11 papers
Pullback metrics and homological obstructions to BLD remetrization of branched covers
Deguang Zhong
We characterize admissible BLD remetrizations of branched self-covers of spheres by linear local contractibility of their canonical pullback length metrics. A relative homology obs…
Affine balancing of complex dilatations: affine invariants and optimal distortion of harmonic quasiconformal mappings
Zhi-Gang Wang, Deguang Zhong
For orientation-preserving planar harmonic mappings, postcomposition by real-affine mappings induces automorphisms on the space of complex dilatations. Motivated by this fact and t…
Monotone Sobolev functions: approximation, critical points, and level sets
Deguang Zhong
We give an affirmative answer to the planar local smoothing problem in Question~1.7 of D.~Ntalampekos and positive and negative answers to the basic approximation and level-set par…
An all-dimensional Lewy theorem for pluriharmonic mappings via Milnor monodromy
Deguang Zhong, Zhi-Gang Wang
Hengartner asked whether a pluriharmonic mapping between equidimensional complex Euclidean domains is locally one-to-one precisely at the points where its real Jacobian is nonzero.…
Odd branching obstructs multiplicity-one integral current structures
Deguang Zhong
Giuliano Basso asked whether every -dimensional integral current space admits an integral current structure with the same characteristic set and multiplicity one almost everywhe…
A counterexample to an open problem of Dorff
Zhi-Gang Wang, Deguang Zhong
The classical Pólya-Schoenberg conjecture, proved by Ruscheweyh-Sheil-Small, asserts that the convolution of two normalized convex univalent functions is again convex. This propert…
Meromorphic solutions of first-order differential equations with rational exponential coefficients
Deguang Zhong, Fanning Meng, Wenjun Yuan
We study first-order differential equations , where $R\in\C(t,w)$. We prove that a meromorphic solution on the whole complex plane is algebraic over $\C(e^z)$ unless $…
Sharp mean-width and Jacobian bounds for Euclidean and hyperbolic harmonic maps
Deguang Zhong, David Kalaj
We prove a sharp mean-width inequality for monotone zonal operators and derive global and differential bounds for Euclidean and hyperbolic-harmonic self-maps of the unit ball. Let…
Global convergence and monotonicity of Newton iteration for the inverse Grötzsch modulus
Deguang Zhong
Let \[ μ(r)=\fracπ{2}\frac{\Kc(r')}{\Kc(r)}, \qquad r'=\sqrt{1-r^2},\qquad 0<r<1, \] be the Grötzsch modulus function. Problems 3.38(a)--(b) in a survey of Vuorinen ask whether New…
Critical Sobolev thresholds for openness and discreteness of Monge-Ampere gradient mappings
Deguang Zhong
Let be a domain and let $u\in W^{2,n}_{\loc}(Ω)$ satisfy \[ \det D^2u\geqδ>0\qquad\text{a.e. in }Ω. \] Guerra and Tione asked whether the gradient mapping must…
Quasiregular values from generalized manifold with controlled geometry
Deguang Zhong
The main aim of this paper is to establish the Reshetnyak's theorem for quasiregualr values from generalized -manifold with suitable controlled geometry to Euclidean space $\mat…