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

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.

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.

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.

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.

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.