We show that, under a division condition, the tangential Cauchy--Riemann cohomology of a compact Lie group with a left-invariant CR structure can be computed on a suitable maximal torus. As a consequence, we conclude that the tangential Cauchy--Riemann cohomology is finite-dimensional. We also show that, for a class of CR structures, this division condition is necessary for the total cohomology to be finite-dimensional. The proof combines Fourier analysis on compact Lie groups, highest-weight representations and Lie algebra cohomology. This not only generalizes but provides completely new proofs for the analogous result due to Pittie and for its extensions to Levi-flat CR structures, obtained by Jacobowitz and Jahnke.

This paper extends a previous work in which the rank-2 co-Higgs bundles on a Hopf surface are classified based on the data of the underlying vector bundle. The aim of the paper is to study the Poisson 3-folds that can be constructed from these co-Higgs bundles by describing their symplectic leaves.

The integral representation of completely monotone functions of several real variables as Laplace or Stieltjes-Fantappié transforms of positive measures opens a Hilbert space path toward their finite-point interpolation by simpler functions. We combine, within a non-commutative Radon transform framework, the matrix pencil realization of the positive semi-definite Hankel kernel associated with the sampling of a completely monotone function with Weyl's operational calculus and Fantappiè's analytic calculus. The interpolation is achieved by finitely determined entire or rational functions, respectively, which are directionally completely monotone. In our relaxation scheme, the original positive measure is approximated by a sequence of specific Wigner distributions, which can also be regarded as analytic functionals. Throughout the interpolation process, tight bounds are enforced on the modulus or the real part of the holomorphic extension to the underlying tube domain.

In this paper, we investigate finite and infinite decomposition formulas for zero mean curvature (ZMC) graphs in three-dimensional spaces with different metrics, including Euclidean space, Lorentz--Minkowski space, and isotropic 3-space. We first derive new Euler--Ramanujan-type identities yielding decompositions for the conjugate of Scherk's first minimal surface in terms of dilated catenoids. We then employ Weierstrass factorisation and power series techniques to obtain infinite decompositions for a broad class of ZMC graphs in isotropic 3-space into helicoids, logarithmoids of revolution, and Enneper surfaces. We further extend these decomposition results to larger families of ZMC surfaces across these spaces by considering surfaces arising from the López--Ross transformation, Bonnet rotation, and a one-parameter family of metric deformations. We also investigate finite decompositions, establishing finite analogues of Scherk tower decompositions in both Euclidean and isotropic settings. In addition, we prove a theorem characterising finite decompositions of isotropic minimal surfaces. Finally, we discuss applications of the resulting decomposition theory to lamellar structures.

The coefficients $A_n(\alpha,\beta,\omega)$ in the Maclaurin expansion $(1+\omega z)^{\alpha}(1-z)^{-\beta}=\sum_{n=0}^{\infty} A_n(\alpha,\beta,\omega)z^n$ are considered for $|\omega|=1$ and $\alpha,\beta\in(0,1]$. D. A. Brannan conjectured in a 1973 paper that $|A_n(\alpha,\beta,\omega)|\le A_n(\alpha,\beta,1)$ for every positive odd integer $n$. The present author recently established the conjecture outside a small neighbourhood of $\omega=-1$. The remaining range is treated here by combining compound Laplace integral representations with two types of local approximation: a Meijer $G$ function approximation for $n|\arg(-\omega)|$ bounded, and a modified Watson approximation for the complementary range. The resulting lower bounds reduce the problem to numerical positivity checks for explicit functions on compact parameter sets. These computations verify the inequality for all $\alpha,\beta\in(0,1]$ and all odd integers $n\ge5$, and hence, together with Brannan's result for $n=3$, complete the proof of his conjecture.

In this paper, we study the sharp bounds of the first three logarithmic inverse coefficients for Janowski convex class $\mathcal{C}(A, B)$. We also derive sharp upper and lower bounds of $\bigl|\,\gamma_2 \,\bigr|-\bigl|\,\gamma_1\,\bigr|$ and $\bigl|\,\Gamma_2 \,\bigr|-\bigl|\,\Gamma_1\,\bigr|$ for functions in the class $\mathcal{C}(A, B)$. Furthermore, a sharp estimate for the second Hankel determinant associated with the logarithmic inverse coefficients for functions in $\mathcal{C}(A, B)$ is obtained.

An entire slice regular function $f:\mathbb H\to\mathbb H$ can omit four prescribed quaternionic values only in the affine-dependent case. More precisely, four affinely independent omitted values force $f$ to be constant, while the converse follows from the plane-omission theorem of Bisi--Winkelmann. The proof passes to the real-symmetric stem function. For each omitted value a quadratic zero-divisor criterion gives a zero-free entire function $Q_j$, and the component normal to the affine span is governed by a square-discriminant identity. Finite-order data are excluded by Hadamard factorization and a rigidity argument on the real axis. In the general case, logarithmic Bloch--Ochiai places the $Q$-curve in a translated algebraic torus. The Laurent-square case reduces to the finite-order contradiction, and the nonsquare case is excluded by an even-ramification argument together with the level-one truncated Second Main Theorem of Noguchi--Winkelmann--Yamanoi.

