Papers

Given a plurisubharmonic function $\varphi>-\infty$ on the unit ball $\mathbb{B}^n$ and a constant $0<p\leq2$, Guan-Zhou's optimal $L^p$ extension theorem yields a sharp lower bound for the weighted $p$-Bergman kernel. We show that the equality holds at some point if and only if $\varphi$ is pluriharmonic on $\mathbb{B}^n$. This provides an analogue of the equality part of Suita's conjecture. As an application, we prove a Forelli-type result. Let $\varphi$ be a real-valued function on $\mathbb{B}^n$. If every slice of $\varphi$ through the origin is harmonic and if $\varphi$ is plurisubharmonic in a neighborhood of the origin, then $\varphi$ is pluriharmonic on $\mathbb{B}^n$.

We show that every left-invariant elliptic structure of minimal rank on an odd-dimensional compact Lie group admits an algebraic decomposition analogous to those obtained by Pittie in the complex case and by Charbonnel and Khalgui for left-invariant CR structures of hypersurface type. We then use this decomposition to prove that the cohomology associated with such an elliptic structure can be reduced to a computation on a suitable maximal torus.

In 1918, Gross proved that a regular local inverse of a meromorphic function in the plane can be continued analytically along every ray from its centre except for directions in a set of Lebesgue measure zero. Eremenko asked whether the same conclusion holds for an implicit function defined by an entire relation in two variables. We prove that there exist an entire function $F$ of two variables and a regular implicit germ $\varphi$, defined by $F(z,\varphi(z))=0$, whose analytic continuation fails on every ray from its centre. In fact, along each ray, the modulus of the continuation tends to infinity as the singular point is approached.

In his thesis work, the first author studies neighborhoods of singular elliptic curves of type $I_0^*$ (Kodaira's classification) and provides the analytic classification. This is related via a double covering construction to the analytic invariants found by Touzet, Voronin and the second author for the analytic classification of neighborhoods of smooth elliptic curves. After recalling these constructions, we explain in this note how this construction is conjecturally related to the Riemann-Hilbert correspondence at infinity for Painlevé VI case when considering neighborhood of Okamoto divisor.

We develop several tools to study the local and global structure of the Teichmüller and moduli spaces for complex nilmanifolds. As a starting point, we prove a structure theorem for holomorphic maps between complex nilmanifolds, which allows for an explicit description of the Teichmüller and moduli spaces. We then introduce two period maps, and give general criteria for the existence of a complex structure on these spaces. To illustrate the theory, we prove a global Torelli theorem for various classes of nilmanifolds, including principal torus bundles over tori and almost abelian nilmanifolds. In particular, we show that there are nilmanifolds of arbitrary dimension and nilpotency index for which the Teichmüller space admits the structure of a complex manifold.

We generalise a mixing result due to Pommerenke for non-autonomous dynamical systems of forward compositions of the boundary values of centred inner functions; the proof depends on a precise bound on the orbits of such forward compositions in the unit disc. This enables us to give new criteria for such systems to have positive metric entropy, lower entropy and topological entropy. In the opposite direction, we construct a forward composition of centred degree two Blaschke products with topological entropy zero.

We formulate and prove a simple criterion for the sphericity of a CR manifold (that is, for its equivalence to its model surface). We discuss the differences between the general case, in which the model surface is weighted homogeneous of arbitrary degree, and the case of Levi-nondegenerate manifolds, whose model surface is quadratic.

We prove Koebe's conjecture for John domains by showing that every planar John domain is conformally equivalent to a circle domain. Our proof is based on a geometric property of John domains, which we prove to be sufficient for uniformization by a circle domain. As a consequence, we obtain an alternative proof of Koebe's conjecture for inner uniform and Gromov hyperbolic domains, recovering results established in earlier work of the authors. Finally, we derive a necessary and sufficient condition for a domain to satisfy Koebe's conjecture by using Schramm's transboundary modulus and well-behaved exhaustions of circle domains constructed by the second author and Rajala.

