Related papers: Lamarle Formula in 3-Dimensional Lorentz Space
We prove that the mean curvature of a smooth surface in $\mathbb{R}^n$, $n\geq 2$, arises as the limit of a sequence of functions that are intrinsically related to the difference between an $n$- and $1$-dimensional fractional Laplacian of a…
Eynard's formulation of Hermitean 1-matrix models in terms of intrinsic quantities of an associated hyperelliptic Riemann surface is rephrased as a Lagrangean field theory of a scalar particle propagating on the hyperelliptic surface with…
We investigate the relationship among characteristic curves on developable surfaces. In case parameter curves coincide with these curves, we show that the base curve of a developable surface could be either a plane curve, a circular helix,…
The spatial distribution of galaxies is a highly complex phenomenon currently impossible to predict deterministically. However, by using a statistical $\textit{bias}$ relation, it becomes possible to robustly model the average abundance of…
The relation between level lines of Gaussian free fields (GFF) and SLE(4)-type curves was discovered by O. Schramm and S. Sheffield. A weak interpretation of this relation is the existence of a coupling of the GFF and a random curve, in…
Timelike sectional curvature bounds play an important role in spacetime geometry, both for the understanding of classical smooth spacetimes and for the study of Lorentzian (pre-)length spaces introduced in \cite{kunzinger2018lorentzian}. In…
The goal of this paper is to extend to two-dimensional optimal control systems with scalar input the classical notion of Gaussian curvature of two-dimensional Riemannian surface using the Cartan's moving frame method. This notion was…
This paper gives a detailed derivation of the surface of a tri-axial ellipsoid. The general result is in terms of the elliptic integrals of the first and second kind. It is in checked for all special cases included and the corresponding…
One of the most important tasks in mathematics and physics is to connect differential geometry and nonlinear differential equations. In the study of nonlinear optics, integrable nonlinear differential equations such as the nonlinear…
We prove new results on existence of solutions for the prescribed gaussian curvature problem on the euclidean sphere S^2. Those results are achieved by relating this problem with the holomorphic triples theory on Riemann surfaces. We think…
We study both diffuse and sharp liquid-vapor interfaces. The equilibrium equation of fluids is derived by using the principle of virtual work in a domain including the interfaces. For diffuse interfaces, the surface tension coefficient…
We investigate the universal property of curvatures in surface models which display a flat phase and a rough phase whose criticality is described by the Gaussian model. Earlier we derived a relation between the Hessian of the free energy…
The hypersurface is one of the most important objects in a space. Many authors studied diffrent geometric aspects of hypersurfaces in a space. In this paper, we define three types of 2-ruled hypersurfaces in a Walker 4-manfold E 41 . We…
In this paper we introduce the notion of contact angle for an immersed surface in three dimensional sphere. We deduce formulas for the Laplacian and for the Gaussian curvature, and we classify minimal surfaces in $S^3$ with constant contact…
Averaged inhomogeneous cosmologies lie at the forefront of interest, since cosmological parameters like the rate of expansion or the mass density are to be considered as volume-averaged quantities and only these can be compared with…
In this paper, we study Mannheim surface offsets in dual space. By the aid of the E. Study Mapping, we consider ruled surfaces as dual unit spherical curves and define the Mannheim offsets of the ruled surfaces by means of dual geodesic…
By introducing the divergence of a vector potential into the Lagrangian, a Lagrangian framework is developed to incorporate surface energy into elasticity. Besides the Euler-Lagrange equation and natural boundary condition, a new boundary…
We point out how some recent developments in the theory of constant scalar curvature K\"ahler metrics can be used to clarify the existence issue for such metrics in the special case of geometrically ruled complex surfaces.
How do we characterize the shape of a surface? It is now well understood that the shape of a surface is determined by measuring how curved it is at each point. From these measurements, one can identify the directions of largest and smallest…
Let G be a n-dimensional Lie group (n>2) with a bi-invariant Riemannian metric. We prove that if a surface of constant Gaussian curvature in G can be expressed as the product of two curves, then it must be flat. In particular, we can…