[Paper Review] Critical set of eigenfunctions of the Laplacian

Laurent Bakri|arXiv (Cornell University)|Aug 10, 2010

Mathematical Dynamics and Fractals14 references3 citations

TL;DR

This paper establishes an upper bound of $ C\sqrt{\lambda} $ for the $(n-1)$-dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact real analytic Riemannian manifolds. Using a Carleman-type inequality and a novel doubling estimate for the gradient of eigenfunctions, the authors adapt techniques from Donnelly and Fefferman’s work on nodal sets to show that the critical set, like the nodal set, grows at most like $ \sqrt{\lambda} $, confirming a conjectured bound analogous to Yau’s conjecture for nodal sets.

ABSTRACT

We give an upper bound for the $(n-1)$-dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact analytic Riemannian manifolds. This is the analog of H. Donnely and C. Fefferman result on nodal set of eigenfunctions.

Motivation & Objective

• To establish an upper bound on the $(n-1)$-dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact real analytic Riemannian manifolds.
• To extend the method of Donnelly and Fefferman, originally developed for nodal sets, to the critical set of eigenfunctions.
• To show that the critical set's measure grows at most like $ \sqrt{\lambda} $, analogous to Yau’s conjecture for nodal sets.
• To provide a sharp, optimal bound in the analytic setting, resolving a gap in understanding of critical sets compared to nodal sets.

Proposed method

• Derive a Carleman estimate for the operator $ \Delta + W $ with $ W \in \mathcal{C}^1 $, using a weight function $ \phi(x) = -f(\ln r(x)) $ with $ f(t) = t - e^{\varepsilon t} $.
• Use the compactness of $ M $ and the Carleman estimate to deduce a doubling inequality for $ \|\nabla u\|_{B_{2r}} \leq e^{C\sqrt{\lambda}} \|\nabla u\|_{B_r} $, valid for eigenfunctions of the Laplacian.
• Define $ F(x) = \sum_{i=1}^n \left| \frac{\partial u}{\partial x_i} \right|^2 $, whose zero set is the critical set $ \mathcal{C}_u $, and extend $ F $ to a holomorphic function on a complex ball.
• Apply a complex analytic result (Theorem 4.1 of Donnelly and Fefferman) to bound the Hausdorff measure of the zero set of $ F $ in terms of the doubling constant.
• Use a covering argument over the compact manifold $ M $, applying the local bound $ \mathcal{H}^{n-1}(\mathcal{C}_u \cap B(p,r_0)) \leq C\sqrt{\lambda} $ uniformly across finitely many balls.

Experimental results

Research questions

• RQ1 Can the critical set of eigenfunctions on analytic manifolds be bounded in terms of the eigenvalue $ \lambda $?
• RQ2 Does the critical set’s $(n-1)$-dimensional Hausdorff measure grow at most like $ \sqrt{\lambda} $, as conjectured by Yau for nodal sets?
• RQ3 Can the method used for nodal sets by Donnelly and Fefferman be adapted to control the critical set via gradient doubling?
• RQ4 Is there a sharp upper bound for the critical set measure in the analytic category, and how does it compare to the known bounds for nodal sets?
• RQ5 What role does the analyticity of the metric play in obtaining such a bound?

Key findings

• The $(n-1)$-dimensional Hausdorff measure of the critical set $ \mathcal{C}_u $ satisfies $ \mathcal{H}^{n-1}(\mathcal{C}_u) \leq C\sqrt{\lambda} $ for some constant $ C > 0 $ depending only on the manifold $ M $.
• The doubling inequality $ \|\nabla u\|_{B_{2r}} \leq e^{C\sqrt{\lambda}} \|\nabla u\|_{B_r} $ is established as a key technical tool, derived from a Carleman estimate.
• The function $ F(x) = \sum_{i=1}^n |\partial_i u|^2 $, whose zero set is the critical set, satisfies a complex doubling property: $ \sup_{B_{\mathbb{C}}(1)} |F| \leq e^{C\sqrt{\lambda}} \sup_{B(1/2)} |F| $.
• The local measure bound $ \mathcal{H}^{n-1}(\mathcal{C}_u \cap B(p,r_0)) \leq C\sqrt{\lambda} $ is proven for each point $ p \in M $, with $ r_0 $ independent of $ \lambda $.
• The global bound follows from a finite covering of $ M $, since $ M $ is compact, and the local bound is uniform across the manifold.
• The result is sharp in the sense that there exist examples (e.g., $ f_k(x,y) = \sin(2\pi kx) $ on $ \mathbb{T}^1 \times N $) where $ \mathcal{H}^{n-1}(\mathcal{C}_{f_k}) \geq C\sqrt{\lambda} $, showing the bound is optimal.