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. Naser had earlier claimed the conclusion in complex dimension two, and a recent theorem of Kalaj supplies a complete proof in that dimension. We prove the result in every complex dimension. The main ingredient is a singularity-theoretic characterization which is of independent interest: if $q\colon(\C^n,0)\to(\C,0)$ is a nonconstant holomorphic germ, then $dq(0)\ne0$ if and only if one (equivalently, every) phase hypersurface \( \{\Repart(e^{-i\theta}q)=0\} \) is a rational homology $(2n-1)$-manifold at the origin. The difficult direction combines local conical structure, Alexander duality, the Milnor fibration, and A'Campo's vanishing theorem for the Lefschetz number of local monodromy. The argument requires neither an isolated critical point nor a reducedness hypothesis. It follows that the real critical set of any pluriharmonic self-dimensional mapping agrees exactly with its local noninjectivity set. As a quantitative application, combining our theorem with Kalaj's boundary regularity theorem shows that, in every complex dimension, a quasiconformal pluriharmonic homeomorphism of the unit ball onto a bounded $C^1$-Dini domain is bi-Lipschitz.

We establish an exact additive formula expressing the Lelong number of a plurisubharmonic function in terms of the limiting difference between its spherical and solid means. Building on this identity, we derive several equivalent criteria for the vanishing of the Lelong number, formulated in terms of fixed-scale increments and the asymptotic agreement of radial maxima, spherical and solid means, and radial regularizations. We also give a pointwise $L^1$ mean-oscillation formulation of the VMO characterization for plurisubharmonic functions with zero Lelong number, together with a streamlined proof of the corresponding global result of Biard and Wu. Finally, we establish a nonlinear characterization in terms of the relative extremal depth and the Bedford-Taylor capacity of deep sublevel sets.

For every $2\le m\le n$, we construct Hölder continuous functions on the closed unit ball that are minimal in the Cegrell class $\F_m$, although their Hessian measures are not maximal in the $m$-subharmonic ordering. This answers a question posed by the second author. We also obtain a minimality criterion based on partial pluricomplex energy and an explicit family of Monge–Ampère examples on a product domain. For $m=1$, minimality and maximality are equivalent on a ball.

Let $f$ be a holomorphic automorphism of a compact Kähler manifold $X$ with simple cohomological action. We prove that the saddle periodic points of $f$ equidistribute with respect to the equilibrium measure. In particular, our result applies to every holomorphic automorphism of a compact Kähler surface with positive topological entropy.

We use hyperbolic area and covering geometry to study wandering Fatou components and singular orbits. We give proofs without quasiconformal deformation of the rational no-wandering theorem and the known no-wandering results in Bergweiler's Question~9. The framework also excludes wandering for entire functions with compact singular sets whose derived sets lie in the Fatou set. For Bergweiler's Question~8, every wandering component of a transcendental meromorphic map has a subsequence of iterates converging locally uniformly to infinity. For Bergweiler's Question~4, we establish hyperbolic separation near singular-free Baker cycles after deleting any finite set of original singular values as orbit generators. For Bergweiler's Question~11, corresponding finite-deletion boundary estimates hold along wandering tails. In higher dimension, a relative-area theorem excludes bounded wandering components of polynomial skew products over bounded bases carrying an absolutely continuous invariant probability of full support. Applied to elliptic polynomial skew products over Siegel rotation domains, it yields a local no-wandering theorem without conditions on the fiber critical orbits or continuity of fiber Julia sets. A global criterion covers regular skew products on $\mathbb P^2$ whose bounded base Fatou components eventually enter periodic Siegel disks, including $(\lambda z+z^d,q(z,w))$ for Brjuno $\lambda$ and polynomial $q$ of total degree at most $d$ with a nonzero $w^d$ coefficient.

