Searcharxiv⌕ Search

arXiv subjects

Francisco Urbano

Publications and source records attributed to Francisco Urbano.

14 recordsLinked to original sources
arXiv2022

Index of compact minimal submanifolds of the Berger spheres

The stability and the index of compact minimal submanifolds of the Berger spheres $\mathbb{S}^{2n+1}_τ,\, 0<τ\leq 1$, are studied. Unlike the case of the standard sphere ($τ=1$), where there are no stable compact minimal submanifolds, the Berger spheres have stable ones if and only if $τ^2\leq 1/2$. Moreover, there are no stable compact minimal $d$-dimensional submanifolds of $\mathbb{S}^{2n+1}_τ$ when $1 / (d+1) < τ^2 \leq 1$ and the stable ones are classified for $τ^2=1 / (d+1)$ when the submanifold is embedded. Finally, the compact orientable minimal surfaces of $\mathbb{S}^3_τ$ with index one are classified for $1/3\leqτ^2\leq 1$.

math.DG↗
arXiv2016

On hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$

We classify the homogeneous and isoparametric hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$. In the classification, besides the hypersurfaces $\mathbb{S}^1(r)\times\mathbb{S}^2,\,r\in (0,1]$, it appears a family of hypersurfaces with three different constant principal curvatures and zero Gauss-Kronecker curvature. Also we classify the hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$ with at most two constant principal curvatures and, under certain conditions, with three constant principal curvatures.

math.DG↗
arXiv2013

Minimal surfaces in S^2xS^2

A general study of minimal surfaces of the Riemannian product of two spheres S^2xS^2 is tackled. We stablish a local correspondence between (non-complex) minimal surfaces of S^2xS^2 and certain pair of minimal surfaces of the sphere S^3. This correspondence also allows us to link minimal surfaces in S^3 and in the Riemannian product S^2xR. Some rigidity results for compact minimal surfaces are also obtained.

math.DG↗
arXiv2010

On stable compact minimal submanifolds

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable compact minimal surfaces of the product of a 2-sphere and any Riemann surface are the complex ones.

math.DG↗
arXiv2009

Compact stable constant mean curvature surfaces in the Berger spheres

In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (a close to 0) there are unstable constant mean curvature spheres. Also, we classify the orientable compact stable constant mean curvature surfaces in S^3(a), 1/3 <= a < 1 proving that they are spheres or the minimal Clifford torus in S^3(1/3). This allows to solve the isoperimetric problem in these Berger spheres.

math.DG↗
arXiv2008

Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2

Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in S^2xS^2 and H^2xH^2 are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of S^2xR and H^2xR is established. Using that, surfaces with vanishing Hopf differentials (in particular spheres with parallel mean curvature vector) are classified and a rigidity result for constant mean curvature surfaces of S^2xR and H^2xR is proved.

math.DG↗
arXiv2007

On Hamiltonian stationary Lagrangian spheres in non-Einstein Kaehler surfaces

Hamiltonian stationary Lagrangian spheres in Kaehler-Einstein surfaces are minimal. We prove that in the family of non-Einstein Kaehler surfaces given by the product $Σ_1\timesΣ_2$ of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example is defined when the surfaces $Σ_1$ and $ Σ_2$ are spheres.

math.DG↗
arXiv2006

Minimal Lagrangian surfaces in $S^2 \times S^2$

We deal with the minimal Lagrangian surfaces of the Einstein-Kähler surface $S^2 \times S^2$, studying local geometric properties and showing that they can be locally described as Gauss maps of minimal surfaces in $S^3 \subset R^4$. We also discuss the second variation of the area and characterize the most relevant examples by their stability behaviour.

math.DG↗
arXiv2005

Hamiltonian stability and index of minimal Lagrangian surfaces of the complex projective plane

We show that the Clifford torus and the totally geodesic real projective plane RP^2 in the complex projective plane CP^2 are the unique Hamiltonian stable minimal Lagrangian compact surfaces of CP^2 with genus less than or equal to 4, when the surface is orientable, and with Euler characteristic greater than or equal to -1, when the surface is nonorientable. Also we characterize RP^2 in CP^2 as the least possible index minimal Lagrangian compact nonorientable surface of CP^2.

math.DG↗
arXiv2004

Hamiltonian-minimal Lagrangian submanifolds in complex space forms

Using Legendrian immersions and, in particular, Legendre curves in odd dimensional spheres and anti De Sitter spaces, we provide a method of construction of new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces, including explicit one parameter families of embeddings of quotients of certain product manifolds. In addition, new examples of minimal Lagrangian submanifolds in complex projective and hyperbolic spaces also appear. Making use of all of them, we get Hamiltonian-minimal and special Lagrangian cones in complex Euclidean space too.

math.DG↗
arXiv2004

On a new construction of special Lagrangian immersions in complex Euclidean space

We construct new special Lagrangian submanifolds in complex Euclidean space using a pair of minimal Legendrian submanifolds in odd-dimensional spheres and certain Lagrangian surface belonging to a family that can be considered as a generalization of the special Lagrangian surfaces in complex Euclidean plane. Our examples include those invariant under the standard action of SO(p+1)xSO(q+1) on C^n = C^(p+1) x C^(q+1), n=p+q+2.

math.DG↗
arXiv2000

A Willmore functional for compact surfaces of complex projective plane

We propose the study of a conformally invariant functional for surfaces of complex projective plane which is closely related to the classical Willmore functional. We show that minimal surfaces of complex projective plane are critical for this functional and construct some minima for it via the twistors spaces of complex projective plane. Also, we find lower bounds for this functional and for its restriction to the class of Lagrangian surfaces and characterize the complex lines and the Lagrangian totally geodesic surfaces and the Whitney spheres as the only attaining those bounds.

math.DG↗