Assume that $M$ is a smooth closed manifold and $E,F$ are fixed smooth vector bundles over $M.$



> 
> Is there a number $C,$ such that for any elliptic operator $\mathcal{D}:\Gamma(E)\to\Gamma(F)$
>  $$\dim\ker\mathcal{D}\leqslant C.$$
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> 
> 


For simplicity, we may assume that $\Gamma(E)=\Gamma(F)=C^\infty(M).$



Is there an upper bound on dimension of kernel of elliptic operator for a fixed closed manifold M
There are, at least, many examples where this is false. For instance, Hitchin showed (I think this was his thesis) that there is a sequence of metrics $\{g\_k\}\_{k=1}^{\infty}$ on the three sphere such that the kernel of the Dirac operator associated to $g\_{k}$ has dimension at least $k.$ It is stated as a conjecture in these notes <http://www.mathematik.uni-regensburg.de/ammann/talks/11BerlinSFB.pdf> that this is a generic phenomena for spin manifolds of dimension at least three.


In the case where $E=F=M\times \mathbb{R},$ I don't have any examples in mind off the top of my head, but I could imagine one could construct a family of metrics $\{g\_{k}\}$ and a family of functions $f\_{k}:M\rightarrow \mathbb{R}$ such that
the sequence of elliptic operators
\begin{align}
D\_{k}=\Delta\_{g\_{k}} + f\_{k}
\end{align}
has arbitrarily large kernel. 


With this in mind, it's hard to imagine a circumstance where what you are asking is true. A better question might be: for some sensible elliptic operator that depends on a parameter, like the Dirac operator on a spin manifold depending on a Riemannian metric, can you find sensible geometric bounds on the Riemannian metric under which this kernel has a uniform bound. 


