In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on singularities with a torus action of complexity $1$, threefold singularities, and hypersurface singularities. Given a real value $v>0$, we prove that the class of K-semistable threefold singularities with normalized volume at least v forms a bounded family. Analogous statements are proved in the case of n-dimensional complexity-$1$ and n-dimensional hypersurface singularities for arbitrary n. In the general case of klt singularities, i.e. without the assumption on K-semistability, we show that, up to special degenerations, the normalized volume bounds singularities with a complexity-$1$ torus action. We exhibit a $3$-dimensional example which shows that this last statement is optimal.

We analyze the existence of Kähler–Einstein metrics of positive curvature in the neighborhood of a germ of a log terminal singularity (X,p). This boils down to solving a Dirichlet problem for certain complex Monge–Ampère equations. We establish a Moser–Trudinger inequality $(MT)_{\gamma}$ in subcritical regimes $\gamma<\gamma_{\rm crit}(X,p)$ and show the existence of smooth solutions in those cases. We show that the expected critical exponent $\tilde{\gamma}_{\rm crit}(X,p)=(({n+1})/{n}) \widehat{\mathrm{vol}}(X,p)^{1/n}$ can be expressed in terms of the normalized volume, an important algebraic invariant of the singularity.

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.

Li introduced the normalized volume of a valuation due to its relation to K-semistability. He conjectured that over a Kawamata log terminal (klt) singularity there exists a valuation with smallest normalized volume. We prove this conjecture and give an explicit example to show that such a valuation need not be divisorial.