Stability Results for the First Eigenvalue of the Laplacian on Domains in Space Forms1

We studied the two known works on stability for isoperimetric inequalities of the first eigenvalue of the Laplacian. The earliest work is due to A. Melas who proved the stability of the Faber–Krahn inequality: for a convex domain contained in n with λ close to ¯λ, the first eigenvalue of the ball B of the same volume, the domain must be close to the ball B with respect to the Hausdorff distance. Later, Y. Xu studied the stability of the Szeg¨o–Weinberger inequality for convex domains in n and n where n denotes hyperbolic space. Our work consists of extending A. Melas’ result to the spaces of constant curvature 2 and 2 and Y. Xu’s result to domains contained in the polar cap Bπ/4 in n.