Papers

For every $\rho$ in (0, 1/2), we construct an entire function F of order $\rho$ whose logarithmic derivative has zero as a deficient value. This disproves an old conjecture of W. H. J. Fuchs.

We prove the existence, uniqueness, and partial regularity outside a proper analytic subset for the Kähler current solving the $J$-equation at the birational minimal slope. This confirms Datar–Mete–Song's conjecture 1.5 in \cite{DMS26}. We introduce an analytic threshold, and prove its equivalence to the birational threshold introduced by Datar–Mete–Song. One of the key tools is the approximation of subsolution by Bergman's kernels, which is motivated by the work of Demailly on the approximation of plurisubharmonic functions with analytic singularities. As an application of the Bergman kernel approximation, we combine the results of Fang–Ma \cite{FM26} to give an analytic characterization of the $J$-null locus of a semistable pair $(\alpha,\beta)$. This removes the technical assumptions in \cite{L26b}.

While the Jacobian of a planar harmonic mapping keeps a constant sign, the oriented and the multiplicity-weighted surface elements agree up to that sign, and the two natural moment hierarchies coincide. The nearest classification results in the literature are stated under exactly this hypothesis; this paper works past it. Once the Jacobian changes sign, the critical circle acquires a caustic as its image, part of the plane is covered several times with opposite orientations, and the hierarchies part company: integrating a radial weight against the oriented surface element $n_z$ lets overlapping sheets cancel, giving the degree-weighted moments $J_n$, while integrating against $|n_z|$ lets every sheet count positively, giving the multiplicity-weighted moments $J_n^{\pm}$. These are the classical degree and area formulas for the same map, the second weighted by the multiplicity function (Banach indicatrix). It is the second family, the one the caustic threatens, for which a closed form is no longer automatic. Our main result is that it has one anyway: $J_n^{\pm}$ is the total variation of the very same explicit primitive whose endpoint difference gives $J_n$. Crossing the caustic therefore costs only finitely many extra evaluation points, and they are the same points for every order $n$; for a monotone profile there is at most one interior breakpoint, at the critical radius itself. The family carrying this is the two-monomial planar harmonic mapping $F(w)=w+iC\bar w^k$, $w=p(u)e^{iv}$, arising as the planar projection of a Fourier-perturbed surface. On the degree-weighted side $J_n$ has a boundary-only closed form for every $n\ge 0$, the zeroth being the oriented $z$-flux. Neither weighted integral is a new object, and the caustic has been located before in the literature on harmonic trinomials; what is new is the closed evaluation of the multiplicity-weighted hierarchy across it.

In this paper, we extend some Bohr-Rogosinski type inequalities of analytic functions in $\mathbb{C}$ to the cases of holomorphic mappings with values in higher-dimensional complex space. First, we obtain the general Bohr-Rogosinski radius for holomorphic mappings with values in the closure of the unit polydisc in $\mathbb{C} ^n$. Furthermore, we consider the corresponding problems for holomorphic mappings with value in the closure of the unit ball of a JB$^*$-triple. All the radius are optimal.

We confirm a conjecture posed by T. Iwaniec in 1982: the $L^p$ norm of the Beurling–Ahlfors transform is $p-1$ if $p\geq 2$, and $(p-1)^{-1}$ if $1<p\leq 2$.

We prove that transcendental entire functions do not have bounded-orbit wandering domains. This answers a long-standing open question. In fact, we prove a more general theorem, establishing the absence of certain wandering domains even for locally defined analytic functions. The proof is based on a recent remarkable argument of Ye, who obtained an elementary proof of Sullivan's classical theorem that rational maps do not have wandering domains.

For every \(n\geq2\), we prove that the complement in \(\C^n\) of a periodic closed set with compact holomorphically convex quotient in \((\C^*)^n\) is Oka. The proof combines periodic divergence-free transport with Fatou–Bieberbach basins. Applications include complements of convex tubes \(\R^n+\ii B\), products of closed annuli, and, more generally, products of planar compact sets whose complements have at most one bounded component. In particular, \(\C^2\setminus\R^2\) is Oka.

We prove Ebenfelt's SOS conjecture, which implies the Huang–Ji–Yin gap conjecture.

In this paper, we investigate the growth, distortion, pre-Schwarzian and Schwarzian norms of functions in the exponentially convex class \(\mathcal C_{e^\lambda}\), \(0<\lambda\le\pi/2\), defined by \(1+zf''(z)/f'(z)\prec e^{\lambda z}\). By representing the associated Schwarz function explicitly, we derive parameter-dependent estimates for \(f\), \(f'\), and the pre-Schwarzian derivative. We further obtain Schwarzian norm estimates under both the general normalization and the additional condition \(f''(0)=0\). The corresponding extremal problems are analyzed through suitable Schwarz functions, and the dependence of the resulting bounds on the exponential parameter is made explicit.

