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In this paper, we apply our previous estimates in Chen and Cheng [ On the constant scalar curvature Kähler metrics (I): a priori estimates , Preprint] to study the existence of cscK metrics on compact Kähler manifolds. First we prove that the properness of K-energy in terms of L1 geodesic distance d1 in the space of Kähler potentials implies the existence of cscK metrics. We also show that the weak minimizers of the K-energy in (E1, d1) are smooth cscK potentials. Finally we show that the non-existence of cscK metric implies the existence of a destabilized L1 geodesic ray where the K-energy is non-increasing, which is a weak version of a conjecture by Donaldson. The continuity path proposed by Xiuxiong Chen [Ann. Math. Qué. 42 (2018), pp. 69–189] is instrumental in the above proofs. [ABSTRACT FROM AUTHOR]

In this paper, we give an affirmative answer to Gromov's conjecture (Geom. Funct. Anal. 28 (2018), pp. 645-726, Conjecture E) by establishing an optimal Lipschitz lower bound for a class of smooth functions on connected orientable open 3-manifolds with uniformly positive sectional curvatures. For rigidity we show that if the optimal bound is attained the given manifold must be a quotient space of R2 × (−c,c) with some doubly warped product metric. This gives a characterization for doubly warped product metrics with positive constant curvature. As a corollary, we also obtain a focal radius estimate for immersed toruses in 3-spheres with positive sectional curvatures. [ABSTRACT FROM AUTHOR]

In this paper we address the issue of positive scalar curvature on oriented nonspin compact manifolds whose universal cover is also nonspin. We provide a conjecture for an obstruction to such curvature in this venue that takes into account all the data known to date. The conjecture is proved for a wide class of closed manifolds based on their fundamental group structure. [ABSTRACT FROM AUTHOR]