Let $({M},g)$ be a connected and non-compact Riemannian manifold without boundary. If $L:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a linear second order elliptic operator on some smooth $\mathbb{R}$-bundle $E$ over ${M}$, is it then true that


$$Lu=0$$


for $u\in\Gamma^{\infty}\_{c}(E)$ implies that $u=0$, or in other words, there are no compactly-supported smooth homogeneous solutions? I think such a result could be proven via some ``unique continuation property of elliptic system''. However, while looking in the literature, I didn't find a suitable version for this situation.


Examples of such operator $L$ I have in mind is for example the connection Laplacian $$\Delta\_{C}:=g^{ij}\nabla\_{i}\nabla\_{j}:\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})\to\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})$$
with Levi-Civita connection $\nabla$ or closely related, the Hodge-de Rham Laplacian $$\Delta\_{H}:=\mathrm{d}\delta+\delta\mathrm{d}:\Omega^{k}({M})\to\Omega^{k}({M})$$
with exterior derivative $\mathrm{d}$ and codifferential $\delta$.



Compactly-supported harmonic tensors
The unique continuation is valid for generalized Laplacians. This follows from Hörmander's result in



> 
> *Hörmander, Lars*, [**Uniqueness theorems for second order elliptic differential equations**](https://doi.org/10.1080/03605308308820262), Commun. Partial Differ. Equations 8, 21-64 (1983). [ZBL0546.35023](https://zbmath.org/?q=an:0546.35023) [MR686819](https://mathscinet.ams.org/mathscinet-getitem?mr=686819)
> 
> 
> 


The operators you mentioned are such generalized Laplacians. Here are a few details.


Suppose that $E$ is a metric vector bundle over the Riemann manifold $(M,g)$. Denote by $\Delta\_M$ the scalar Laplacian determined by the metric $g$. Fix a connection on $E$ compatible with the metric on $E$ and set $\Delta\_E=\nabla^\*\nabla: C^\infty(E)\to C^\infty(E)$.


For any $u\in C^\infty(E)$ we have


$$\Delta\_M |u(x)|\_E^2=2\big\langle \Delta\_E u(x),u(x)\big\rangle\_E-2\vert \nabla u(x)\vert\_E^2.$$


A second order partial differential operator $L$ is called a generalized Laplacian if its principal symbol satisfies


$$
\sigma\_L(\xi)=-|\xi|\_g^2\cdot \mathbf{1}\_{E\_x},\;\;\forall x\in M,\;\;\forall \xi\in T^\*\_xM. $$


Any generalized Laplacian $L:C^\infty(E)\to C^\infty(E)$ has the form


$$ L=\nabla^\*\nabla+ T=\Delta\_E+T,$$


for some a connection $\nabla$ on $E$ compatible with the metric on $E$ and an endomorphism $T$ of the bundle $E$; see [Proposition 10.1.34 here](https://www3.nd.edu/%7Elnicolae/Lectures.pdf).


If $Lu=0$, then $\Delta\_E u=-Tu$ and we deduce


$$\Delta\_M |u(x)|\_E^2=-2\big\langle Tu(x),u(x)\big\rangle\_E-2\vert \nabla u(x)\vert\_E^2.$$


At this point you can invoke the above results of Hörmander for the scalar function $|u(x)|\_E^2$ to obtain the unique continuation.