In this paper, we prove that every rationally connected projective manifold admits a smooth uniformly RC-positive Hermitian metric on its tangent bundle, answering Yang's question and giving a characterization of rational connectedness by uniform RC-positivity. We find an example to show that RC-positivity alone does not characterize rational connectedness. We also prove that uniform RC-positivity of the tangent bundle is preserved under blow-ups along connected smooth centers on compact complex manifolds. In the non-Kähler setting, we construct such metrics on all Hopf and Kato surfaces and obtain classification results for compact complex surfaces.

We address Problems~2.2(i), 2.2(ii), and~2.4 in Dinh–Sibony's list. For a saturated holomorphic curve foliation on a compact Kähler manifold with finitely many singularities, every positive directed $dd^c$-closed current has finite Poincaré mass. Thus linearizability is not needed for the geometric ergodic theorem in this setting. We construct a projective surface with dense hyperbolic leaves and a diffuse extremal current whose Nevanlinna averages converge at a uniform reciprocal logarithmic rate. This is an existence result for a chosen foliation, not a rate theorem for an arbitrary prescribed foliation. Under the hypotheses of the first result, every hyperbolic leaf has a logarithmic lower bound for its Nevanlinna characteristic, even if no diffuse directed current exists. Positive-dimensional singular sets are not treated.

Let $f$ be an analytic function in the unit disc whose modulus is bounded by $1$. We present a new proof of Bohr's theorem, which asserts that its majorant function $Mf(r)$ is bounded by $1$ for $0\leq r \leq 1/3$ and that $1/3$ is optimal. The argument also yields a sharp upper bound for $Mf(r)$ in terms of $\lvert f(0) \rvert$, complementing a result due to Bombieri.

We classify compact Kähler surfaces with semipositive anticanonical bundle. In particular, we resolve a question of Demailly–Peternell–Schneider: any blowup of $\mathbb{P}^2$ at 9 points on a smooth cubic has semipositive anticanonical bundle.

Alpöge's construction produces a period family of compact complex threefolds fibred over $\mathbb{P}^1$, with complex two-tori as general fibres. We give a direct proof that every admissible member is Oka, using only the complex-analytic construction. Du–Guo–Kusakabe–Xie independently established the Oka property and determined the exact biholomorphism classes in this family. Assuming Alpöge's identification of the underlying smooth manifolds with $S^6$, these results yield uncountably many pairwise nonbiholomorphic Oka complex structures on the six-sphere.

We characterize Griffiths positivity of vector bundles on smooth projective varieties by a finite-dimensional convex condition on first jets of sections of positive twists. A nonzero positive matrix-valued measure annihilated by the adjoint Levi operator gives the dual obstruction. We also give an explicit bundle on an abelian surface that satisfies the first-level condition, but whose complete untwisted evaluation has a differential kernel.

We construct a rank-two ample holomorphic vector bundle on an abelian surface that admits no smooth Griffiths-semipositive Hermitian metric, thereby giving a counterexample to the Griffiths conjecture. The proof combines a curvature obstruction that persists under deformations with the openness of ampleness in an irreducible moduli space of stable sheaves.

We establish lower bounds for the critical CR GJMS operator on closed pseudo-Hermitian manifolds. In dimension $2n+1$, assuming embeddability, we obtain a lower bound involving the Heisenberg Sobolev norm of order $n+1$, modulo an $L^{2}$ term, on the orthogonal complement of the kernel. In dimension three, we prove that the CR Paneitz operator is bounded below without assuming embeddability. The proofs combine the Heisenberg calculus with a square-root construction based on the joint functional calculus of the sub-Laplacian and the Reeb vector field on the Heisenberg group. As an application, we show that the infinitely many negative eigenvalues of the CR Paneitz operator on the Rossi sphere have zero as their only accumulation point. Consequently, this operator does not have closed range.

For the semistable $J$-equation, we prove that the numerical $J$-null locus coincides with the ambient $C^2$-singular locus of Murakami's weak solution. Consequently, this singular locus is a proper analytic subset with finitely many irreducible components, and the weak solution is locally smooth outside the $J$-null locus. The proof relies on two analytic ingredients. First, we establish a regularization theorem for singular $J$-subsolutions, showing that Demailly's global regularization preserves quantitative strict cone conditions. Thus, singular strict subsolutions with prescribed analytic poles can be replaced by smooth strict subsolutions away from their pole sets. Second, we derive relative a priori estimates adapted to these subsolutions: a relative $L^\infty$-estimate from determinant control and a weighted second order estimate yielding uniform $C^2$-bounds on compact subsets of the regular locus.

