The inner radius of nodal domains in high dimensions (Q6608705)

Let \((M,g)\) be a smooth closed Riemannian manifold of dimension \(d\). Consider on \(M\) an eigenfunction \(u_\lambda\) of the positive Laplace-Beltrami operator \(-\Delta_g\) corresponding to an eigenvalue \(\lambda\). A nodal domain \(\Omega_\lambda\) of \(u_\lambda\) is any connected component of the the set \(\{u_\lambda \neq 0\}\).\N\NIt is well known that there exists a positive constant \(c_{\text{up}} = c_{\text{up}}(M,g)\) independent of \(\lambda\) or \(u_\lambda\) such that every ball of radius bigger than \(c_{\text{up}}(M,g)\lambda^{-1/2}\) contains a zero of \(u_\lambda\) [\textit{J. E. D'Atri} and \textit{W. Ziller}, Naturally reductive metrics and Einstein metrics on compact Lie groups. Providence, RI: American Mathematical Society (AMS) (1979; Zbl 0404.53044); \textit{J. Brüning}, Math. Z. 158, 15--21 (1978; Zbl 0349.58012)]. On the other hand, the Faber-Krahn inequality [\textit{G. Faber}, Münch. Ber. 1923, 169--172 (1923; JFM 49.0342.03); \textit{E. Krahn}, Math. Ann. 94, 97--100 (1925; JFM 51.0356.05)] shows that the volume of every nodal domain \(\Omega_k\) is bounded from below by \(c_{FK}\lambda^{-d/2}\) for some positive constant \(c_{FK} = c_{FK}(M)\).\N\NOne is naturally led to look for the largest positive number \(r\) such that every nodal domain contains a ball of radius \(r\). In two dimensions it is known that once can inscribe a ball of radius \(c\lambda^{-1/2}\) in every nodal domain [\textit{D. Mangoubi}, Can. Math. Bull. 51, No. 2, 249--260 (2008; Zbl 1152.58027)]. This paper is concerned with lower bounds for the inner radius in higher dimensions. In particular, let \((M,g)\) be of dimension \(d\) of at least three. let \(x_{\max}\) be a point where \(|u_\lambda(x_{\max})| = \max_{\Omega_\lambda} |u_\lambda|\). The authors show that the ball of radius \(c_{\text{lo}} \lambda^{-1/2} (\log \lambda)^{-(d-2)/2}\) centered at \(x_{\max}\) is contained in \(\Omega_\lambda\), where \(c_{\text{lo}} = c_{\text{lo}}(M,g)\) is a constant that only depends on \((M,g)\).\N\NLower bounds of the form \(\lambda^{-c(d)}\) on the inner radius of nodal domains were obtained in [\textit{D. Mangoubi}, Can. Math. Bull. 51, No. 2, 249--260 (2008; Zbl 1152.58027); Commun. Partial Differ. Equations 33, No. 9, 1611--1621 (2008; Zbl 1155.35404)], where \(c(d) \to \infty\) as the dimension \(d\) increases. A \(c\lambda^{-1}\) lower bound was obtained by \textit{B. Georgiev} [J. Geom. Anal. 29, No. 2, 1546--1554 (2019; Zbl 1428.35253)] under the assumption of a real analytic metric.\N\NAn important starting point for this work is the exsitence of an almost inscribed ball \(B\) of radius \(r = \delta \lambda^{-1/2}\) for some \(\delta > 0\), guaranteed by \textit{B. Georgiev} and \textit{M. Mukherjee} [Anal. PDE 11, No. 1, 133--148 (2018; Zbl 1378.35208)]. The idea is to then show that the complementary set \(B \setminus \Omega_\lambda\) cannot approach the center too much. The argument is based on a Remez-type inequality due to \textit{A. Logunov} and \textit{E. Malinnikova} [ICM 2018, 2391--2411 (2018; Zbl 1453.35061); IAS/Park City Math. Ser. 27, 1--34 (2020; Zbl 1467.35124)].