Related papers: Application of integral geometry to minimal surfac…
We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms of the underlying conformal geometry.…
An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for…
We refurbish our axiomatics of differential geometry introduced in [Mathematics for Applications,, 1 (2012), 171-182]. Then the notion of Euclideaness can naturally be formulated. The principal objective in this paper is to present an…
The conformal structure on minimal surfaces plays a key role in studying the properties of minimal surfaces. Here we extend the results of uniformization of surfaces with boundary to get the (weak) uniformization results for triple junction…
This paper develops some of the methods of the "Italian School" of algebraic geometry in the context of infinitesimals. The results of this paper have no claim to originality, they can be found in Severi, we have only made the arguments…
In this paper, we systematically investigate the geometry and topology of manifolds with integral radial curvature bounds, and obtain many interesting and important conclusions.
A method for computing singular or nearly singular integrals on closed surfaces was presented by J. T. Beale, W. Ying, and J. R. Wilson [Comm. Comput. Phys. 20 (2016), 733--753, arXiv:1508.00265] and applied to single and double layer…
An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total…
This exposition gives an introduction to the theory of surfaces in Laguerre geometry and surveys some results, mostly obtained by the authors, about three important classes of surfaces in Laguerre geometry, namely L-isothermic, L-minimal,…
It was shown by Ramanathan \cite{R} that any compact oriented non-simply-connected minimal surface in the three-dimensional round sphere admits at most a finite set of pairwise noncongruent minimal isometric immersions. Here we show that…
The author has removed this paper, following the publication of a more complete version.
Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more recently, the Seiberg-Witten invariants.…
This article presents a novel resolution to the problem of spline interpolation versus least-squares fitting on smooth Riemannian manifolds utilizing the method of gradient flows of networks. This approach represents a contribution to both…
This note completes a talk given at the conference Curves over Finite Fields: past, present and future celebrating the publication the book {\em Rational Points on Curves over Finite Fields by J.-P. Serre and organised at Centro de ciencias…
The first part of these notes is a self-contained introduction to generalized complex geometry. It is intended as a `user manual' for tools used in the study of supersymmetric backgrounds of supergravity. In the second part we review some…
The algorithms given in Karney, J. Geodesy 87, 43-55 (2013), to compute geodesics on terrestrial ellipsoids are extended to apply to ellipsoids of revolution with arbitrary eccentricity. For the direct and inverse geodesic problems, this…
Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area…
This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…
This note contains a correction to the paper, ``Local contribution to the Lefschetz fixed point formula'', Inv. Math. 111 (1993), pp. 1-33.
The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact…