For a C^{1,1} function whose Hessian has uniformly positive or negative determinant almost everywhere, the index of the Hessian is constant almost everywhere, proving Šverák's 1992 conjecture.
$C^{1,\frac{1}{3}-}$ very weak solutions to the two dimensional Monge-Amp\'ere equation
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For any $\theta<\frac{1}{3}$, we show that very weak solutions to the two-dimensional Monge-Amp\`ere equation with regularity $C^{1,\theta}$ are dense in the space of continuous functions. This result is shown by a convex integration scheme involving a subtle decomposition of the defect at each stage. The decomposition diagonalizes the defect and, in addition, incorporates some of the leading-order error terms of the first perturbation, effectively reducing the required amount of perturbations to one.
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Constancy of the index for gradient mappings
For a C^{1,1} function whose Hessian has uniformly positive or negative determinant almost everywhere, the index of the Hessian is constant almost everywhere, proving Šverák's 1992 conjecture.