Sharp estimates on the first eigenvalue of the $$p$$ p -Laplacian with negative Ricci lower bound

Abstract

We complete the picture of sharp eigenvalue estimates for the p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator ∆ p when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.

Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations

“…The result for non-negative Ricci curvature was proved previously (for any p > 1) by Valtorta [74], and the special case of convex domains in Euclidean space was proved independently in [43]. The result for arbitrary K was proved by Valtorta and Naber [67]. The methods outlined above can also be applied to a range of fully nonlinear equations satisfying certain concavity assumptions.…”

“…Nevertheless, the optimal lower bound for p-eigenvalues as a function of diameter and lower Ricci curvature bound for p > 2 can be proved using methods similar to the height-dependent gradient estimates discussed above. In this case, however, we apply the estimates not to the evolution equation but instead to the eigenfunction equation, and the argument is therefore much closer to that originally used in the works of [25,60,63,64] for p = 2 and by Valtorta in [74] and Valtorta and Naber in [67]. Indeed the key gradient estimate we prove is essentially the same as in [67], but the argument is made much easier by the use of multi-point maximum principles rather than direct estimation of the gradient.…”

“…The optimal lower bound on the p-eigenvalue for given diameter then follows, since it was shown in [67] that the lowest eigenvalue for a one-dimensional p-eigenfunction with given diameter occurs in the case where the range is symmetric.…”

“…In this setting (as in [25,67,74]) it is useful to expand our class of 'onedimensional' solutions to include all 'warped product' solutions in the model spaceM K : For given K ∈ R and h > 0, let Σ h be a hypersurface with constant principal curvatures equal to h in the model spaceM K of constant sectional curvature K, and let ν(z) be the outward unit normal to…”

First eigenvalue of thep-Laplacian under integral curvature condition

“…The first nontrivial Neumann eigenvalue for M is given by Though the regularity theory of the p-Laplacian is very different from the usual Laplacian, many of the estimates for the first eigenvalue of the Laplacian (when p = 2) can be generalized to general p. Matei [11] generalized Cheng's first Dirichlet eigenvalue comparison of balls [5] to the p-Laplacian. For closed manifolds with Ricci curvature bounded below by (n − 1)K, Matei for K > 0 [11], Valtora for K = 0 [17] and Naber-Valtora for general K ∈ R [12] give a sharp lower bound for the first nontrivial eigenvalue. Andrews-Clutterbuck [1], [2] also gave a proof using modulus of continuity argument.…”