Volume 40 – 2011 – The International Meeting on Differential Geometry and Partial Differential Equations

Artigos

Some characterizations of Euclidean domains in the steady state and hyperbolic spaces

Under suitable restrictions on the image of the Gauss mapping and on the values of the mean curvature, we extend the technique developed by Colares jointly with the second author in [7], in order to establish characterization results concerning the Euclidean domains of the steady state space Hn+1 and of the hyperbolic space Hn+1. As applications of such characterizations, we obtain rigidity theorems for the spacelike hyperplanes of Hn+1 and for the horospheres of Hn+1.
Autores:C.P. Aquino, H. F. de Lima, M. A. L. Velásquez

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Orbital stability of periodic travelling wave solutions of the modified Boussinesq equation

This paper is concerned with stability of periodic travelling wave solutions of the modified Boussinesq equation. It will be shown that the constants and a nontrivial class of these solutions are nonlinearly stable in the energy space for a range of their speeds of propagation and periods.
Autor:Lynnyngs Kelly Arruda

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Wiener Measures on Riemannian Manifolds and the Feynman-Kac Formula

This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schr¨odinger operators with L∞-potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.
Autores:Christian Bär, Frank Pfäffle

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Immersion of almost Ricci solitons into a Riemannian manifold

The principal aim of this short paper is to study immersions of an almost Ricci soliton or a Ricci soliton (Mn, g, X, λ) into a Riemannian manifold Mfn+p. First we shall present some obstruction results in order to obtain a minimal immersion under conditions on the sectional curvature of Mfn+p. When Mfn+p is a space form Mfn+pc of sectional curvature c, the pinching λ ≥ (n−1)(c+H2) gives that such an immersion is umbilical. Finally, concerning to Ricci solitons we shall show that a shrinking Ricci soliton immersed into a space form with constant mean curvature must be the Gaussian soliton or its traceless tensor associated to the second fundamental form has supremum strictly positive.
Autores:Abdênago Barros, José N. Gomes, Ernani Ribeiro Jr

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Some generalizations of Calabi compactness theorem

In this paper we obtain generalized Calabi-type compactness criteria for complete Riemannian manifolds that allow the presence of negative amounts of Ricci curvature. These, in turn, can be rephrased as new conditions for the positivity, for the existence of a first zero and for the nonoscillatory-oscillatory behaviour of a solution g(t) of g′′ + Kg = 0, subjected to the initial condition g(0) = 0, g′(0) = 1. A unified approach for this ODE, based on the notion of critical curve, is presented. With the aid of suitable examples, we show that our new criteria are sharp and, even for K ≥ 0, in borderline cases they improve on previous works of Calabi, Hille-Nehari and Moore.
Autores:Bruno Bianchini, Luciano Mari, Marco Rigoli

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Energy and volume of vector fields on spherical domains (RETRACTED)

The main result of this work can be obtained from the paper with the same title and by the same authors which appeared in Pacific Journal of Mathematics 257 (2012), 1-7.

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A superlinear type problem for a p-laplacian perturbation

In this work we investigate existence and multiplicity of positive solutions for the superlinear type problem where Ω ⊂ R N is an open bounded domain, q > p > 1 and f changes sign.
Autores:F. O. de Paiva, H. R. Quoirin

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A survey on Beurling-Selberg majorants and some consequences of the Riemann hypothesis

This article briefly describes the recent advances on the Beurling-Selberg extremal problem in harmonic analysis and its connection with the theory of the Riemann zeta-function. In particular, under the Riemann hypothesis, this extremal tool provides improved bounds for the size of ζ(s) in the critical strip, for the argument function S(t) and for its antiderivative, the function S1(t).
Autor:Emanuel Carneiro

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Carleman Inequality and Null Controllability for Parabolic Equations

This paper is concerned with a detailed exposition on the Carleman inequality for a parabolic equation. Specifically, it represents only a part of the work of A. V. Fursikov & O. Yu Imanovilov [7] for the particular model pt − ∆p + f(p) = h of the heat equation. Moreover, we study the null controllability employing fixe points for multi-valued mapping.
Autores:J. Limaco, H. R. Clark, S. B. de Menezes, L. A. Medeiros

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On the Ricci curvature equation and the Einstein equation for diagonal tensors

We consider the pseudo-euclidean space (Rn, g), with n ≥ 3. We provide necessary and sufficient conditions for a diagonal tensor to admit a metric ¯g, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Examples of complete metrics are included.
Autores:Romildo Pina, Keti Tenenblat

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The r−mean curvature equation of a graph and scalar flat hypersurfaces revisited

Among the results we discuss in this work we will see how to transform non-singular analytic curves Σ in C2 into strictly convex scalar flat 3-dimensional hypersurfaces.
Autor:Sebasti˜ao C. de Almeida

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Stability of Non Liquid Bridges

We give a new, simpler proof characterizing the stable non liquid bridges. Numerical examples are given which show that the conditions imposed on the functionals in these theorems are essential. We also show that arbitrary convex non liquid bridges are always stable.
Autores:Josu Arroyo, Miyuki Koiso, Bennet Palmer

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Shadow boundaries in space forms

We study shadow boundaries of submanifolds of Riemannian mani-folds admitting a closed conformal vector field. As applications we give a method to find a principal direction in a compact hypersurface and a characterization of totally umbilical hypersurfaces in space forms.
Autores:Oscar Palmas, Gabriel Ruiz-Hernández

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