Let $(M,g)$ be a simply-connected compact surface with boundary $\partial M$ and metric $g$. Let $N$ denote the outward unit normal on $\partial M$, $\nabla$ the Levi-Civita connection and $\Delta_g$ the corresponding connection Laplacian (i.e. the trace of the Hessian).
I am interested in the properties of the vector field $X$ that minimises $\int_M \langle \nabla X, \nabla X\rangle dV$ subject to the boundary condition $\nabla_N X=0$, where $\langle,\rangle$ denotes the extension of g to arbitrary tensor fields. Because of these boundary conditions, $\int_M \langle \nabla X, \nabla X\rangle dV=-\int_M \langle \Delta_g X, X\rangle dV$ and we just need the lowest eigenvalue of the Laplacian with these boundary conditions.
I am not sure this boundary problem is well posed, but if so, this eigenvalue cannot be zero in general, since there are no parallel vector fields on a curved surface by the Gauss-Bonnet theorem. It is my hope that a general solution to this boundary problem exists, and that the lowest eigenvalue can be expressed in terms of the Gauss curvature of the surface. I would be very grateful for an answer to this question, and ideally a reference that deals with this problem in more detail, since all books I can find deal with the Hodge Laplacian.