arXiv subjects

M. P. Dussan

Publications and source records attributed to M. P. Dussan.

4 recordsLinked to original sources

arXiv2023

Spacelike minimal surfaces in R^4_1 through of a $θ$-Family

In this paper we introduce a $θ$-family of spacelike surfaces in the Lorentz-Minkowski space R^4_1 based in two complex valued functions $a(w), μ(w)$, which when they are holomorphic we will be dealing with a family of spacelike minimal surfaces. The $θ$-family is such that it connects spacelike minimal surfaces in R^3_1 to spacelike minimal surfaces in R^3. We study the family through of the curvature and we prove that the family preserves planar points and moreover, that the existence of planar points corresponds to the existence of solutions of equation |a_w(w)|^2 =0. We also show that if a pair of surfaces are associated through of a $θ$-family then they can not be complete surfaces. As applications we focus to one type of graph surfaces in R^4_1 and we prove that if the imaginary part of $a(w)$ is zero at least in a point then the surface cannot assume local representations of that type of graph. Several explicit examples are given.

math.DG↗

arXiv2023

Geodesic Lines in Fields of Velocity

This work is a purely syntactic geometric exploration of some few elements, which are our axioms, that in last instance it is the set of differential equations whose solutions give the geodesic lines of the Schwarzschild spacetime. We observe that non new physics principles or postulates will be introduced in this work. We only link the Bohr's atoms model with the Einstein's relativity through of a common geometric syntax. To obtain this common syntax, we will define the {\it extended Lorentz group}, which is defined to preserve the volume form of the Minkowski spacetime. The Schwarzschild spacetime will be defined as a manifold associated to a set of radial fields of velocities within of the four-dimensional Minkowski vectorially space form. Our procedure includes a comparison of the Newtonian and the Schwarzschild times along geodesic lines. Our constructions have strong influence of the Einstein paper about the energy content produced by fields, as well as by the Schrödinger digression about the annihilation of matter. We define the orbital associated to the Kepler's laws as a set of elliptical orbits, which have equal eccentricity and equal major semi-axis. Then identifying the eccentricity with the relativistic velocity we will obtain a thermodynamic equivalence between the increasing of mass in kinetic form in special relativity theory and an adiabatic process with degree of freedom equal to 2. The eccentricity will be the needed velocity to move the revolution ellipsoid and so to obtain a contraction of its major axis such that it converts into a sphere with radius given by the minor semi-axis. Therefore we can associate to the each class of equal eccentricity orbital an unique timelike unit vector, which is called {\it the observer} of class.

math.GM↗

arXiv2021

Minimal spacelike surfaces and the graphic equations in R^4_1

In this paper we study an extension of the Bernstein Theorem for minimal spacelike surfaces of the four dimensional Minkowski vector space form and we obtain the class of those surfaces which are also graphics and have non-zero Gauss curvature. That is the class of entire solutions of a system of two elliptic non-linear equations that is an extension of the equation of minimal graphic of $\mathbb R^3$. Therefore, we prove that the so-called Bernstein property does not hold in general for the case of graphic spacelike surfaces in $\mathbb R^4_1$. In addition, we also obtain explicitly the conjugated minimal spacelike surface, and identify the necessary conditions to extend continuously a local solution of the generalized Cauchy-Riemann equations.

math.DG↗

arXiv2019

Timelike surfaces in the de Sitter space $\mathbb S^3_1(1)\subset \mathbb R^4_1$

This paper studies timelike minimal surfaces in the De Sitter space $\mathbb S^3_1(1) \subset \mathbb R^4_1$ via a complex variable. Using complex analysis and stereographic projection of lightlike vectors we obtain a representation formula. Real and complex special quadrics in $\mathbb CP^3$ are identified with the grassmannians of spacelike and timelike oriented 2-planes of $\mathbb R^4_1$, and the normal frame is written in terms of certain complex valued functions $x$ and $y$, which may be considered holomorphic functions as a special case. Then several results describing the analytic restrictions via solutions of certain PDE in complex variable, are shown. Finding solutions allows us to identify explicitly the representation of the associated surfaces. Moreover, using our technique we find a new kind of complex function which we call quasi-holomorphic and which satisfy a generalized version of the Cauchy-Riemann equations. Our technique allows the explicit construction of many families of minimal timelike surfaces in $\mathbb S^3_1(1)$ whose intrinsic Gauss map will also belong to the same class of surfaces.

math.DG↗