The singly periodic, genus one helicoid was conjectured to be the limit of a one parameter family of doubly periodic minimal surfaces referred to as Perturbed Genus One Scherk Surfaces. Using elliptic functions, we show such surfaces exist, solving a two-dimensional period problem by perturbing a one-dimensional problem. Using flat structures associated to these minimal surfaces, we verify the conjecture.

Sullivan’s construction of minimal models for topological spaces is refined for the case of a simply connected closed Riemannian manifold, $(M,\lt,\gt)$, to define a unique finitely generated field extension, $\mathbf{k}$ of $\mathbf{Q}$, baptized the harmonic field of $(M,\lt,\gt)$, and a morphism, $m:(\Lambda V,d)\rightarrow A_{DR}(M)$, from a Sullivan model defined over $\mathbf{k}$. The Sullivan model and the morphism are determined up to isomorphism, and the natural extension of $H(m)$ to $H(\Lambda V,d)\otimes_{\mathbf{k}}\mathbf{R}$ is an isomorphism; in particular, $(\Lambda V,d)$ is isomorphic to a rational Sullivan model for $M$ tensored with $\mathbf{k}$. Examples are constructed to show that every finitely generated extension field of $\mathbf{Q}$ occurs as a harmonic field of such a Riemannian manifold.

The anisotropic fractional $s$-perimeter with respect to a convex body $K$ in $\mathbb{R}^n$ is shown to converge as $s\to 1^-$ to the anisotropic perimeter with respect to the moment body of $K$. For anisotropic fractional $s$-seminorms on $BV(\mathbb{R}^n)$, the corresponding result is established (generalizing results of Bourgain, Brezis, and Mironescu and Dávila). Minimizers of the anisotropic fractional iso-perimetric inequality with respect to $K$ are shown to converge to the moment body of $K$ as $s\to 1^-$. Anisotropic fractional Sobolev inequalities are established.

We define an associative $\mathbb{C}[[h]]$-algebra $\mathcal{A}\mathcal{S}_h(\Sigma)$ generated by regular isotopy classes of arcs and links over a punctured surface $\Sigma$ which is a deformation quantization of the Poisson algebra $\mathcal{C}(\Sigma)$ of arcs and loops on $\Sigma$ endowed with a generalization of the Goldman bracket. We then construct a Poisson algebra homomorphism from $\mathcal{C}(\Sigma)$ to the algebra of smooth functions on the decorated Teichmüller space endowed with a natural extension of the Weil-Petersson Poisson structure described by Mondello. The construction relies on a collection of geodesic lengths identities in hyperbolic geometry which generalize Penner’s Ptolemy relation, the trace identities and Wolpert’s cosine formula. As a consequence, we derive an explicit formula for the geodesic lengths functions in terms of the edge lengths of an ideally triangulated decorated hyperbolic surface.

The hermitian analog of Aleksandrov’s area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily invariant area measures is described explicitly.