Let $F$ be a totally real field of degree $n$ and discriminant $\Delta_F$, and let $X_1(\eta)$ be the cover of the Hilbert modular variety parametrizing abelian varieties with real multiplication by $\mathcal O_F$, together with a torsion point having annihilator $\eta$. Let $L=K_{\overline X_1(\eta)}+D$ be the log-canonical bundle on a smooth toroidal compactification, and let $H_L$ be an associated multiplicative height. We prove that rational points on $X_1(\eta)$ become sparser as $|\mathrm{Nm}(\eta)|\to\infty$, with $(\eta,\Delta_F)=1$. More precisely, for every number field $K$, set $$ N_{\eta,K}(B)=\#\{x\in X_1(\eta)(K):H_L(x)\leq B\}. $$ If $|\mathrm{Nm}(\eta)|\ge 5^n$ and $(\eta,\Delta_F)=1$, we prove $$ \limsup_{B\to\infty}\frac{\log\max\{1,N_{\eta,K}(B)\}}{\log B} \leq\delta_{\eta,K,n},\qquad \delta_{\eta,K,n}\ll_{[K:\mathbb Q],n}|\mathrm{Nm}(\eta)|^{-1/(2n)}. $$ In particular, $\delta_{\eta,K,n}\to0$ uniformly when $n$ and $[K:\mathbb Q]$ are bounded and $|\mathrm{Nm}(\eta)|\to\infty$. The main geometric result is a uniform lower bound, growing with the level, for the log-canonical degree of subvarieties of $X_1(\eta)$. Combining this estimate with recent progress derived from determinant-method, due to Ellenberg–Lawrence–Venkatesh and Brunebarbe–Maculan, we obtain the sparsity result above. We also prove that, for sufficiently large $|\mathrm{Nm}(\eta)|$, every subvariety of $X_1(\eta)$ is of general type, and establish a higher-dimensional generalization of the geometric torsion theorem of Bakker–Tsimerman. Namely, for a family of abelian varieties with real multiplication over a quasi-projective base of arbitrary dimension, we bound the torsion subgroup of its Mordell–Weil group in terms of the canonical volume of the base, uniformly in the totally real multiplication field of fixed degree.

We construct a function $\Phi $ such that the orbit under the representation of PSL(2,Z) is an orthonormal basis for the Bergman space with weight $\alpha =12$. Moreover, we show that $\Phi$ is effectively computable as a holomorphic function on the upper half-plane (in the precise sense of computable analysis), by providing an effective procedure. This constructs a wandering and tracelike vector for PSL(2,Z), whose abstract existence was proved by Sir Vaughan Jones in his last paper, where the corresponding construction was left as a problem. The function is built using an orthonormalization and modularization method, and it displays modular reminiscencies, despite not being modular itself. provides a computable implementing vector for the abstract anti-isomorphism between the von Neumann algebra $M_{12}(\Gamma )$ and its commutant, which is generated, in Rădulescu's sense, by cusp-form Toeplitz operators, while Voiculescu's results provide a random matrix model for $M_{12}(\Gamma )$.

Yang introduced a tempered xi function $\widehat\xi(s)$ by replacing the hyperbolic cosine in a classical integral representation of the Riemann xi function by a hyperbolic sine, and conjectured that every zero of $\widehat\xi$ lies on the critical line $\Re s=1/2$. We disprove this conjecture. After the change of variables $x=e^{2t}$, one has \[ \widehat\xi\!\left(\tfrac12+iz\right)=iS(z), \qquad S(z)=\int_0^\infty K(t)\sin(zt)\,dt, \] where $K$ is positive, smooth, and doubly exponentially decaying. Two integrations by parts give $S(y)=K(0)/y+O(y^{-2})$ for real $|y|\to\infty$, with $K(0)>0$; hence $S$ has only finitely many real zeros. On the other hand $S$ is an entire function of order at most one. If it had only finitely many zeros in the complex plane, Hadamard factorization would force $S(z)=P(z)e^{az+b}$, which is incompatible with $S(y)\to0$ as $y\to\pm\infty$. Thus $S$ has infinitely many nonreal zeros, and consequently $\widehat\xi$ has infinitely many zeros off the critical line. The proof is unconditional and does not use the Riemann Hypothesis or numerical zero finding. A short numerical illustration is included.

