Critical Sobolev thresholds for openness and discreteness of Monge-Ampere gradient mappings

Abstract

Let $\Omega\subset\R^n$ be a domain and let $u\in W^{2,n}_{\loc}(\Omega)$ satisfy \[ \det D^2u\geq\delta>0\qquad\text{a.e. in }\Omega. \] Guerra and Tione asked whether the gradient mapping $Du$ must be open and discrete. We give a negative answer in every dimension $n\geq4$, in a form that isolates the sharp regularity mechanism. First, a local Pogorelov model is chosen to solve the exact equation $\det D^2u=1$ a.e. It is convex, belongs to $C^{1,1-2/n}\cap W^{2,n}_{\loc}$, and its Hessian is positive definite off a line. Nevertheless, its gradient collapses that line to one point. In fact the gradient is neither open nor discrete, its branch set is exactly the collapsed line, and its Jacobian equals one a.e. We then develop a $k$-dimensional version of the construction. If $1\leq k<n/2$, there are convex potentials with a $k$-dimensional flat contact set, uniformly positive and bounded Hessian determinant, and exact critical exponent \[ p_{n,k}=\frac{n(n-k)}{2k}. \] Their Hessians belong to $L^p_{\loc}$ precisely for $p<p_{n,k}$, lie in the weak endpoint space $L^{p_{n,k},\infty}_{\loc}$, and fail to belong to any finite-index Lorentz endpoint. This matches the critical minimum-set theorem of Collins and Mooney. At the regularity required in the question, the construction produces branch sets of every integer dimension $k<n/3$. We also give a topological reformulation of the problem. In the convex branch, gradient fibers are exactly contact sets with supporting affine functions, and openness and discreteness are equivalent to strict convexity. For a general gradient under the hypotheses above, critical Sobolev mapping theory already supplies continuity and sense preservation. The question asks whether the monotone factor in the Eilenberg--Whyburn monotone--light factorization is trivial.