Geometric Reinterpretation of Partial Differential Equations and Applications

We obtain improved regularity estimates on solutions of partial differential equations by combining the method of Fuchsian Reduction with geometric transformations. Examples include the meron problem and the regularity of the conformal radius. In each case, Reduction needs to be combined with a reinterpretation of the underlying geometry. We argue that the geometric meaning assigned to a problem has an influence, positive or negative, on the range of methods envisioned for its solution, and that the Euler–Poisson–Darboux (EPD) equation cannot be properly understood within a single geometric framework. This explains the central position of EPD-like equations.