Abstract:
In this note we prove that any compact Riemannian manifold of dimension $ngeq 4$ which is non-conformal to the standard n-sphere and has positive Yamabe invariant admits infinitely many conformal metrics with nonconstant positive scalar curvature on which the classical sharp Sobolev inequalities admit extremal functions. In particular we show the existence of compact Riemannian manifolds with nonconstant positive scalar curvature for which extremal functions exist. Our proof is simple and bases on results of the best constants theory and Yamabe problem.

Abstract:
We concerns here with the continuity on the geometry of the second Riemannian L^p-Sobolev best constant B_0(p,g) associated to the AB program. Precisely, for 1 <= p <= 2, we prove that B_0(p,g) depends continuously on g in the C^2-topology. Moreover, this topology is sharp for p = 2. From this discussion, we deduce some existence and C^0-compactness results on extremal functions.

Abstract:
We develop a comprehensive study on sharp potential type Riemannian Sobolev inequalities of order 2 by means of a local geometric Sobolev inequality of same kind and suitable De Giorgi-Nash-Moser estimates. In particular we discuss questions like continuous dependence of optimal constants and existence and compactness of extremal maps. The main obstacle arising in the present setting lies at fairly weak conditions of regularity assumed on potential functions.

Abstract:
We introduce the notion of pseudohermitian k-curvature, which is a natural extension of the Webster scalar curvature, on an orientable manifold endowed with a strictly pseudoconvex pseudohermitian structure (referred here as a CR manifold) and raise the k-Yamabe problem on a compact CR manifold. When k=1, the problem was proposed and partially solved by Jerison and Lee for CR manifolds non-locally CR-equivalent to the CR sphere. For k > 1, the problem can be translated in terms of the study of a fully nonlinear equation of type complex k-Hessian. We provide some partial answers related to the CR k-Yamabe problem. We establish that its solutions with null Cotton tensor are critical points of a suitable geometric functional constrained to pseudohermitian structures of unit volume. Thanks to this variational property, we establish a Obata type result for the problem and also compute the infimum of the functional on the CR sphere. Furthermore, we show that this value is an upper bound for the corresponding one on any compact CR manifolds and, assuming the CR Yamabe invariant is positive, we prove that such an upper bound is only attained for compact CR manifolds locally CR-equivalent to the CR sphere. In the Riemannian field, recent advances have been produced in a series of outstanding works.

Abstract:
Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems of metrics in a same conformal class that fixes the boundary and satisfy certain integral conditions on the the scalar curvatures and the mean curvatures on the boundary. No condition on the first eigenvalues of operators is need.

Abstract:
Let $(M^{n+1},g,e^{-f}d\mu)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-\'{E}mery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded $f$-minimal hypersurfaces in $M$ with uniform upper bounds on $f$-index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in $\mathbb{R}^{n+1}$ with $2\leq n\leq 6$. We also prove some estimates on the $f$-index of $f$-minimal hypersurfaces, and give a conformal structure of $f$-minimal surface with finite $f$-index in three-dimensional smooth metric measure space.

Abstract:
We prove a positive mass theorem for $n$-dimensional asymptotically flat manifolds with a non-compact boundary if either $3\leq n\leq 7$ or if $n\geq 3$ and the manifold is spin. This settles, for this class of manifolds, a question posed in a recent paper by the first author in connection with the long-term behavior of a certain Yamabe-type flow on scalar-flat compact manifolds with boundary.

Abstract:
We show that the limit at infinity of the vector-valued Brown-York-type quasi-local mass along any coordinate exhaustion of an asymptotically hyperbolic $3$-manifold satisfying the relevant energy condition on the scalar curvature has the conjectured causal character. Our proof uses spinors and relies on a Witten-type formula expressing the asymptotic limit of this quasi-local mass as a bulk integral which manifestly has the right sign under the above assumptions. In the spirit of recent work by Hijazi, Montiel and Raulot, we also provide another proof of this result which uses the theory of boundary value problems for Dirac operators on compact domains to show that a certain quasi-local mass, which converges to the Brown-York mass in the asymptotic limit, has the expected causal character under suitable geometric assumptions.

Abstract:
The
soybean crop has great economical importance in Brazil and in the world. In
order to make the crop production profitable, several factors must be considered.
The objective of this study was to evaluate the impact of spacing between
soybean crop rows (Glycine max). The
experiment was installed in the Mutuca farm
(Arapoti—PR, southern Brazil), in the crop seasons of
2012/2013 (four seeding seasons) and 2013/2014 (two seeding seasons), in a
completely randomized blocks design. We used four treatments and six
replicates. The treatments were the spacing between rows as follow: 0.25, 0.50,
0.75 and 1.00 m. The variables evaluated were: initial and final plant
population, plant height, number of internodes, viable internodes, pods per
plant, grains per pod, mass of thousand grains and crop productivity. We
concluded that the reduction of the spacing between rows significantly
increased, in most part of the crop seasons, the number of pods per plants and
the crop productivity.

Abstract:
O objetivo foi estudar a introdu o de 8% de gr os e subprodutos (farelo ou torta) da canola em dietas para cordeiros. Para a avalia o das características quantitativas da carca a, foram utilizadas 24 carca as de cordeiros, utilizando delineamento inteiramente casualizado. As dietas com média de 15,4% de PB na MS e 80,2% de NDT foram compostas por 40% de feno de capim-Tifton e 60% de concentrado composto por milho em gr o, farelo de soja, canola em gr o integral, farelo de canola, torta de canola e mistura mineral. A utiliza o de gr os e subprodutos da canola na dieta de borregos terminados em confinamento n o influenciou (p > 0,05) as características quantitativas da carca a. Em rela o aos rendimentos dos cortes, n o houve efeito dos tratamentos para nenhuma das variáveis analisadas. Assim, a introdu o de 8% de gr os e subprodutos (farelo ou torta) da canola possibilitaram bons resultados podendo ser recomendados nas formula es de dietas para cordeiros. The aim of this work was to evaluate the introduction of 8% grains and by-products (meal or cake) of canola in the diets of lambs. To evaluate quantitative carcass characteristics, 24 Santa Ines lambs were used in a completely randomized design. Diets with averages of 15.4% of CP in DM and 80.2% of TDN were composed for 40% Tifton hay and 60% concentrate based on corn grain, soybean meal, whole grain canola, canola meal, canola cake and mineral mixture. The use of whole grains and by-products of canola in the diet of lambs finished in feedlot did not influence (p > 0.05) quantitative carcass characteristics. For cut dressing in relation to the CCW, no effect was observed for the analyzed variables among treatments. It was concluded that the introduction of grains and by-products of canola allow for satisfactory results, and could be recommended in the formulations of lamb diets.