arXiv subjects

Graham Andrew Smith

Publications and source records attributed to Graham Andrew Smith.

3 recordsLinked to original sources

arXiv2026

Weil--Petersson homeomorphisms, minimal lagrangian diffeomorphisms, and maximal surfaces in anti-de Sitter space

In this paper, we study the class of Weil--Petersson circle homeomorphisms from the point of view of three-dimensional anti-de Sitter space $\mathbf{AdS}^{2,1}$. We show that a homeomorphism $\varphi:\mathbf{RP}^1\to\mathbf{RP}^1$ is Weil--Petersson if and only if its graph, viewed as a curve in the boundary at infinity of $\mathbf{AdS}^{2,1}$, is the asymptotic boundary of a complete maximal spacelike surface in $\mathbf{AdS}^{2,1}$ with finite renormalized area. As an application, we obtain the following AdS-independent result in Teichm\"uller theory: a homeomorphism is Weil--Petersson if and only if its minimal lagrangian extension to $\mathbf{H}^2$ has square-integrable Beltrami differential. We also provide two further new technical characterizations, which we believe to be of independent interest, and which are essential for the proofs of our main results.

math.DG↗

arXiv2024

On the elliptic sinh-Gordon equation with integrable boundary conditions II -- Baker-Akhiezer theory

We study finite-type solutions of the elliptic sinh-Gordon equation along the strip $\Bbb{R}\times[0,L]$ with Durham conditions on each boundary component. We determine necessary rationality criteria for these conditions to be satisfied on both components. These rationality criteria are analogous to the necessary and sufficient criteria for finite-type solutions to be periodic. However, in contrast to the periodic case, they are not in themselves sufficient for Durham conditions to be satisfied on both boundary components: certain additional criteria must also be satisfied at 4 specific points of the spectral curve. Nevertheless, in the special case where the Durham conditions imposed on the two boundary components are complementary to one another in a sense that we will clarify, these rationality criteria are, indeed, sufficient. This study is a necessary preliminary for any general construction of finite-type solutions of the elliptic sinh-Gordon equation subject to Durham boundary conditions.

math.AP↗

arXiv2023

On a convexity property of the space of almost fuchsian immersions

We study the space of Hopf differentials of almost fuchsian minimal immersions of compact Riemann surfaces. We show that the extrinsic curvature of the immersion at any given point is a concave function of the Hopf differential. As a consequence, we show that the set of all such Hopf differentials is a convex subset of the space of holomorphic quadratic differentials of the surface. In addition, we address the non-equivariant case, and obtain lower and upper bounds for the size of this set.

math.DG↗