XVII School of Differencial Geometry – Preface
Authors:Jaime Ripoll, Walcy Santos
Characterizations of linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces
We deal with complete linear Weingarten hypersurfaces immersed in a locally symmetric Lorentz space, whose sectional curvatures are supposed to obey some standard controls. In this setting, under suitable boundedness on the norm of the traceless part of the second fundamental form, we are able to show that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures one of which is simple
Authors:Cícero P. Aquino, Henrique F. de Lima, Marco Antonio L. Velásquez
Equivariant deformations of Hamiltonian stationary Lagrangian submanifolds
We prove an equivariant deformation result for Hamiltonian stationary Lagrangian submanifolds of a K¨ahler manifold, with respect to deformations of its metric and almost complex structure that are compatible with an isometric Hamiltonian group action. This yields existence of Hamiltonian stationary Lagrangian submanifolds in possibly non-Kähler symplectic manifolds whose metric is arbitrarily close to a Kähler metric
Authors:Renato G. Bettiol, Paolo Piccione, Bianca Santoro
Minimal graphs in PSL2(R, τ )
In this paper we study the existence and non-existence of vertical and horizontal graphs in the space PSL2(R, τ ). We prove that there is no entire horizontal minimal graph. On the other hand, there are entire vertical minimal graphs having prescribed continuous boundary values.
Authors:Abigail Folha, Carlos Pen˜afiel
On harmonic diffeomorphisms from conformal annuli to Riemannian annuli
In this paper we construct a harmonic diffeomorphism from C* onto the quotient of the hyperbolic plane H2 by the group < ψ > generated by some parabolic or hyperbolic isometry ψ. It draws its inspiration to a large extent from [CR]. The proof we give here is based on the theory of minimal surfaces: we construct an entire minimal graph Σ c (H/ < ψ >) × R over H/ < ψ >, such that Σ is conformally C*. This solves the problem since the vertical projection from Σ onto H/ < ψ > is a harmonic diffeomorphism when Σ is minimal
Authors:Martin Leguil, Harold Rosenberg
The asymptotic Plateau problem for convex hypersurfaces of constant curvature in Hyperbolic space
We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in Hn+1 satisfying f(κ) = σ ∈ (0, 1) with a prescribed asymptotic boundary Γ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if Γ is (Euclidean) starshaped, the solution is unique and also (Euclidean) starshaped while if Γ is mean convex the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature of our approach is a “global interior curvature estimate”
On symmetries of singular implicit ODEs
We study implicit ODEs, cubic in derivative, with infinitesimal symmetry at singular points. Cartan showed that even at regular points the existence of nontrivial symmetry imposes restrictions on the ODE. Namely, this algebra has the maximal possible dimension 3 iff the web of solutions is flat. For cubic ODEs with flat 3-web of solutions we establish sufficient conditions for the existence of non-trivial symmetries at singular points and show that under natural assumptions such a symmetry is semi-simple, i.e. is a scaling is some coordinates. We use this symmetry to find first integrals of the ODE.
Author:Sergey I. Agafonov
Stability Properties of Rotational Catenoids in the Heisenberg Groups
In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids Ca (minimal surfaces invariant by rotations) in the Heisenberg group with a left-invariant metric. We show that these catenoids have Morse index at least 3 and we bound the index from above in terms of the parameter a. We also show that the index of Ca tends to infinity with a. Finally, we study the rotationally symmetric stable domains on the higher dimensional catenoids.
Authors:Pierre Bérard, Marcos P. Cavalcante
On Contact Normal Parallel Spacelike Submanifolds in a semi-Riemannian Sasakian space form
In this paper we study the contact normal, spacelike parallel submanifold Mn with parallel mean curvature vector in a semi-Riemannian Sasakian space form M?2m+1 q (c) with m > n and codimension p > q. We use a Simons type inequality to obtain a gap theorem.
Authors:Aldir Brasil, Maxwell Mariano, Rodrigo R. Montes
The boundary term from the Analytic Torsion of a cone over a m-dimensional sphere
We present a direct proof that the Anomaly Boundary term of J. Bru¨ning and X. Ma [BM1, BM2] generalizes to the cases of the cone over a m-dimensional sphere.
On the study of Existence of solutions for a class of equations with critical Sobolev exponent on compact Riemannian Manifold
We study the existence of solutions for a class of non-linear differential equation with critical Sobolev’s exponent on the compact riemannian manifold (Mn, g), n > 6.
Author:Carlos Rodrigues da Silva