Do harmonic frames always exist locally?

Let $M$ be a smooth Riemannian manifold of dimension $d$. Let $1 < k <d$ be an integer. Consider the exterior power bundle $\Lambda_{k}(T^*M)$.

Do closed and co-closed frames for $\Lambda_{k}(T^*M)$ always exist locally?

In other words, let $p \in M$. Does there exist a neighbourhood $U$ of $p$, which admits a frame of $k$-forms, all of which are closed and co-closed?

For $k=1$, the answer is positive, due to the existence of harmonic coordinates. Note that I do not require the frame to come "from coordinates" in any way; In Euclidean space, however, we can use coordinates of course; Take $dx^{i_1} \wedge \dots \wedge dx^{i_k}$, where $x_i$ are the standard coordinates.

Edit:

Let us specialize to even dimension $d$, and let $k=\frac{d}{2}$.

Then, for a generic metric $g$, there are no coordinate systems where even one wedge is harmonic: $\delta(\mathrm{d}x^{i_1}\wedge\cdots\wedge\mathrm{d}x^{i_n}) =0$. This implies harmonic frames, if exist generically, cannot be induced in general by coordinates. Moreover, if such a frame exists, then each member in it cannot be decomposable, since a closed decomposable form can always be expressed locally as the wedge of coordinate differentials.

In fact, such a harmonic frame $\omega^i$ must have the following property:

There are no non-zero decomposable elements, spanned by $\omega^i$ using constant coefficients. (i.e. there do not exist real numbers $a_i$ such that $\sum a_i \omega^i \neq 0$ is decomposable). Indeed, if such numbers existed, then $\sum a_i \omega^i$ would be decomposable and harmonic, which is generically impossible as mentioned above.

For a "generic" frame of $k$-forms, this property probably holds. (Here is an example for such "strongly non-decomposable frames").

$$\newcommand{\M}{\mathcal{M}}$$ $$\newcommand{\ep}{\epsilon}$$

The answer is positive. Closed and co-closed frames always exist locally.

I will sketch here one proof: (For other approaches, see here and here). We want to prove that around every point $$p \in \M$$ there exist a local frame for $$\bigwedge^k(T^*\M)$$ whose elements are closed and co-closed.

For the Euclidean metric this is immediate: We have the standard (constant) frame $$dx^I=dx^{i_1} \wedge \ldots dx^{i_k}$$. Since every metric is locally close to being Euclidean on small neighbourhoods, the idea is to use an approximation argument:

Given a Riemannian metric $$g$$, we denote the space of $$g$$-harmonic forms of degree $$k$$ by $$H^k_{g}$$.

We view $$H^k_{g}$$, as a subspace of $$\Omega^k(\M)$$ which is "changing continuously" with the metric $$g$$. Suppose $$g_{\ep} \to g_0$$ in the $$C^1$$-norm where $$g_0$$ is the Euclidean metric; Then $$H^k_{g_{\ep}} \to H^k_{g_0}$$ in the following sense: there exist a family of bases of $$H^k_{g_{\ep}}$$, which converges to a basis of $$H^k_{g_{0}}$$ in $$C^1$$; this basis of $$H^k_{g_{0}}$$ forms a local frame for $$\bigwedge^k(T^*\M)$$. Since being a frame is an open condition, those bases for $$H^k_{g_{\ep}}$$ are local frames for sufficiently small $$\ep$$.

For the full details, see Appendix A in my paper here.

Some more details:

Even though the claim is local, and the approximation scheme is also inspired by a local phenomena, the implementation of the proof is based on a combination of local and global arguments. The reason is that on a closed manifold, being closed and co-closed is equivalent to being harmonic, and the dimension of the space of harmonic forms is a finite number which is a topological invariant of the manifold; it does not depend on the chosen metric.

Thus, given a family of metrics $$g_{\ep} \to g_0$$ on a closed manifold $$\M$$, we consider the behaviour of the finite-dimensional subspaces $$H^k_{g_{\ep}}$$ (all of the same dimension) as $$\ep \to 0$$.

That is, we look at the map $$g \to H^k_{g}=\ker \Delta_g$$. It turns out that this map is continuous in some appropriate sense; this relies on a certain "stability property of kernels of linear operators". It turns out that a crucial factor in the existence of such a stability phenomenon is the assumption that all the kernels have the same finite dimension. The convergence of kernels does not always hold when the dimensions are not equal or infinite.