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We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold$X$. Given a big (1, 1)-cohomology class$α$on$X$(i.e. a class that can be represented by a strictly positive current) and a positive measure$μ$on$X$of total mass equal to the volume of$α$and putting no mass on pluripolar sets, we show that$μ$can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in$α$. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if$μ$has$L$^{1+$ε$}-density with respect to Lebesgue measure. If$μ$is smooth and positive everywhere, we prove that$T$is smooth on the ample locus of$α$provided$α$is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.

We consider left-invariant distances d on a Lie group G with the property that there exists a multiplicative one-parameter group of Lie automorphisms (0,∞)→Aut(G), λ↦δλ, so that d(δ_λx,δ_λy)=λd(x,y), for all x,y∈G and all λ>0. First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties of their infinitesimal generator. Third, we show that an admissible left-invariant distance on a Lie group with at least one nontrivial dilating automorphism is bi-Lipschitz equivalent to one that admits a one-parameter group of dilating automorphisms. Moreover, the infinitesimal generator can be chosen to have spectrum in [1,∞). Fourth, we characterize the automorphisms of a Lie group that are a dilating automorphisms for some admissible distance. Finally, we characterize metric Lie groups admitting a one-parameter group of dilating automorphisms as the only locally compact, isometrically homogeneous metric spaces with metric dilations of all factors. Such metric spaces appear as tangents of doubling metric spaces with unique tangents.

Surface registration between cortical surfaces is crucial in medical imaging for performing systematic comparisons between brains. Landmark-matching registration that matches anatomical features, called the sulcal landmarks, is often required to obtain a meaningful 1-1 correspondence between brain surfaces. This is commonly done by parameterizing the surface onto a simple parameter domain, such as the unit sphere, in which the sulcal landmarks are consistently aligned. Landmark matching surface registration can then be obtained from the landmark aligned parameterizations. For genus-0 closed brain surfaces, the optimized spherical harmonic parameterization, which aligns landmarks to consistent locations on the sphere, has been widely used. This approach is limited by the loss of bijectivity under large deformations and the slow computation. In this paper, we propose FLASH, a fast algorithm to compute the optimized spherical harmonic parameterization with consistent landmark alignment. This is achieved by formulating the optimization problem to ℂ and thereby linearizing the problem. Errors introduced near the pole are corrected using quasiconformal theories. Also, by adjusting the Beltrami differential of the mapping, a diffeomorphic (1-1, onto) spherical parameterization can be effectively obtained. The proposed algorithm has been tested on 38 human brain surfaces. Experimental results demonstrate that the computation of the landmark aligned spherical harmonic parameterization is significantly accelerated using the proposed algorithm.