We consider a 1-parameter family of strictly convex hypersurfaces in Rn+1 moving with speed −Kαν, where ν denotes the outward-pointing unit normal vector and α⩾1/(n+2). For α>1/(n+2), we show that the flow converges to a round sphere after rescaling. In the affine invariant case α=1/(n+2), our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.

An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichm¨uller and moduli spaces. The tensor is evaluated on the gradients of geodesiclengths for disjoint geodesics. A precise lower bound for sectional curvature in terms of the surface systole is presented. The curvature tensor expansion is applied to establish continuity properties at the frontier strata of the augmented Teichm¨uller space. The curvature tensor has the asymptotic product structure already observed for the metric and covariant derivative. The product structure is combined with the earlier negative sectional curvature results to establish a classification of asymptotic flats. Furthermore, tangent subspaces of more than half the dimension of Teichm¨uller space contain sections with a definite amount of negative curvature. Proofs combine estimates for uniformization group exponential-distance sums and potential theory bounds.

In this paper we consider proper holomorphic embeddings of finitely connected planar domains into ℂ^{$n$}that approximate given proper embeddings on smooth curves. As a side result we obtain a tool for approximating a $\mathcal{C}^{\infty}$ diffeomorphism on a polynomially convex set in ℂ^{$n$}by an automorphism of ℂ^{$n$}that satisfies some additional properties along a real embedded curve.

We determine a global structure of the moduli space of self-dual metrics on 3CP2 satisfying the following three properties: (i) the scalar curvature is of positive type, (ii) they admit a non-trivial Killing field, (iii) they are not conformal to the LeBrun’s self- dual metrics based on the ‘hyperbolic ansatz’. We prove that the moduli space of these metrics is isomorphic to an orbifold R3/G, where G is an involution of R3 having two-dimensional fixed locus. In particular, the moduli space is non-empty and connected. We also remark that Joyce’s self-dual metrics with torus symmetry appear as a limit of our self-dual metrics. Our proof of the result is based on the twistor theory. We first determine a defining equation of a projective model of the twistor space of the metric, and then prove that the projective model is always birational to a twistor space, by determining the family of twistor lines. In determining them, a key role is played by a classical result in algebraic geometry that a smooth plane quartic always possesses twenty-eight bitangents.

We study generalized Hopf–Cole transformations motivated by the Schrödinger bridge problem, which can be seen as a boundary value Hamiltonian system on the Wasserstein space. We prove that generalized Hopf–Cole transformations are symplectic submersions in the Wasserstein symplectic geometry. Many examples, including a Hopf–Cole transformation for the shallow water equations, are given. Based on this transformation, energy splitting inequalities are provided.