The goal of this article is to study the pinching problem proposed by S.-T. Yau in 1990 replacing sectional curvature by one weaker condition on biorthogonal curvature. Moreover, we classify 4-dimensional compact oriented Riemannian manifolds with nonnegative biorthogonal curvature. In particular, we obtain a partial answer to Yau Conjecture on pinching theorem for 4-dimensional compact manifolds.

In this paper we obtain an estimate for the Ricci curvature and a criterion for the vanishing of the homology groups of compact submanifolds of spheres and Euclidean spaces. This criterion depends on the results of Lawson and Simons [LS], Leung [Le2] and Xin [X] on stable currents. As consequences we obtain a topological version of theorems of Cheng and Nakagawa [CK], Alencar and do Carmo [AC], and Xu [Xu], on hypersurfaces in spheres with constant mean curvature. c an isometric immersion of a connected n-dimensional Riemannian manifold M n into a complete, simply connected n m-dimensional manifold Q nm c with constant sectional curvature c, where n b 2 and m b 1. LetH, H and S denote the mean curvature vector of the immersion, its norm and the square of the length 313

The purpose of this paper is to prove a Hitchin-Thorpe inequality for a fourdimensional compact almost Ricci soliton. Moreover, we prove that under a suitable integral condition, a four-dimensional compact almost Ricci soliton is isometric to standard sphere. Finally, we prove that under a simple condition, a four-dimensional compact Ricci soliton with harmonic self-dual part of Weyl tensor is either isometric to a standard sphere S 4 or is Kaehler-Einstein.

Background:The use of deoxycholic acid to reduce localized fat deposits is a procedure that has been in use for about 30 years. Its effectiveness as treatment is due to emulsification of phospholipids and therefore, solubilization of the biological membranes with resulting fat necrosis. The purpose of the study was to assess the effectiveness and the safety of an injectable solution containing sodium deoxycholtate 1.25% (DB125), used as intralipotherapy.Methods:The effectiveness and safety of DB125 solution have been assessed with a multicentre observational prospective study carried out between February and October 2017. The 221 selected patients presented with various forms and degrees of localized fat in several areas. Intralipotherapy treatments were performed 6 weeks apart and until the clinical result was obtained. Aesthetic outcomes were evaluated by the authors using preoperative and postoperative photographic documentation and by the patients with their level of satisfaction by filling out an anonymous form. Major adverse events were reported by each doctor who performed the treatment.Results:Two hundred twenty-one patients treated in 273 cases of different localized fat deposits. One hundred eighty-five patients who could be assessed for final results gave the effectiveness of the treatment an average score of 7.4. The failure percentage of the treatment was 3.8%. The medical evaluation showed treatment success in 93.5% of cases. Adverse events can be divided into 2 groups: minor adverse events, which are very frequent and major ones, which are extremely rare. For both groups, the adverse events can be ascribed to localized problems in the treatment area.Conclusion:Studies have shown that the second-generation solution containing sodium deoxycholate 1.25% is effective and safe to treat different localized fat deposits. The high degree of effectiveness shown in the study was not associated with a lesser degree of handling because, at the doses indicated and with the use of intralipotherapy, the occurrence of adverse events was minimal.