Einstein-like geometric structures on surfaces

An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. They generalize Weyl structures, and a pair of AH structures is induced on a co-oriented non-degenerate immersed hypersurface in flat affine space. The author has defined for AH structures Einstein equations, which specialize on the one hand to the usual Einstein Weyl equations and, on the other hand, to the equations for affine hyperspheres. Here these equations are solved for Riemannian signature AH structures on compact orientable surfaces, the deformation spaces of solutions are described, and some aspects of the geometry of these structures are related. Every such structure is either Einstein Weyl (in the sense defined for surfaces by Calderbank) or is determined by a pair comprising a conformal structure and a cubic holomorphic differential, and so by a convex flat real projective structure. In the latter case it can be identified with a solution of the Abelian vortex equations on an appropriate power of the canonical bundle. On the cone over a surface of genus at least two carrying an Einstein AH structure there are Monge-Amp`ere metrics of Lorentzian and Riemannian signature and a Riemannian Einstein K"ahler affine metric. A mean curvature zero spacelike immersed Lagrangian submanifold of a para-K"ahler four-manifold with constant para-holomorphic sectional curvature inherits an Einstein AH structure, and this is used to deduce some restrictions on such immersions.

A Schwarz lemma for Kahler affine metrics and the canonical potential of a proper convex cone

This is an account of some aspects of the geometry of Kahler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for Kahler affine metrics of Yau s Schwarz lemma for volume forms. By a theorem of Cheng and Yau, there is a canonical Kahler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an n -dimensional cone, a rescaling of the canonical potential is an n -normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Ampère metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-Kahler space.

Evaluation of the surface free energy of plant surfaces: toward standardizing the procedure

Plant surfaces have been found to have a major chemical and physical heterogeneity and play a key protecting role against multiple stress factors. During the last decade, there is a raising interest in examining plant surface properties for the development of biomimetic materials. Contact angle measurement of different liquids is a common tool for characterizing synthetic materials, which is just beginning to be applied to plant surfaces. However, some studies performed with polymers and other materials showed that for the same surface, different surface free energy values may be obtained depending on the number and nature of the test liquids analyzed, materials' properties, and surface free energy calculation methods employed. For 3 rough and 3 rather smooth plant materials, we calculated their surface free energy using 2 or 3 test liquids and 3 different calculation methods. Regardless of the degree of surface roughness, the methods based on 2 test liquids often led to the under- or over-estimation of surface free energies as compared to the results derived from the 3-Liquids method. Given the major chemical and structural diversity of plant surfaces, it is concluded that 3 different liquids must be considered for characterizing materials of unknown physico-chemical properties, which may significantly differ in terms of polar and dispersive interactions. Since there are just few surface free energy data of plant surfaces with the aim of standardizing the calculation procedure and interpretation of the results among for instance, different species, organs, or phenological states, we suggest the use of 3 liquids and the mean surface tension values provided in this study.