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A Sharp Bound for the Ratio of the First Two Dirichlet Eigenvalues of a Domain in a Hemisphere of S^n

For a domain $Ω$ contained in a hemisphere of the $n$-dimensional sphere $\SS^n$ we prove the optimal result $λ_2/λ_1(Ω) \le λ_2/λ_1(Ω^{\star})$ for the ratio of its first two Dirichlet eigenvalues where $Ω^{\star}$, the symmetric rearrangement of $Ω$ in $\SS^n$, is a geodesic ball in $\SS^n$ having the same $n$-volume as $Ω$. We also show that $λ_2/λ_1$ for geodesic balls of geodesic radius $θ_1$ less than or equal to $π/2$ is an increasing function of $θ_1$ which runs between the value $(j_{n/2,1}/j_{n/2-1,1})^2$ for $θ_1=0$ (this is the Euclidean value) and $2(n+1)/n$ for $θ_1=π/2$. Here $j_{ν,k}$ denotes the $k^{th}$ positive zero of the Bessel function $J_ν(t)$. This result generalizes the Payne-Pólya-Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of $\SS^n$ and having a fixed value of $λ_1$ the one with the maximal value of $λ_2$ is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for $λ_2/λ_1$. Various other results for $λ_1$ and $λ_2$ of geodesic balls in $\SS^n$ are proved in the course of our work.