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We construct a fundamental piece of the boundary of the maximal globallyhyperbolic development (MGHD) of Cauchy data for the multi-dimensionalcompressible Euler equations, which is necessary for the local shockdevelopment problem. For an open set of compressive and generic $H^7$ initialdata, we construct unique $H^7$ solutions to the Euler equations in the maximalspacetime region below a given time-slice, beyond the time of the firstsingularity; at any point in this spacetime, the solution can be smoothly anduniquely computed by tracing both the fast and slow acoustic characteristicsurfaces backward-in-time, until reaching the Cauchy data prescribed along theinitial time-slice. The future temporal boundary of this spacetime region is asingular hypersurface, containing the union of three sets: first, aco-dimension-$2$ surface of ``first singularities'' called the pre-shock;second, a downstream hypersurface called the singular set emanating from thepre-shock, on which the Euler solution experiences a continuum of gradientcatastrophes; third, an upstream hypersurface consisting of a Cauchy horizonemanating from the pre-shock, which the Euler solution cannot reach. We developa new geometric framework for the description of the acoustic characteristicsurfaces which is based on the Arbitrary Lagrangian Eulerian (ALE) framework,and combine this with a new type of differentiated Riemann variables which arelinear combinations of gradients of velocity, sound speed, and the curvature ofthe fast acoustic characteristic surfaces. With these new variables, weestablish uniform $H^7$ Sobolev bounds for solutions to the Euler equationswithout derivative loss and with optimal regularity.

The geometry of maximal development and shock formation for the Euler equations in multiple space dimensions