We present a non existence result of complete, Einstein hypersurfaces tangent to the Reeb vector field of a regular Sasakian manifold which fibers onto a complex Stein manifold.

This paper is a continuation of our previous work on double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the Möbius function, and so on. We consider the analytic behaviour around the non-positive integer points on singularity sets which are points of indeterminacy. In particular, we show a certain reciprocity law of their residues. Also on this occasion we correct some inaccuracies in our previous paper.

In this paper, we prove the logarithmic Harnack inequalities for $L^p$-Log-Sobolev function on $n$-dimensional weighted Riemannian manifolds with $m$-Bakry-Émery Ricci curvature bounded below by $-K$ $(m\ge n, K\ge0)$. Under the assumption of nonnegative $m$-Bakry-Émery Ricci curvature, we obtain a global Li-Yau type gradient estimate and a Hamilton type estimate for the positive solutions to the weighted parabolic $p$-Laplace equation with logarithmic nonlinearity. As applications, the corresponding Harnack inequalities are derived.

Let $M^n$ be an $n$-dimensional complete and locally conformally flat hypersurface in the unit sphere $\mathbf{S}^{n+1}$ with constant scalar curvature $n(n-1)$. We show that if the total curvature $\left(\int_{M}|H|^{n}dv \right)^{\frac{1}{n}}$ of $M$ is sufficiently small, then $M^n$ is totally geodesic.

We obtain rigidity results concerning complete noncompact solitons of the mean curvature flow related to a nonsingular Killing vector field $K$ globally defined in a semi-Riemannian space, which can be modeled as a warped product whose base corresponds to a fixed integral leaf of the distribution orthogonal to $K$ and the warping function is equal to $|K|$. Our approach is based on a suitable maximum principle dealing with a notion of convergence to zero at infinity. As application, we study the uniqueness of solutions for the mean curvature flow soliton equation in these ambient spaces.

We classify the ribbon structures of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$. Our result generalizes Kauffman and Radford's classification result of the ribbon elements of the Drinfeld double of a finite-dimensional Hopf algebra. Our result implies that $\mathcal{Z}(\mathcal{C})$ is a modular tensor category in the sense of Lyubashenko if $\mathcal{C}$ is a spherical finite tensor category in the sense of Douglas, Schommer-Pries and Snyder.