Lower Bounds for Nodal Sets of Eigenfunctions

We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions. 0. Introduction Let M be a smooth closed Riemannian manifold and the Laplace operator. A function u is said to be an eigenfunction with eigenvalue λ if u = −λ u. (0.1) With our convention on the sign of , the eigenvalues are non-negative and go to infinity. One of the most fundamental questions about eigenfunctions is to understand the sets where they vanish; these sets are called nodal sets. Nodal sets are (n − 1)-dimensional manifolds away from (n−2)-dimensional singular sets where the gradient also vanishes, so it is natural to estimate the (n − 1)-dimensional Hausdorff measure Hn−1. The main result of this short note is the following lower bound: Theorem 1. Given a closed n-dimensional Riemannian manifold M, there exists C so that Hn−1({u = 0}) ≥ C λ 3−n 4 . (0.2) In particular, Theorem 1 gives a uniform lower bound in dimension n = 3. In [Y], S.-T. Yau conjectured the lower bound C √ λ in all dimensions. This was proven for surfaces by Bruning, [B], and Yau, independently, and for real analytic metrics by Donnelly and Fefferman in [DF], but remains open in the smooth case. The authors were partially supported by NSF Grants DMS 0606629, DMS 0906233, and NSF FRG grants DMS 0854774 and DMS 0853501. 778 T. H. Colding, W. P. Minicozzi II The Donnelly-Fefferman argument leads to exponentially decaying lower bounds in the smooth case; see also [HL]. In [M], Mangoubi considered eigenfunctions on a ball in a manifold and proved lower bounds for the volume of the subset of the ball where the function is positive and the subset where it is negative. Combined with the isoperimetric inequality, one can get the lower bound C λ 3−n 2 − 1 2n for the measure of the nodal set on the entire manifold. Recently, Sogge and Zelditch, [SZ], proved the lower bound C λ 7−3n 8 . Their argument is completely different and is based in part on a beautiful new integral formula relating the L1 norm of |∇u| on the nodal set and the L1 norm of u on M . Finally, note that some papers use λ2 for the eigenvalue (i.e., u = −λ2 u); with that convention, our bound is C λ 3−n 2 . 1. Finding Good Balls Throughout M will be a fixed closed manifold with Laplace operator and u will be an eigenfunction with eigenvalue λ and ∫ M u 2 = 1. We will always assume that λ ≥ 1, since our interest is in what happens when λ goes to infinity. The first step is to fix a scale r depending on λ: Lemma 1. There exists a > 0 so that u has a zero in every ball of radius a 3 √ λ . This lemma is standard, but we will include a proof since it is so short. Proof. If u does not vanish on B r 3 , then Barta’s theorem implies that the lowest Dirichlet eigenvalue on this ball is at least λ. Let φ be a function that is identically one on B r 6 and that cuts off linearly to zero on the annulus B r 3 \B r 6 . It follows that λ ≤ ∫ |∇φ|2 ∫ φ2 ≤ 36r −2 Vol(B r 3 ) Vol(B r 6 ) ≤ C r2 , (1.1) where C comes from the Bishop-Gromov volume comparison (see p. 275 of [G]) and depends only on n, the Ricci curvature of M , and an upper bound for r . Since this is impossible for r2 > C λ−1, the lemma follows. From now on, we set r = a λ− 1 2 with a given by Lemma 1. Next we use a standard covering argument to decompose the manifold M into small balls of radius r . If B is a ball in M , then we write 2B for the ball with the same center as B and twice the radius. Lemma 2. There exists a collection {Bi } of balls of radius r in M so that M ⊂ ∪i Bi and each point of M is contained in at most CM = CM (M) of the double balls 2Bi ’s. Proof. Choose a maximal disjoint collection of balls B r 2 (pi ). It follows immediately from maximality that the double balls Bi = Br (pi ) cover M . Suppose that p ∈ M is contained in balls B2r (p1), . . . , B2r (pk). In particular, the disjoint balls B r 2 (p1), . . . , B 2 (pk) are all contained in B3r (p) so that k ∑ i=1 Vol(B r 2 (pi )) ≤ Vol(B3r (p)). (1.2)