In this paper, we study the class \Sigma_{H}^{k}(p) of sense-preserving univalent harmonic mappings in the unit disk \mathbb{D} that possess a simple pole at p\in[0,1) and admit a k-quasiconformal extension to the extended complex plane for k\in[0,1). In 2024, Bhowmik and Satpati established an area theorem and derived a sufficient condition for the k-quasiconformal extension of harmonic mappings belonging to \Sigma_{H}^{k}(p) without logarithmic terms. Motivated by their work, we investigate the corresponding problem when a logarithmic singularity is present. Our main contributions are two-fold: we first prove a generalized area theorem for all mappings in \Sigma_{H}^{k}(p); we then obtain a sufficient condition for sense-preserving univalent harmonic mappings in \mathbb{D} to admit explicit k-quasiconformal extensions. These results extend the aforementioned work to the setting where logarithmic singularities are allowed.

The existence of small-time asymptotics for the heat kernel of the Kohn Laplacian on a general CR manifold has remained an open problem. In this paper, we resolve the problem in the non-degenerate case. More precisely, let $X$ be a compact oriented CR manifold of dimension $2n+1$, $n \ge 1$, with a nondegenerate Levi form of constant signature $(n_-, n_+)$. Suppose that condition $Y(q)$ holds at each point of $X$, we establish the small-time asymptotics of the heat kernel of Kohn Laplacian. Suppose that condition $Y(q)$ fails, we establish the small-time asymptotics of the kernel of the difference of the heat operator and Szegő projector. As an application we define the analytic torsion on compact oriented nondegenerate CR manifolds and study its dependence on changes of the metrics. Let $L^k$ be the $k$-th power of a CR complex line bundle $L$ over $X$. We establish the asymptotics, as $k \to \infty$, of the analytic torsion with values in $L^k$, under a variant of spectral gap condition. Furthermore, when $X$ admits a transversal CR $S^1$-action, we establish the small-time asymptotics of the $S^1$-equivariant heat kernel of the Kohn Laplacian with values in $L^k$. As an application we define the $S^1$-equivariant Quillen metric with values in $L^k$ and study its dependence on changes of the metrics. Finally, we establish the asymptotics, as $k \to \infty$, of the $S^1$-equivariant analytic torsion with values in $L^k$.

This paper investigates the differential-geometric and topological properties of the Cayley condition in Poncelet porism for triangles, defined as the locus of pairs of non-degenerate conics that admit a Poncelet triangle. While the algebraic condition for this porism, established by Cayley, is classical, the geometric nature of the set of solutions has remained largely unexplored. We demonstrate that this Cayley set is a smooth, connected, 9-dimensional complex manifold. This is proven by showing it is an open subset of a smooth algebraic variety endowed with a trivial fiber bundle structure over the space of non-degenerate conics. To further analyze its structure, we construct the moduli space of transversely intersecting conic pairs under the action of $\mathbb{P}GL_3(\mathbb{C})$ and identify it with an open subset of $\mathbb{CP}^2/S_3$. We compute the fundamental group of a generic orbit. The elliptic j-invariant is then introduced as a holomorphic map on this space of conics, which factors through this moduli space. We analyze the subset of the Cayley set where this map is a submersion - its regular part - which corresponds to excluding points whose j-invariant is one of the critical values $0$ or $1728$. We prove that this regular part is the total space of a fiber bundle over $\mathbb{C} \setminus \{0,1728\}$. This structure allows for the computation of the fundamental group via the long exact sequence of homotopy. Finally, we provide a principal bundle formulation for the Poncelet correspondence itself over orbits of conic pairs.

Let $X$ be a compact Kähler manifold. We study subgroups $G \le \mathrm{Aut}(X)$ of biholomorphic automorphisms of zero entropy when $\mathrm{Aut}^0(X)$ is compact (e.g. when $\mathrm{Aut}^0(X)$ is trivial). We show that the virtual derived length $\ell_{\mathrm{vir}}(G)$ of $G$ satisfies $\ell_{\mathrm{vir}}(G) \le \dim X -κ(X)$, where $κ(X)$ is the Kodaira dimension of $X$. Modulo the main conjecture of our previous work concerning the essential nilpotency class, we obtain the same upper bound $c_{\mathrm{vir}}(G) \le \dim X -κ(X)$ for the virtual nilpotency class $c_{\mathrm{vir}}(G)$, together with a geometric description of the $G$-action on $X$ when the equality holds.