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A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of…
Motivated by geometry processing for surfaces with non-trivial topology, we study discrete harmonic maps between closed surfaces of genus at least two. Harmonic maps provide a natural framework for comparing surfaces by minimizing…
We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We…
We present uniqueness results for enclosing ellipses of minimal area in the hyperbolic plane. Uniqueness can be guaranteed if the minimizers are sought among all ellipses with prescribed axes or center. In the general case, we present a…
We give bounds on geodesic distances on the Stiefel manifold, derived from new geometric insights. The considered geodesic distances are induced by the one-parameter family of Riemannian metrics introduced by H\"uper et al. (2021), which…
Let $\rho_\Sigma=h(|z|^2)$ be a metric in a Riemann surface $\Sigma$, where $h$ is a positive real function. Let $\mathcal H_{r_1}=\{w=f(z)\}$ be the family of univalent $\rho_\Sigma$ harmonic mapping of the Euclidean annulus…
While there may be many Thurston metric geodesics between a pair of points in Teichm\"uller space, we find that by imposing an additional energy minimization constraint on the geodesics, thought of as limits of harmonic map rays, we select…
One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincar\'e…
This is the third paper in a series in which we prove Thurston's conjectural duality between best Lipschitz maps and transverse measures. In the second paper we found a special class of best Lipschitz maps between hyperbolic surfaces…
Chapter 1 is a short history of non-Euclidean geometry, which synthesises my readings of mostly secondary sources. Chapter 2 presents each of the main models of hyperbolic geometry, and describes the tesselation of the upper half-plane…
We determine the asymptotic behavior of the optimal Lipschitz constant for the systole map from Teichmuller space to the curve complex.
Any subanalytic germ $(X,0) \subset (\mathbb R^n,0)$ is equipped with two natural metrics: its outer metric, induced by the standard Euclidean metric of the ambient space, and its inner metric, which is defined by measuring the shortest…
A geometric construction is provided that associates to a given flat front in $\mathbb{H}^3$ a pair of minimal surfaces in $\mathbb{R}^3$ which are related by a Ribaucour transformation. This construction is generalized associating to a…
We study various regularity properties of minimizers of the $\Phi$--perimeter, where $\Phi$ is a norm. Under suitable assumptions on $\Phi$ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is…
In many singular metric spaces, the regularity of a shortest-length curve is unknown. Algebraic varieties, or more generally sets defined by finitely many polynomial or real analytic equalities or inequalities, all locally partition into…
Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through…
We consider a surface $M$ immersed in $\mathbb{R}^3$ with induced metric $g=\psi\delta_2$ where $\delta_2$ is the two dimensional Euclidean metric. We then construct a system of partial differential equations that constrain $M$ to lift to a…
We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak…
The Dirichlet Laplacian between two parallel hypersurfaces in Euclidean spaces of any dimension in the presence of a magnetic field is considered in the limit when the distance between the hypersurfaces tends to zero. We show that the…
Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly,…