Francisco Urbano

New examples of minimal Lagrangian tori in the complex projective plane

A new family of minimal Lagrangian tori in the complex projective plane is constructed. This family is characterized by its invariability by a one-parameter group of holomorphic isometries of the complex projective plane.

Examples of unstable Hamiltonian-minimal Lagrangian tori in C2

A new family of Hamiltonian-minimal Lagrangian tori in the complex Euclidean plane is constructed. They are the first known unstable ones and are characterized in terms of being the only Hamiltonian-minimal Lagrangian tori (with non-parallel mean curvature vector) in C2 admitting a one-parameter group of isometries.

Twistor holomorphic Lagrangian surfaces in the complex projective and hyperbolic planes

We completely classify all the twistor holomorphic Lagrangian immersions in the complex projective plane ℂℙ2, i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over ℂℙ2 are holomorphic. This classification provides a one-parameter family of examples of Lagrangian spheres in ℂℙ2.

SECOND VARIATION OF SUPERMINIMAL SURFACES INTO SELF-DUAL EINSTEIN FOUR-MANIFOLDS

The index of a compact orientable superminimal surface of a selfdual Einstein four-manifold M with positive scalar curvature is computed in terms of its genus and area. Also a lower bound of its nullity is obtained. Applications to the cases M =

Index of Lagrangian submanifolds of ℂℙ n and the Laplacian of 1-forms

4 and M = CIp2 are given, characterizing the standard Veronese immersions and their twistor deformations as those with lowest index.

On the Gauss curvature of compact surfaces in homogeneous 3-manifolds

Inequalities, involving the two first eigenvalues of the Laplacian acting on 1-forms of minimal Lagrangian submanifolds of the complex projective space, are obtained. The Clifford torus in the complex projective plane is characterized by its index.

Willmore Surfaces of ℝ4 and the Whitney Sphere

Compact flat surfaces of homogeneous Riemannian 3-manifolds with isometry group of dimension 4 are classified. Non-existence results for compact constant Gauss curvature surfaces in these 3-manifolds are established.

Second variation of one-sided complete minimal surfaces.

We make a contribution to the study of Willmore surfaces infour-dimensional Euclidean space ℝ4 by making useof the identification of ℝ4 with two-dimensionalcomplex Euclidean space ℂ2. We prove that theWhitney sphere is the only Willmore Lagrangian surface of genus zero inℝ4 and establish some existence and uniquenessresults about Willmore Lagrangian tori in ℝ4≡ ℂ2.

On stable compact minimal submanifolds

The stability and the index of complete one-sided minimal surfaces of certain three-dimensional Riemannian manifolds with positive scalar curvature are studied.

A characterization of the Lagrangian pseudosphere

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.