In this paper, we generalize our apriori estimates on cscK(constant scalar curvature K\"ahler) metric equation \cite{cc1} to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of K\"ahler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the $K$-energy in $(\mathcal{E}^1, d_1)$ are smooth. The continuity path proposed in \cite{chen15} is instrumental in this proof.

We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau. We define the positive MV-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

For a compact connected set$X$⊆$ℓ$^{∞}, we define a quantity$β$′($x$,$r$) that measures how close$X$may be approximated in a ball$B$($x$,$r$) by a geodesic curve. We then show that there is$c$>0 so that if$β$′($x$,$r$)>$β$>0 for all$x$∈$X$and$r$<$r$_{0}, then $\operatorname{dim}X>1+c\beta^{2}$ . This generalizes a theorem of Bishop and Jones and answers a question posed by Bishop and Tyson.