Abstract:
Let M^3 be a closed CR 3-manifold. In this paper we derive a Bochner formula for the Kohn Laplacian in which the pseudo-hermitian torsion plays no role. By means of this formula we show that the non-zero eigenvalues of the Kohn Laplacian are bounded below by a positive constant provided the CR Paneitz operator is non-negative and the Webster curvature is positive. Our lower bound for the non-zero eigenvalues is sharp and is attained on S^3. A consequence of our lower bound is that all compact CR 3-manifolds with non-negative CR Paneitz operator and positive CR Yamabe constant are embeddable. Non-negativity of the CR Paneitz operator and positivity of the CR Yamabe constant are both CR invariant conditions and do not depend on conformal changes of the contact form. In addition we show that under the sufficient conditions above for embeddability, the embedding is stable in the sense of Burns and Epstein. We also show that for the Rossi example for non-embedability, the CR Paneitz operator is negative. For CR structures close to the standard structure on $S^3$ we show the CR Paneitz operator is positive on the space of pluriharmonic functions with respect to the standard CR structure on $S^3$.

Abstract:
We show that every complete nontrivial gradient Yamabe soliton admits a special global warped product structure with a one-dimensional base. Based on this, we prove a general classification theorem for complete nontrivial locally conformally flat gradient Yamabe solitons.

Abstract:
In this paper, we introduce the concept of quasi Yamabe gradient solitons, which generalizes the concept of Yamabe gradient solitons. By using some ideas in [7,8], we prove that $n$-dimensional $(n\geq3)$ complete quasi Yamabe gradient solitons with vanishing Weyl curvature tensor and positive sectional curvature must be rotationally symmetric. We also prove that any compact quasi Yamabe gradient solitons are of constant scalar curvature.

Abstract:
In this paper, we introduce a new combinatorial curvature on two and three dimensional triangulated manifolds, which transforms in the same way as that of the smooth scalar curvature under scaling of the metric and could be used to approximate the Gauss curvature on two dimensional manifolds. Then we use the flow method to study the corresponding constant curvature problem, which is called combinatorial Yamabe problem.

Abstract:
We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.

Abstract:
We classify Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are sol-solitons. In particular, we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not to be algebraic Ricci solitons. We classify Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are solvsolitons. In particular, we obtain new solitons on $G_{2}$, $G_{5}$, and $G_{6}$, and we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not to be algebraic Ricci solitons.

Abstract:
We completely describe paracontact metric three-manifolds whose Reeb vector field satisfies the Ricci soliton equation. While contact Riemannian (or Lorentz\-ian) Ricci solitons are necessarily trivial, that is, $K$-contact and Einstein, the paracontact metric case allows nontrivial examples. Both homogeneous and inhomogeneous nontrivial three-dimensional examples are explicitly described. Finally, we correct the main result of [AGAG-D-13-00189], concerning three-dimensional normal paracontact Ricci solitons.

Abstract:
We suggest and study the stable disk- and cigar-shaped gap solitons of a dipolar Bose-Einstein condensate of $^{52}$Cr atoms localized in the lowest band gap by three optical-lattice (OL) potentials along orthogonal directions. The one-dimensional version of these solitons of experimental interest confined by an OL along the dipole moment direction and harmonic traps in transverse directions is also considered. Important dynamics of (i) breathing oscillation of a gap soliton upon perturbation and (ii) dragging of a gap soliton by a moving lattice along axial $z$ direction demonstrates the stability of gap solitons. A movie clip of dragging of three-dimensional gap soliton is included.

Abstract:
We describe three-dimensional Lorentzian homogeneous Ricci solitons, showing that all types (i.e. shrinking, expanding and steady) exist. Moreover, all non-trivial examples have non-diagonalizable Ricci operator with one only eigenvalue.

Abstract:
We consider a Yamabe type problem on a family $A_\epsilon$ of annulus shaped domains of $\R^3$ which becomes "thin" as $\epsilon$ goes to zero. We show that, for any given positive constant $C$, there exists $\epsilon_0$ such that for any $\epsilon < \epsilon_0$, the problem has no solution $u_\epsilon$ whose energy is less than $C$. Such a result extends to dimension three a result previously known in higher dimensions. Although the strategy to prove this result is the same as in higher dimensions, we need a more careful and delicate blow up analysis of asymptotic profiles of solutions $u_\epsilon$ when $\epsilon$ goes to zero.