We construct the moduli space of marked polystable $\mathrm{SL}_r(\mathbb C)$-Higgs bundles via analytic methods. The resulting space is normal and Hausdorff, and its local models are products of a Teichmüller neighbourhood with a quotient of a quadratic cone. These models preserve the curve parameter and identify the stabilizers. We prove agreement with the complex and unitary gauge-quotient topologies and establish the coarse moduli property for semistable families over reduced analytic bases. The mapping class group and Higgs scaling act holomorphically. Besides, on the stable locus, the moduli spaces that we consider agree with the joint moduli constructed by Collier, Toulisse and Wentworth.

We study harmonic mappings between the source domain in the $\alpha$-Grushin plane and the target domain in the $\beta$-Grushin plane for $\alpha$ not necessarily equal to $\beta$ and $\alpha\in [0,1)$. Our harmonic mappings arise as a Euler–Lagrange system of equations related to the Dirichlet energy and are studied in weak and strong forms. The key results enclose the second order regularity of harmonic mappings, the Bochner identity, the weak Harnack estimates for coordinate functions of a harmonic mapping, as well as the Caccioppoli estimates and the Liouville type theorem. Furthermore, we discuss several examples of (Grushin) harmonic mappings and, in particular, relate them to holomorphic mappings in the plane.

We study the relation between canonical positivity, rigidity of a prescribed Hermitian metric, and the curvature not detected by Chern holomorphic sectional curvature. A compact Brody-hyperbolic complex manifold admitting an independent smooth plurinegative Hermitian metric is projective with ample canonical bundle. In Fujiki's class, nonpositive constant Chern holomorphic sectional curvature forces the original metric to be Kähler. On compact complex threefolds, without an auxiliary metric or a Fujiki-class assumption, a nonzero constant forces Kählerness. We also give an alternative proof of Chen–Zheng's Chern-flatness theorem for zero-HSC threefolds, by proving local rank-two differential compatibility and localizing their curvature bound under $\partial\eta=0$. In contrast, for every complex dimension at least seven we give compact balanced zero-HSC metrics with nonzero Chern curvature. Their octonionic curvature formula also implies the vanishing of all four Chern–Ricci contractions, and an explicit invariant volume form is Chern parallel. Finally, an auxiliary semipositive plurinegative $(1,1)$-form, positive definite at some point, restores flat Kählerness under zero HSC and vanishing first Chern–Ricci tensor. A mixed Ricci integral criterion explains the role of this last curvature hypothesis.

We introduce the chordal isometric circle of a Möbius map, and use this to give a factorization of any Möbius map as the composition of a chordal isometry and either a reflection, or a rotary reflection, across a circle. We then use this to find the chordal, and spherical, Lipschitz constants of a Möbius map, and compare this with related results in the literature.

For the unicritical family $f_{c,j}(z)=z^j+c$, the Multibrot sets $M_j$ converge geometrically to the closed unit disk as $j\to\infty$, but this limit does not quantify their finite-degree area behavior. We estimate Area($M_j$) using escape-time pixel counting while separating the effects of the iteration threshold $K$ and spatial sampling density $\rho$. The finite-iteration sets form a nested decreasing sequence, while grid refinement is treated empirically. Across $K\in\{25,100,500,1000\}$ and $\rho\in\{128,256,512,1024\}$, the final grid refinement changes the reported areas by at most $1.1\times10^{-3}$. At $K=1000$ and $\rho=1024$, the estimates are well described by $C(j)\approx\pi-2.49216j^{-0.57103}$. Independently, Laurent coefficients of the exterior conformal map and Gronwall's area formula yield finite-truncation upper bounds $U_{1000}(j)\approx\pi-2.07405j^{-0.643427}$. Across all tested degrees, the grid estimates remain below the corresponding conformal bounds, and both sequences exhibit similar power-law decay of the area deficit from $\pi$.

We study the asymptotic behavior of Bergman kernels for high tensor powers of semipositive line bundles over Hermitian manifolds. At points where the curvature degenerates, the classical asymptotic expansion may fail. In this paper, we establish a full local asymptotic expansion and rapid off-diagonal decay near degenerate points at which the metric admits a local decoupled model. More generally, we make the following two spectral hypotheses: a localized mild spectral gap for the Kodaira Laplacian and a spectral gap for the rescaled local model. Under these assumptions, we prove a localization property and the rapid off-diagonal decay for the Bergman kernel. Furthermore, if the metric has a local quasi-homogeneous structure, we obtain a full local asymptotic expansion in the $C^\infty$-topology. As an application, we study pull-backs of positive line bundles under branched coverings. Near a smooth ramification hypersurface, the resulting asymptotic expansion reflects the branching order. Finally, for certain non-quasi-homogeneous models, we still obtain localization and leading-order asymptotics.

