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Curvature of the second kind and a conjecture of Nishikawa

https://doi.org/10.4171/CMH/545

Cao, Xiaodong; Gursky, Matthew; Tran, Hung (January 2023, Commentarii Mathematici Helvetici)

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Aubry-André Anderson model: Magnetic impurities coupled to a fractal spectrum

https://doi.org/10.1103/PhysRevB.106.165123

Wu, Ang-Kun; Bauernfeind, Daniel; Cao, Xiaodong; Gopalakrishnan, Sarang; Ingersent, Kevin; Pixley, J. H. (October 2022, Physical Review B)

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Four-manifolds of pinched sectional curvature

https://doi.org/10.1016/j.aim.2018.07.003

Cao, Xiaodong; Tran, Hung (January 2022, Pacific journal of mathematics)

In this paper, we study closed four-dimensional manifolds. In particular, we show that under various pinching curvature conditions (for example, the sectional curvature is no more than 5 6 of the smallest Ricci eigenvalue), the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it must be self-dual or anti-self-dual under the same conditions. Similarly, if restricting to an Einstein metric, then it must be either the complex projective space with its Fubini-Study metric, the round sphere, or their quotients. Furthermore, we also classify Einstein manifolds with positive intersection form and an upper bound on the sectional curvature.
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