p-Fundamental tone estimates of submanifolds with bounded mean curvature

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

“…Geometric estimates on fundamental tones and the bottom of the spectrum of the Laplacian on Riemannian manifolds have been obtained in various contexts (see, e.g., [6,7,8,13,22] for p = 2 and φ constant, [16,23] for p ∈ (1, ∞) and φ constant) and for other results [1,2,17,18,21,25,33] and [3,14,19,22,24,32,34]. In this paper we present generalisations of some of these results to the setting of smooth metric measure spaces.…”

“…In this final section we utilise Theorem 2.2 to give lower bounds on the first eigenvalue of the weighted p-Laplacian on geodesic balls whose radius does not exceed the injectivity radius. To this end let us first recall the following Hessian comparison theorem (see [12] or [6,16,23]).…”

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

“…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.…”

“…(3. 19) Let B r (x) be a geodesic ball centered at x with radius r in (M, g, φ). By the variational characterization discussed above we have…”

“…See also similar results [26,35,36,39] for the p-Laplacian or drifting Laplacian eigenvalue problems. For applications, following closely the idea of Hessian comparison estimates applied in [11,19,25] on different problems, we discuss lower bound estimates for the first eigenvalues on the weighted geodesic ball whose radius does not exceed the injectivity radius and submanifolds having bounded weighted mean curvature. In fact, our estimates extend the results of [11,25,38] when the supremum of the sectional curvature inside the ball is nonnegative.…”