Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of non-positive curvature (Q2806008)

Let \((M,g)\) be a complete Riemannian manifold with non-positive curvature. Let \(\Omega\subset M\). Assume that a tubular neighborhood of \(\Omega\) is embedded. Let \(\Omega_t\) be a smooth 1-parameter family of domains so that \(\Omega_s\subset\Omega_t\) for \(s<t\) with \(\Omega_0=\Omega\). Let \(\lambda(t)\) be the first Dirichlet eigenvalue of \(D_t\); this is a decreasing function of \(t\) and the rate of decrease is estimated in the article. The authors obtain a reverse Hölder inequality generalizing results of \textit{G. Chiti} [Boll. Unione Mat. Ital., VI. Ser., A 1, 145--151 (1982; Zbl 0484.35067); Z. Angew. Math. Phys. 33, 143--148 (1982; Zbl 0508.35063)]. The first section contains a historical introduction to the problem and a statement of results. Section 2 introduces some geometric preliminaries. Section 3 discusses rearrangements of the first eigenfunction and reverse Hölder inequalities. Section 4 studies the evolution of the first Dirichlet eigenvalue as the domain evolves.