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

Abstract

Abstract. We show that a problem by Yau in [15] can not be true in general. The counterexamples are constructed based on the recent work of Wu and Zheng [14].

Longtime existence of the Kähler-Ricci flow on ℂⁿ

“…Remark A.1. By the proof of [22,23], it is easy to see that similar to (b) in the above theorem, g has nonpositive bisectional curvature if and only if ξ ′ ≤ 0.…”

Geometry of positive scalar curvature on complete manifold

“…In fact, Yau proposed a more general version of this problem that is involved with the σ k , k = 1, 2, • • • , n of Ricci tensor in [40]. Unfortunately, Yang [38] constructs a counterexample on Kähler manifold to prove that the general version of Yau's Problem 1.5 does not hold for k = 1, 2, • • • , n − 1, Xu [37] obtains an estimate involved with the integral of scalar curvature towards the Problem 1.5 in the case of three-dimensional Riemannian manifold by using the monotonicity formulas of Colding and Minicozzi [5]. However, Problem 1.5 remains open.…”

Integral of Scalar Curvature on Non-parabolic Manifolds

“…Motivated to get a generalization of the Cohn-Vossen's inequality, Yau [Yau92, Problem 9] posed the following question: Given a n-dimensional complete manifold (M n , g) with Rc ≥ 0, let B r (p) be the geodesic ball around p ∈ M n and σ k be the k-th elementary symmetric function of the Ricci tensor, is it true that lim r→∞ r −n+2k B r (p) σ k < ∞? In 2013, Bo Yang [Yan13] constructed examples, which answered the above question for k > 1 negatively. However, the interesting case k = 1 is still open, where σ 1 is the scalar curvature R. We formulate it in the following question separately.…”