We relate the theory of $q$-rational and $q$-real numbers introduced by Morier-Genoud and Ovsienko to the classical theory of Kleinian groups and their Teichmüller spaces. This provides a geometric point of view on several recent results about realisations of $q$-rationals for particular values of $ q \in \mathbb{C} $. As an application we resolve a conjecture of Bapat, Becker, and Licata on the topology of the $q$-deformed Farey tessellation.

In 1918, Gross proved that a regular local inverse of a meromorphic function in the plane can be continued analytically along every ray from its centre except for directions in a set of Lebesgue measure zero. In 1977, Nagasaka asked whether this exceptional set must have logarithmic capacity zero. We prove that there exist a transcendental entire function and a regular local inverse for which the exceptional set contains a compact set of positive logarithmic capacity. In fact, for every $0<s<1$, such a function and inverse can be chosen so that this compact set has positive $s$-dimensional Hausdorff measure.

In 2012, Iwaniec, Kovalev and Onninen proposed an upper Nitsche–Grötzsch type estimate for harmonic $K$-quasiconformal homeomorphisms between circular annuli: in normalized form, every such map $h:A(1,s)\to A(1,S)$ should satisfy \[ S \leq \frac{K+1}{2}s-\frac{K-1}{2s}. \] The radial and spiral-radial one-mode models explain why this estimate is natural. We show that this one-mode evidence does not extend to the unrestricted non-radial class. For every $1<s<S$, we construct a non-radial harmonic orientation-preserving diffeomorphism $h:A(1,s)\to A(1,S)$ with \[ \norm{\omega_h}_{\infty}<\frac{S-s}{S-s^{-1}}, \] strictly below the conjectural threshold. Thus, the dilatation lower bound predicted by the conjecture fails, even for smooth harmonic diffeomorphisms with prescribed circular boundary components. The construction uses a small high-frequency reparametrization of the outer boundary: it lowers the first Fourier-mode dilatation by order $t^2$, while the compensating high modes are exponentially damped at the inner boundary. We also record structured regimes in which the one-mode estimate survives, including an inner-boundary anti-conformal energy condition, a Fourier leakage criterion, and a low-frequency spectral stability result. In the minimal-surface interpretation, the upper radial model is helicoidal and vertical-periodic rather than single-valued catenoidal; accordingly, the counterexamples give vertical-periodic minimal annuli with slope below the expected helicoidal threshold, and lead to a non-radial extremal problem for harmonic annulus diffeomorphisms.

We study the existence and nonexistence of meromorphic solutions of nonlinear differential equations involving differential polynomials and finite sums of exponential-type functions with entire coefficients of controlled growth. Employ ing Nevanlinna theory, we obtain structural results showing that any admissible meromorphic solution must reduce to an exponential-type function governed by the dominant exponential term. The obtained lemma and main result strengthen several existing results in nonlinear complex differential equations with exponential terms.

In this paper, we study the existence, nonexistence, and growth be haviour of solutions of certain nonlinear differential and differential-difference equa tions involving the term fnf'. Using tools from Nevanlinna theory, we establish several nonexistence results for finite-order entire solutions of equations involving ff'. In particular, we describe all the possible entire solutions of the type f2f' in which right hand side involves exponential polynomials. The results obtained by us improve several known results in the literature and we have them demonstrated through illustrative examples.

We prove sharp metric entropy estimates for spaces of Kähler potentials. In complex dimension $n$, normalized potentials have Kolmogorov entropy of order $\e^{-n}$ in the background $L^1$ metric. On a polarized manifold, a relative-entropy sublevel has the same order in the Mabuchi–Darvas $d_1$ metric, including its full finite-energy closure. The upper bound is $C_X(1+B)^{n+1}\e^{-n}$ for entropy budget $B$. For toric potentials, the sharp exponent is $n/2$.

This paper is the final installment in our series on the geometric theory of complex Monge-Ampere equations. We study singular Kahler metrics on compact normal Kahler spaces with klt singularities whose volume densities lie in $L^p$ for some $p>1$. We show that a lower bound for the Ricci current in the sense of pluripotential theory is equivalent to a synthetic Ricci lower bound in the sense of RCD theory. As a consequence, every Kahler class on a compact normal Kahler space with klt singularities admits an RCD structure, which makes differential geometric analysis available on singular complex spaces. As an application, we show that the fundamental group of a compact Kahler Calabi-Yau space with klt singularities is almost Abelian. We further establish compactness results for singular Kahler-Einstein spaces.

