arXiv subjects
Publications and source records attributed to Jackeline Conrado.
A correspondence is established between a class of minimal immersed surfaces of $\mathbb{S}^3(2)$ and area-minimizing unit vector fields defined on the antipodally punctured unit sphere $\mathbb{S}^2\backslash\{N,S\}$. As a consequence, we establish a stability relation for Lawson cylinders in $\mathbb{S}^3(2)$.
We establish in this paper a sharp lower bound for the area of a unit vector field $V$ defined on some spherical annuli in the Euclidean sphere $\mathbb{S}^2$.
We prove the existence of multiple solutions to the Allen--Cahn--Hilliard (ACH) vectorial equation (with two equations) involving a triple-well (triphasic) potential with a small volume constraint on a closed parallelizable Riemannian manifold. More precisely, we find a lower bound for the number of solutions depending on some topological invariants of the underlying manifold. The phase transition potential is considered to have a finite set of global minima, where it also vanishes, and a subcritical growth at infinity. Our strategy is to employ the Lusternik--Schnirelmann and infinite-dimensional Morse theories for the vectorial energy functional. To this end, we exploit that the associated ACH energy $Γ$-converges to the weighted multi-perimeter for clusters, which combined with some deep theorems from isoperimetric theory yields the suitable setup to apply the photography method. Along the way, the lack of a closed analytic expression for the multi-isoperimetric function for clusters imposes a delicate issue. Furthermore, using a transversality theorem, we also show the genericity of the set of metrics for which solutions to the ACH system are nondegenerate.
We provide a lower value for the volume of a unit vector field tangent to an antipodally Euclidean sphere $\mathbb{S}^2$ depending on the length of an ellipse determined by the indexes of its singularities.
We provide a lower value for the volume of a unit vector field tangent to an antipodally Euclidean sphere $\mathbb{S}^2$ depending on the length of an ellipse determined by the indexes of its singularities. In addition, for minimizing vector fields having specific pair of indexes, we show that their image coincides with the image of minimally immersed Klein bottles.
In these short notes we characterize the loxodromic unit vector fields on antipodally punctured Euclidean spheres as the only ones achieving a lower bound for the volume functional depending on the Poincaré indexes around their singularities.