Applications of PDEs to the study of affine surface geometry

If $\mathcal{M}=(M,\mathrm{\nabla })$ is an affine surface, let $\mathcal{Q}(\mathcal{M}):=\ker (\mathcal{H}+\frac{1}{m-1}{\rho }_s)$ be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let $\tilde{\mathcal{M}}=(M,\tilde{\mathrm{\nabla }})$ be another affine structure on $M$ which is strongly projectively flat. We show that $\mathcal{Q}(\mathcal{M})=\mathcal{Q}(\tilde{\mathcal{M}})$ if and only if $\mathrm{\nabla }=\tilde{\mathrm{\nabla }}$ and that $\mathcal{Q}(\mathcal{M})$ is linearly equivalent to $\mathcal{Q}(\tilde{\mathcal{M}})$ if and only if $\mathcal{M}$ is linearly equivalent to $\tilde{\mathcal{M}}$. We use these observations to classify the flat Type~$\mathcal{A}$ connections up to linear equivalence, to classify the Type~$\mathcal{A}$ connections where the Ricci tensor has rank 1 up to linear equivalence, and to study the moduli spaces of Type~$\mathcal{A}$ connections where the Ricci tensor is non-degenerate up to affine equivalence.