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Third-order affine-invariant (systems of) PDEs in two independent variables as vanishing of the Fubini-Pick invariant (2202.09894v4)
Published 20 Feb 2022 in math.DG
Abstract: In this paper we study $3{\mathrm{rd}}$ order (system of) PDEs in two independent variables $x,y$ and one unknown function $u$ that are invariant with respect to the group of affine transformation $\mathrm{Aff}(3)$ of $\mathbb{R}3={(x,y,u)}$. After proving their relationship with the Fubini-Pick invariant, we derive the aforementioned PDEs by using a general method introduced in [D.V. Alekseevsky, J. Gutt, G. Manno, and G. Moreno: A general method to construct invariant {PDEs} on homogeneous manifolds. Communications in Contemporary Mathematics (2021)], which sheds light on some of their geometrical properties.
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