The first eigenvalue of analytic level surfaces on spheres (Q1901643)

The author establishes a lower bound for the first eigenvalue of the Laplacian on the level set of a real valued real analytic function defined on a sphere. More precisely, let \(f\) be a real valued analytic function on the sphere \(S^{n + 1}\) and let \(V_t = f^{-1} (t)\) be a regular value. Let \(t_0\) be a critical value. Let \(\lambda_1 (V_t) > 0\) be the first eigenvalue of the Laplacian. Theorem: There exist constants (independent of \(t\) and \(t_0)\), \(C = C(f) > 0\) and \(\alpha = \alpha (f)\) so that \(\lambda_1 (V_t) \geq C |t - t_0 |^\alpha\) as \(t \to t_0\). In the case that \(f\) is a polynomial, this was established first by Gromov. However, the extension to the real analytic setting is useful to study the local solvability of a system of vector fields in order to obtain appropriate bounds on the Green's function.