CONFORMAL MAPS OF SMALL DISKS

Area of small disks

This paper considers Riemannian metrics on two dimensional disks where all geodesics are minimizing. A sharp reverse isoperimetric inequality is proven. This in turn yields near optimal bounds for the area of disks as well as near optimal upper bounds on the first nonzero Neumann eigenvalue of the Laplacian in terms only of the radius.

Conformal Maps and Minimal Surfaces

Conformal maps and non-reversibility of elliptic area-preserving maps

It has been long observed that area-preserving maps and reversible maps share similar results. This was certainly known to G.D. Birkhoff [5] who showed that these two types of maps have periodic orbits near a general elliptic fixed point. The KAM theory, developed by Kolmogorov-ArnoldMoser for Hamiltonian systems [9], [1] and area preserving maps [15], has also been extended a great deal to rev...

Conformal Maps and p-Dirichlet Energies

Let Φ : (M1, g1)→ (M2, g2) be a diffeomorphism between Riemannian manifolds and Φ# : D(M2)→ D(M1) the induced pull-back operator. The main theorem of this work is Theorem 4.1 which relates preservation of the p-Dirichlet energies φ 7→ ∫ Mi |dφ| dμi under Φ# to isometric or conformal properties of Φ. More precisely: In case p = n, Φ# preserves the p-Dirichlet energy if and only if Φ is conformal...

New Quadrature Formulas from Conformal Maps

Gauss and Clenshaw–Curtis quadrature, like Legendre and Chebyshev spectral methods, make use of grids strongly clustered at boundaries. From the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal. Nevertheless such methods may “waste” a factor of π/2 with respect to each space dimension. We propose new nonpolynomial quadrature methods that avoid th...