Quadratic growth rigidity for the complex Monge--Ampère equation

We establish normality criteria for families of quasimeromorphic mappings in $\mathbb{R}^n$, $n\geq 2$, based on the avoidance and sharing of continuous functions. We first establish a rescaling principle for sequences of continuous functions and use it to derive a higher-dimensional analogue of Lappan's avoidance criterion. We then obtain normality criteria based on the sharing of continuous functions. In particular, we prove that a family of $K$-quasimeromorphic mappings sharing $q_0(n,K)+1$ mutually disjoint continuous functions is normal, where $q_0(n,K)$ is the corresponding quasiregular Picard constant. We further consider sequences of quasimeromorphic mappings sharing sequences of continuous functions with a normal comparison sequence and establish a corresponding normality criterion, together with an improved bound under an additional separation condition on the limiting functions. Finally, we derive applications to normal quasiregular mappings into $\mathbb{S}^n$.

math.CV↗

arXiv2026

A new class of smooth algebraic varieties with the density property and a counterexample to the Tóth-Varolin Conjecture

We prove that the smooth affine hypersurfaces \[ X_{m,n}=\{ζ^mz_0^n+ (1+z_1ζ)z_2+ζz_3^2=1\}\subset\mathbb{C}^5,\;\; m,n\geq 2, \] have the density property. For every odd integer $d\geq 3$, we identify $X_{d-1,d}$ with a Kaliman modification along a Brieskorn variety of type $(d,2,2,2)$. We prove that the fourfolds $X_{d-1,d}$ are diffeomorphic to $\mathbb{C}^4$ but not biholomorphic to $\mathbb{C}^4$. These, therefore, give counterexamples to the Tóth-Varolin conjecture.

math.CV↗

arXiv2026

Subclass of bi-univalent function associated to Chebyshev polynomial

The Chebyshev polynomials are utilized in this study to define the subclass of the bi-univalent function. Also, Chebyshev polynomial bounds and Fekete-Szego inequalities for functions defined in the classes are established.

math.CV↗