Let $\Omega\subset\mathbb C^n$($n\geq 3$) be a pseudoconvex domain with a smooth strongly pseudoconvex boundary point. We prove that if the Bergman metric of $\Omega$ has constant scalar curvature, then it is necessarily Kähler–Einstein.

Let $\varphi\in L^\infty(\D)$. We study compactness criteria for \(T_\varphi\) on the Bergman space $A^2(\D)$. Axler and Zheng~\cite{AZ1998} established a necessary and sufficient condition for compactness in terms of the Berezin transform. For a general bounded measurable function $\varphi$, however, its Berezin transform $\tilde{\varphi}$ does not readily reveal the intrinsic properties of $\varphi$. Motivated by a characterization in terms of the symbol itself, Zhu~\cite{ZhuSlides} proposed a conjecture on compact Toeplitz operators. In this paper, we characterize compactness of $T_\varphi$ on the unweighted Bergman space in terms of local averages of the symbol. We prove that compactness is equivalent to the vanishing of averages over Bergman disks of any prescribed fixed radius. We also establish an equivalent criterion in terms of Carleson box averages that tend to zero uniformly in the angular variable. Finally, we construct a nonnegative bounded symbol whose Carleson box averages tend to zero at every fixed angle, although the associated Toeplitz operator is not compact.

We prove that every proper subdomain of the Euclidean space equipped with its quasihyperbolic metric is CAT(2). The CAT property leads to a positive resolution of Väisälä's uniqueness, prolongation, quasihyperbolic convexity, and local geodesic conjectures on quasihyperbolic geodesics affirmatively on all dimensions, extending the two-dimensional results by Herron. The sectional curvature bound is optimal as demonstrated by the complement of two points in dimensions at least three. We also show that in any quasihyperbolic domain, quasihyperbolic spheres up to radius $\pi/\sqrt{2}$ are $\mathcal{C}^{1,\frac{1}{2}}$-diffeomorphic to the Euclidean sphere. Consequently, we answer a question by Gehring and Vuorinen on the regularity of such spheres. Similar techniques lead to a positive resolution of Väisälä's conjecture on the Euclidean convexity of quasihyperbolic balls up to radius $\arctan(\sqrt{2})/\sqrt{2}$. We establish that in all dimensions, the quasihyperbolic metric is CAT(0) on convex domains. This leads to a positive answer to a question by Martio and Väisälä on the quasihyperbolic convexity of quasihyperbolic balls on such domains. Finally, we prove that a variable Alexandrov curvature lower bound for a quasihyperbolic domain self-improves to a global lower bound of $-1$ and is equivalent to the concavity of the domain.

Let \(X\) be a smooth open manifold of even dimension, \(T\) be a topological space, and \(\{J_t\}_{t\in T}\) be a continuous family of smooth integrable Stein structures on \(X\). The family is said to be tame if the \(J_t\)-convex hulls of any compact set in \(X\) are upper semicontinuous with respect to \(t\in T\). It was recently shown that tame Stein families have good function-theoretic properties; in particular, they satisfy the Oka principle for families of holomorphic maps to any Oka manifold. We analyse some constructions and operations on Stein manifolds that preserve tameness of families and we exhibit several conceptually new examples of tame Stein families.

We consider a family of uniqueness problems in the setting of weighted $\ell^2$ Fourier spaces on the unit-circle, from the perspective of simultaneous approximation. We develop a general potential theoretical framework, which enables us to extend and unify several classical results by Ahlfors, Beurling, Frostman and Khrushchev, previously known in the more classical regimes. In addition, we prove a surprisingly sharp result that illustrates a deep discrepancy between unilateral and bilateral uniqueness sets.

We study the parameter dependence of weighted Bergman kernels on fixed bounded domains in $\mathbb C^n$. Our main result establishes real-analytic dependence on $t\in(-1,\infty)$ for the kernels associated with $\delta^t\,dV$ on bounded pseudoconvex domains with $C^2$ boundary, where $\delta$ is the Euclidean distance to the boundary. The parameter derivatives satisfy factorial estimates in $C^\ell(S\times S)$ for every $S\Subset\Omega$ and $\ell\ge0$, uniformly on compact parameter intervals. The proof combines weighted $L^2$ estimates for $\bar\partial$ with a holomorphic family of bounded operators and gives a local holomorphic extension with values in a fixed weighted Bergman space. For weight families smooth jointly in space and parameter up to the boundary, we also express all parameter derivatives in terms of iterated weighted Bergman projections and complete exponential Bell polynomials. This formula implies preservation of the Gevrey class $G^s$, $s\ge1$, under uniform parameter estimates up to the boundary. We show that the same derivative formula holds for the weights $-t\log\delta$.