤ c for some constant c ≥ 0, then for any vector field V ∈ X ( ), we have the bound…”

“…Note that Cheung-Leung's result [18] can be regarded as an extended version of McKean lower bound [28]. McKean's lower bound for weighted p-fundamental tone has been recently studied by the first author on a complete noncompact simply connected smooth metric measure space with sectional curvature bounded from above by negative constant [2,6]. Theorem 3.1 extends the result of Zhang and Zhao [38,Theorem 3.1], where they considered the first eigenvalue λ 1 (N) for buckling problem on an n-dimensional compact connected Riemannian manifold N with smooth boundary and Ric(N) ≥ K > 0 and obtained…”

“…First, we state an important lemma on Hessian comparison (see [16] for details), which will help realize the desired estimates. This idea has been applied previously by several authors [6,11,19,38] to different settings.…”

“…From (3.18) the following lower bounds with respect to Laplacian for the distance function ρ and squared distance function ρ 2 can be deduced (see [6,11,38]): ρ(x) ≥ (m -1)μ ρ(x) , ρ 2 (x) ≥ 2(m -1)ρ(x)μ ρ(x) + 2.…”