Critical Sobolev thresholds for openness and discreteness of Monge-Ampere gradient mappings
Abstract
Let Ω⊂Rn be a domain and let u∈ W2,nloc(Ω) satisfy det D2u≥δ>0 a.e. in Ω. Guerra and Tione asked whether the gradient mapping Du must be open and discrete. We give a negative answer in every dimension n≥4, in a form that isolates the sharp regularity mechanism. First, a local Pogorelov model is chosen to solve the exact equation det D2u=1 a.e. It is convex, belongs to C1,1−2/n∩ W2,nloc, 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≤ k<n/2, there are convex potentials with a k-dimensional flat contact set, uniformly positive and bounded Hessian determinant, and exact critical exponent pn,k=(n(n−k))/2k. Their Hessians belong to Lploc precisely for p<pn,k, lie in the weak endpoint space Lpn,k,∞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.
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BibTeX
@misc{zhong2026critical,
title = {{Critical Sobolev thresholds for openness and discreteness of Monge-Ampere gradient mappings}},
author = {Deguang Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.16757},
eprint = {2608.16757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16757},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical Sobolev thresholds for openness and discreteness of Monge-Ampere gradient mappings
AU - Deguang Zhong
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2608.16757
UR - https://arxiv.org/abs/2608.16757
AB - Let Ω⊂R^n be a domain and let u∈ W^2,n_loc(Ω) satisfy det D^2u≥δ>0 a.e. in Ω. Guerra and Tione asked whether the gradient mapping Du must be open and discrete. We give a negative answer in every dimension n≥4, 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∩ 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≤ 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=(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,∞_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.
ER -