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18
result(s) for
"Castro-Infantes, Jesús"
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Curves in the Lorentz-Minkowski plane with curvature depending on their position
Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in the Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in
whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular momentum (with respect to the origin) and geometric linear momentum (with respect to the fixed lightlike geodesic), respectively, we get two abstract integrability results to determine such curves through quadratures. In this way, we find out several new families of Lorentzian spiral, special elastic and grim-reaper curves whose intrinsic equations are expressed in terms of elementary functions. In addition, we provide uniqueness results for the generatrix curve of the Enneper surface of second kind and for Lorentzian versions of some well-known curves in the Euclidean setting, like the Bernoulli lemniscate, the cardioid, the sinusoidal spirals and some non-degenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.
Journal Article
Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
This article is motivated by a problem posed by David A. Singer in 1999 and by the classical Euler elastic curves. We study spacelike and timelike curves in the Lorentz-Minkowski plane 𝕃
whose curvature is expressed in terms of the Lorentzian pseudodistance to fixed geodesics. In this way, we get a complete description of all the elastic curves in 𝕃
and provide the Lorentzian versions of catenaries and grim-reaper curves. We show several uniqueness results for them in terms of their geometric linear momentum. In addition, we are able to get arc-length parametrizations of all the aforementioned curves and they are depicted graphically.
Journal Article
New Plane Curves with Curvature Depending on Distance from the Origin
Motivated by a problem posed by David A. Singer in 1999 and by the classical Bernoulli lemniscate and the Norwich spiral, we study the plane curves whose curvature is expressed in terms of the distance from a point. In terms of the geometric angular momentum, we provide new characterizations of some well-known curves, like the mentioned Bernoulli lemniscate, the inverse Norwich spiral, the anti-clothoid, the cardioid, the sinusoidal spirals and the Cassini ovals. We also find out several new families of plane curves whose intrinsic equations are expressed in terms of elementary functions and we are able to get their arc-length parametrizations and they are depicted graphically.
Journal Article
Genus one H-surfaces with k-ends in H2×R
We construct two different families of properly Alexandrov-immersed surfaces in H2×R with constant mean curvature 0
Journal Article
Genus one$H$ -surfaces with$k$ -ends in$\\mathbb{H}^{2}\\times\\mathbb{R}
We construct two different families of properly Alexandrov-immersed surfaces in H^2 R with constant mean curvature 0
Journal Article
GENUS $1$ MINIMAL k-NOIDS AND SADDLE TOWERS IN $\\mathbb {H}^2\\times \\mathbb {R}
For each
$k\\geq 3$
, we construct a
$1$
-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space
$\\mathbb {H}^2\\times \\mathbb {R}$
with genus
$1$
and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus
$1$
and
$2k$
ends in the quotient of
$\\mathbb {H}^2\\times \\mathbb {R}$
by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature
$-4k\\pi $
. Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus
$1$
in quotients of
$\\mathbb {H}^2\\times \\mathbb {R}$
by the action of a hyperbolic or parabolic translation.
Journal Article
GENUS MINIMAL k -NOIDS AND SADDLE TOWERS IN
For each$k\\geq 3$, we construct a$1$-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space$\\mathbb {H}^2\\times \\mathbb {R}$with genus$1$and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus$1$and$2k$ends in the quotient of$\\mathbb {H}^2\\times \\mathbb {R}$by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature$-4k\\pi $. Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus$1$in quotients of$\\mathbb {H}^2\\times \\mathbb {R}$by the action of a hyperbolic or parabolic translation.
Journal Article
Spherical curves whose curvature depends on distance to a great circle
Motivated by a problem posed by David A. Singer in 1999 and by the elastic spherical curves, we study the spherical curves whose curvature is expressed in terms of the distance to a great circle (or from a point). By introducing the notion of spherical angular momentum, we provide new characterizations of some well known curves, like the mentioned elastic curves, spherical catenaries, loxodromic-type spherical curves, the Viviani's curve, and the spherical Archimedean spirals curves. Furthermore, we show that they may be obtained as critical points of some energy curvature functionals. We also find out several new families of spherical curves whose intrinsic equations are expressed in terms of elementary functions or Jacobi elliptic functions, and we are able to get arc length parametrizations of them.
Paper
Helicoidal minimal surfaces in the 3-sphere: An approach via spherical curves
We prove an existence and uniqueness theorem about spherical helicoidal (in particular, rotational) surfaces with prescribed mean or Gaussian curvature in terms of a continuous function depending on the distance to its axis. As an application in the case of vanishing mean curvature, it is shown that the well-known conjugation between the belicoid and the catenoid in Euclidean three-space extends naturally to the 3-sphere to their spherical versions and determine in a quite explicit way their associated surfaces in the sense of Lawson. As a key tool, we use the notion of spherical angular momentum of the spherical curves that play the role of profile curves of the minimal helicoidal surfaces in the 3-sphere.
Paper
On the asymptotic Plateau problem in \\({\\widetilde{\\mathrm{SL}}_2(\\mathbb{R})}\\)
We prove some non-existence results for the asymptotic Plateau problem of minimal and area minimizing surfaces in the homogeneous space \\({\\widetilde{\\mathrm{SL}}_2(\\mathbb{R})}\\) with isometry group of dimension 4, in terms of their asymptotic boundary. Also, we show that a properly immersed minimal surface in \\({\\widetilde{\\mathrm{SL}}_2(\\mathbb{R})}\\) contained between two bounded entire minimal graphs separated by vertical distance less than \\(\\sqrt{1+4\\tau^2}\\pi\\) have multigraphical ends. Finally, we construct simply connected minimal surfaces with finite total curvature which are not graphs and a family of complete embedded minimal surfaces which are non-proper in \\({\\widetilde{\\mathrm{SL}}_2(\\mathbb{R})}\\).
Paper