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We show that a problem asked by Yau (Open problems in geometry. Chern–a great geometer of the twentieth century, pp. 275–319, 1992) cannot be true in general. The counterexamples are constructed based on the recent work of Wu and Zheng (Examples of positively curved complete Kähler manifolds. Geometry and Analysis, vol. 17, pp. 517–542, 2010).

On a problem of Yau regarding a higher dimensional generalization of the Cohn–Vossen inequality