Let $(M,g)$ be a compact Riemannian manifold with boundary. It is well known that the first positive Steklov eigenvalue $\sigma_1$ of $M$ has the following variational characterization:

$$\sigma_1(M,g) = \inf \left\lbrace \frac{\int_M \vert \nabla u \vert^2 \, \mathrm{d}V}{\int_{\partial M} u^2 \, \mathrm{d}A }: u \in H^1(M) \text{ and } \int_M u \, \mathrm{d} V =0\right\rbrace$$

Moreover, there exists a smooth minimizer $u : M \to \mathbb{R}$, which satisfies the following problem:

$$\cases{\Delta u = 0, \quad \text{on } M \\ \frac{\partial u}{\partial \nu} = \sigma_1 u, \quad \text{on } \partial M}$$

where $\nu$ is the outward unit normal to $\partial M$.

What happens if we now change the definition to consider only circle-valued functions $u:M \to S^1 =\mathbb{R} / \mathbb{Z}$? Is the infimum still positive? Is it achieved by a smooth function satisfying a specific PDE as before?