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.



Properties of connection Laplacian on vector fields
I don't think such a result is possible without any assumptions on the geometry of the boundary.


Consider the scalar case. If $M$ is a closed manifold with $\operatorname{Ric} \geq (n-1)k g$ for some positive constant $k$, then $\lambda\_1(M) \geq nk$. The proof (which is due to Lichnerowicz and is summarized in Obata's paper *Certain conditions for a Riemannian manifold to be isometric with a sphere*) uses integration by parts, so one has to be careful when $\partial M$ is nontrivial. This was studied by Reilly in *Applications of the Hessian operator on a Riemannian manifold*. He proved that the same lower bound for the first (Dirichlet) eigenvalue holds as long as $\partial M$ has nonpositive mean curvature.


The proof is just a computation, so it should be straightforward (though tedious) to see if a similar result holds for the connection Laplacian on vector fields.


