International Journal of Mathematics

We establish some Boju–Funar type compactness criteria for complete Riemannian manifolds via $m$-Bakry–Émery and $m$-modified Ricci curvatures assuming that $m$-Bakry–Émery and $m$-modified Ricci curvatures tend slowly to zero as the distance from a fixed point goes to infinity. Our results are generalizations of Cheeger–Gromov–Taylor type compactness criteria for complete Riemannian manifolds via $m$-Bakry–Émery and $m$-modified Ricci curvatures established by Y. Soylu and the author.

In this paper, we study moduli spaces of low-dimensional complex Lie superalgebras. We discover a similar pattern for the structure of these moduli spaces as we observed for ordinary Lie algebras, namely, that there is a stratification of the moduli space by projective orbifolds. The moduli spaces consist of some families as well as some singleton elements. The different strata are linked by jump deformations, which gives a unique manner of decomposing the moduli space which is consistent with deformation theory.

In this paper, we extend some recent results of Guedj et al. [V. Guedj, C. H. Lu and A. Zeriahi, Plurisubharmonic envelopes and supersolutions. J. Differ. Geom. 113(2) (2019) 273–313] about psh envelopes of bounded functions on bounded domains in ${ℂ}^n$. We also present a result on the regularity of psh envelopes.

Given a crossed module of groupoids $N\to G$, we construct (1) a natural homomorphism from the product groupoid $\mathbb{Z}\times (N⋊G)\rightrightarrows N$ to the crossed product groupoid $N⋊G\rightrightarrows N$ and (2) a transgression map from the singular cohomology $H^{\ast }(G_{•},\mathbb{Z})$ of the nerve of the groupoid $G$ to the singular cohomology $H^{\ast -1}\left(\right.{(N⋊G)}_{•},\mathbb{Z}\left)\right.$ of the nerve of the crossed product groupoid $N⋊G$. The latter turns out to be identical to the transgression map obtained by Tu–Xu in their study of equivariant $K$-theory.

We give a construction of the genus field for Kummer ${\ell }^n$-cyclic extensions of rational congruence function fields, where $\ell$ is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer $\ell$-cyclic extension. Finally, we study the extension ${(K_1{K}_2)}_{𝔤𝔢}/{(K_1)}_{𝔤𝔢}{(K_2)}_{𝔤𝔢}$, for $K_1$, $K_2$ abelian extensions of $k$.

In this paper, we study Finsler metrics expressed in terms of a Riemannian metric, a 1-form, and its norm and find equations with sufficient conditions for such Finsler metrics to become Ricci-flat. Using certain transformations, we show that these equations have solutions and lead to the construction of a large and special class of Einstein metrics.

We prove that discrete compact quantum groups (or more generally locally compact, under additional hypotheses) with coamenable dual are continuous fields over their central closed quantum subgroups, and the same holds for free products of discrete quantum groups with coamenable dual amalgamated over a common central subgroup. Along the way we also show that free products of continuous fields of $C^{\ast }$-algebras are again free via a Fell-topology characterization for $C^{\ast }$-field continuity, recovering a result of Blanchard’s in a somewhat more general setting.

We show that if $𝜃$ is an irrational number in $(0,1)$, ${𝜃}_1$ and ${𝜃}_2$ are in $[0,1),$ $A=\left(\right.\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\left)\right.$ is a matrix of infinite order in SL${}_2(\mathbb{Z})$, either tr$(A)=3$ or tr$(A)=2$ and the greatest common divisor of the entries in $I_2-A^{-1}$ is one, then for any $𝜀>0,$ there exists $\delta >0$ satisfying the following: For any unital simple separable $C^{\ast }$-algebra ${𝒜}$ with tracial rank at most one, any three unitaries $u,v,w$ in ${𝒜}$, if $u,v,w$ satisfy certain trace conditions and

Fractional Yamabe-type equations of the form $A_{s}u=K{u}^{\frac{n+2s}{n-2s}},u>0$ in $\mathrm{\Omega },u=0$ on $\partial \mathrm{\Omega }$, where $\mathrm{\Omega }$ is a bounded domain of ${ℝ}^n,n\ge 2$, $K$ is a given function on $\mathrm{\Omega }$ and $A_s,s\in (0,1)$, is the fractional Laplacian are considered. Bahri’s estimates in the fractional setting will be proved and used to establish a global existence result through an index-counting formula.

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the $\mathrm{\text{Sp}}(4,ℝ)$-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the $2g-3$ exceptional components of the maximal $\mathrm{\text{Sp}}(4,ℝ)$-Higgs bundle moduli space, which correspond to components solely consisting of Zariski dense representations. This also allows a comparison between the invariants for maximal Higgs bundles and the topological invariants for Anosov representations constructed by Guichard and Wienhard.

We extend the notion of quasibounded harmonic functions to the plurisubharmonic setting. As an application, using the theory of Jensen measures, we show that certain generalized Dirichlet problems with unbounded boundary data admit unique solutions, and that these solutions are continuous outside a pluripolar set.