Lower bounds on the Hausdorff measure of nodal sets II
Christopher D. Sogge, Steve Zelditch · Mathematical Research Letters · 2012
We give a very short argument showing how the main identity (0.2) from our earlier paper [12] immediately leads to the best lower bound currently known [2] for the Hausdorff measure of nodal sets in dimensions n ≥ 3.Let (M, g) be a compact smooth Riemannian manifold of dimension n and let e λ be real-valued eigenfunction of the associated Laplacian, i.e.,with frequency λ > 0. Recent papers have been concerned with lower bounds for the (n -1)-dimensional Hausdorff measure, |Z λ |, of the nodal set of e λ ,in dimensions n ≥ 3. When n = 2 the sharp lower bound by the frequency, λ |Z λ |, was obtained by Brüning in [1] and independently by Yau.For all dimensions, in the analytic case, the sharp upper and lower bounds |Z λ | ≈ λ were obtained by Donnelly and Fefferman [4,5].Until recently, the best known lower bound when n ≥ 3 seems to have been e -cλ |Z λ | (see [6]).Using a variation (0.2) of an identity of Dong [3], the authors showed in [12] that this can be improved to be λ 7