Noninjective harmonic maps between closed negatively curved manifolds in every dimension at least three

For every dimension n ≥ 3, this paper constructs two smooth metrics on a closed, connected, orientable n-manifold, both with sectional curvature at most −1. The harmonic representative of the identity homotopy class has negative Jacobian at some point and is therefore not injective. This provides a counterexample to Problem 111 in every dimension n ≥ 3. Repository: https://github.com/opensource120/Some-Answers-to-Yau-s-120-open-problems-in-geometryPDF at the deposited version: https://github.com/opensource120/Some-Answers-to-Yau-s-120-open-problems-in-geometry/blob/b4f17a71d1712073bf82196a01c29ae04292e5c1/papers/problem_111_harmonic_maps.pdfTeX source at the deposited version: https://github.com/opensource120/Some-Answers-to-Yau-s-120-open-problems-in-geometry/blob/b4f17a71d1712073bf82196a01c29ae04292e5c1/papers/problem_111_harmonic_maps.tex