I am interested in the Dirichlet eigenvalues of $-\Delta_{\mathbb S^n}$ on open subsets $D \subsetneq \mathbb S^n$. If $D$ is a hemisphere, then the first eigenvalue is equal to $n$, with the eigenfunction given by $x_1^+$ for some choice of embedding of $\mathbb S^n$ into $\mathbb R^{n+1}$. (The hemisphere is also a nodal domain for an eigenfunction on all of $\mathbb S^n$ with eigenvalue $n$.)
Conversely, if all we know about $D$ is that its first eigenvalue equals $n$, can we conclude $D$ is a hemisphere? What if we know more about $D$, for example, that its surface area is equal to that of the hemisphere?
EDIT: Thinking about it a bit more, if $\phi$ is an eigenfunction on $D$ corresponding to $\lambda = n$, and $f(x) = |x|\phi(|x|)$, then by direct computation, $f$ is harmonic in $\tilde D \subset \mathbb R^{n+1}$, where $\tilde D$ is the union of rays passing through points in $D$. If we could say that $f$ is a homogeneous polynomial (which would be true if $D$ were the whole sphere, by a Liouville theorem), then this should force $\phi$ to be a spherical harmonic on $\mathbb S^n$ with eigenvalue $n$, whose zero set must be a great circle, I believe. However, I don't know how/whether a Liouville-type theorem would apply to subsets of $\mathbb R^{n+1}$ like $\tilde D$.