Simple piecewise geodesic interpolation of simple and Jordan curves with applications

We explicitly construct simple, piecewise minimizing geodesic, arbitrarily fine interpolation of simple and Jordan curves on a Riemannian manifold. In particular, a finite sequence of partition points can be specified in advance to be included in our construction. Then we present two applications of our main results: the generalized Green’s theorem and the uniqueness of signature for planar Jordan curves with finite p -variation for 1⩽p<2.

A comparison of extra-tropical cyclones in recent re-analyses; ERA-Interim, NASA MERRA, NCEP CFSR and JRA-25

Extra-tropical cyclones are identified and compared using data from four recent re-analyses for the winter periods in both hemispheres. Results show the largest differences occur between the older lower resolution JRA25 re-analysis when compared with the newer high resolution re-analyses, in particular in the Southern Hemisphere (SH). Spatial differences between the newest re-analyses are small in both hemispheres and generally not significant except some common regions associated with cyclogenesis close to orography. Intensities are generally related to spatial resolution except NASA-MERRA which has larger intensities for several different measures. Matching storms between re-analyses shows the number matched between ERA-Interim and the other re-analyses are similar in the Northern Hemisphere (NH). In the SH the number matched between JRA25 and ERA-Interim is lower than in the NH, but for NASA-MERRA and NCEP-CFSR the number matched is similar to the NH. The mean separation of the identically same cyclones is typically less than 20 geodesic in both hemispheres for the latest re-analyses, whereas JRA25 compared with the other re-analyses has a broader distribution in the SH indicating greater uncertainty. The instantaneous intensity differences for matched storms shows narrow distributions for pressure while for winds and vorticity the distributions are much broader indicating larger uncertainty typical of smaller scale fields. Composite cyclone diagnostics show that cyclones are very similar between the re-analyses, with differences being related to the intensities, consistent with the intensity results. Overall, results show NH cyclones correspond well between re-analyses, with a significant improvement in the SH for the latest re-analyses, indicating a convergence between re-analyses for cyclone properties.

The Brownian traveller on manifolds

We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the at case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the at case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.