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In a nutshell:

Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form?

Let $\mathbb{T}^n$ be the $n$-Torus. Fix $1<k<n$. Given a Riemannian metric $g$ on $\mathbb{T}^n$, we denote by $H^k_g$ the space of $g$-harmonic forms of degree $k$. Suppose that $g$ is flat on some open neighbourhood $U \subseteq \mathbb{T}^n$.

Let $p \in U$, and consider the evaluation map at $p$, $ev_p:H^k_g \to \bigwedge^k T_p^*\mathbb{T}^n$, given by $ev_p(\omega)=\omega_p$.

Is it true that $ev_p$ is an isomorphism? (Note that the dimensions of both spaces are equal).

Of course, this holds for the standard flat metric on $\mathbb{T}^n$ induced by the Euclidean metric on $\mathbb{R}^n$. (There we essentially have $dx^{i_1} \wedge \dots \wedge dx^{i_k}$ as a global basis for $H^k_g$).

Since $g$ is flat around $p$, we can take isometric coordinates, where $g_{ij}=\delta_{ij}$. So, locally we have all the $dx^{i_1} \wedge \dots \wedge dx^{i_k}$ as harmonic forms. However, I am not sure that all these local forms can be extended into global harmonic forms on $\mathbb{T}^n$. If such an extension is possible, then of course the evaluation map would be surjective.

Is there any reason to believe such an extension is always possible?

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  • $\begingroup$ Note that the space of forms that are harmonic on $U$ is infinite-dimensional, whereas the space of forms that are harmonic on the entire manifold is finite-dimensional. This seems to be a problem that contradicts my intuition that the extension would seem to be possible. $\endgroup$ – Phillip Andreae Sep 20 '18 at 13:54
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Certainly, local extendability is not always possible, even when $(k,n)=(1,2)$. (However, note that, in this case, $ev_p$ is, in fact, always an isomorphism, even without any hypothesis of local flatness.)

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  • $\begingroup$ Yes, you are right of course. We cannot expect arbitrary local extendability, since the space of forms that are harmonic on $U$ is infinite-dimensional, whereas the space of forms that are harmonic on the entire manifold is finite-dimensional. However, this still does not rule out the possibility that we can choose in a clever way some suitable local forms and extend them. So, the general question "Is $ev_p$ always an isomorphism" is still open, right? $\endgroup$ – Asaf Shachar Sep 20 '18 at 14:13

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