### 3-manifolds with(out) metrics of nonpositive curvature.

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In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for metric measure spaces. Our new definition, based only on properties of the distance, allows also us to deal with discrete metric measure spaces. Then we show the stability of our new inequality under convergence of metric measure spaces. This result gives as corollary the stability of the classical Brunn-Minkowski inequality for geodesic spaces. The proof of this stability...

We propose to study a fully nonlinear version of the Yamabe problem on manifolds with boundary. The boundary condition for the conformal metric is the mean curvature. We establish some Liouville type theorems and Harnack type inequalities.

We study the notion of strong $r$-stability for the context of closed hypersurfaces ${\Sigma}^{n}$ ($n\ge 3$) with constant $(r+1)$-th mean curvature ${H}_{r+1}$ immersed into the Euclidean sphere ${\mathbb{S}}^{n+1}$, where $r\in \{1,...,n-2\}$. In this setting, under a suitable restriction on the $r$-th mean curvature ${H}_{r}$, we establish that there are no $r$-strongly stable closed hypersurfaces immersed in a certain region of ${\mathbb{S}}^{n+1}$, a region that is determined by a totally umbilical sphere of ${\mathbb{S}}^{n+1}$. We also provide a rigidity result for such hypersurfaces.

We consider the Monge-Ampère-type equation $det(A+\lambda g)=\mathrm{const}.$, where $A$ is the Schouten tensor of a conformally related metric and $\lambda \>0$ is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique.

We consider an n-dimensional compact Riemannian manifold (M,g) and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ > 0, together with an upper bound on the energy of the vector field ξ, implies that M is isometric to the n-sphere Sⁿ(λ). We also introduce the notion of φ-analytic conformal vector fields, study their properties, and obtain a characterization of n-spheres...

We describe a new link between Perelman’s monotonicity formula for the reduced volume and ideas from optimal transport theory.

In this note we study the Ledger conditions on the families of flag manifold $({M}^{6}=SU\left(3\right)/SU\left(1\right)\times SU\left(1\right)\times SU\left(1\right),{g}_{({c}_{1},{c}_{2},{c}_{3})})$, $({M}^{12}=Sp\left(3\right)/SU\left(2\right)\times SU\left(2\right)\times SU\left(2\right),{g}_{({c}_{1},{c}_{2},{c}_{3})})$, constructed by N. R. Wallach in (Wallach, N. R., Compact homogeneous Riemannian manifols with strictly positive curvature, Ann. of Math. 96 (1972), 276–293.). In both cases, we conclude that every member of the both families of Riemannian flag manifolds is a D’Atri space if and only if it is naturally reductive. Therefore, we finish the study of ${M}^{6}$ made by D’Atri and Nickerson in (D’Atri, J. E., Nickerson, H. K., Geodesic...