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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.

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    $\begingroup$ The existence follows from basic facts about elliptic boundary value problems. I do not believe that the lowest eigenvalue can be expressed in a simple way in terms of the Gaussian curvature alone . To see what could happen start with the simple case when $M$ is the unit flat disk in the plane then slightly perturb the metric and see what happens with the first eigenvalue. The computations may not be too pretty. $\endgroup$ – Liviu Nicolaescu May 9 '16 at 13:54
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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.

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