This paper is the continuation of our earlier work on Holder estimates for Kahler potentials on compact normal Kahler spaces. We establish a geometric counterpart for analytic Holder regularity of Kolodziej. If a singular metric $\omega_\phi$ has Holder continuous potentials with respect to a smooth background metric, then its distance function is Holder continuous with respect to a smooth background distance. The result holds on normal Kahler spaces and yields compactness of the metric completion as well as its identification with the underlying complex space if $\omega_\phi$ is a Kahler current. Under this positivity assumption, our estimates establish the Holder equivalence between the intrinsic canonical Kahler metrics and the extrinsic smooth metrics on Kahler-RCD spaces, particularly on smoothable Kahler-Einstein spaces.

Let $X$ be a compact Kahler manifold. The analytic stability theorem of Kolodziej for complex Monge-Ampere equation states that for any Kahler metrics $\omega$ and $\omega'$ in the same cohomology class, if their volume measures are bounded in $L^p(X)$ (for some $p>1$) and close in $L^1(X)$, then their Kahler potentials are close in $L^\infty(X)$. In this paper, we establish the geometric stability for complex Monge-Ampère equations that $L^1$-closeness of volume measures implies $L^\infty$-closeness for the induced distance functions by $\omega$ and $\omega'$. Consequently, we prove that any non-smooth Kahler current with volume measure bounded in $L^p$ (for some $p>1$) and Ricci current bounded below induces a unique metric space, which turns out to be a compact RCD space homeomorphic to $X$ itself.

We say that the Poisson operator $P_r$ is Schwarz-contractive from a space $X$ to a space $Y$ if $\|P_r\|_{X\to Y}\le r$ holds for all $0<r<1$. The classical Schwarz lemma is the case $H^\infty_0\to H^\infty$, the subscript indicating zero mean. We are concerned with the non-holomorphic case $L^\infty_0 \to L^p$. For real functions Schwarz-contractivity holds up to the sharp exponent $p_{\mathbb R}=4.109\ldots$. For complex functions we prove it for $p\le 3$ and conjecture that the critical exponent is $4$.

We prove that the Hilbert matrix operator $H:A^2_{\ln^{\alpha}} \to A^2$ is bounded when $\alpha\geq 2$ and that $H$ does not map $A^2_{\ln^{\alpha}} $ into $A^2$ when $\alpha<2$, solving an open problem posed by B. Karapetrović and M. Jevtić. The boundedness when $\alpha=2$ is shown as a corollary to the more general statement: the Hilbert matrix $H:\ell_v^p \to \ell_w^p$ is bounded, where $v(n) = (\ln n)^p / n$ and $w(n) = 1/n$.

Starting from Alpöge's torus fibration with exactly three special fibres on a complex six-sphere, we construct a locally miniversal family of compact complex threefolds. Every member is Oka, and the projection over each relatively compact parameter disc is an Oka map. Deleting or simultaneously blowing up finitely many distinct points preserves the Oka property. We also determine the tangent cohomology, the smooth Kuranishi germ, the automorphism group, and the exact biholomorphism relation $X_a\simeq X_{a'}$ if and only if $a'-a\in2\mathbb{Z}$. The Oka arguments use explicit cyclic covers and Kusakabe's localization principle.

We give affirmative answers to both parts of Question 2.6 posed by Grosse-Erdmann (2025) concerning Wiman-Valiron inequalities in the unit disk. For every unbounded analytic function in the disk, we establish the proposed iterated-logarithm inequalities outside exceptional sets of finite logarithmic measure. The corresponding power estimate is a corollary. Both conclusions follow from a variance bound for Khinchin families and a classical estimate for their largest atom. The multiplicative constants in the main inequalities can be chosen absolute. We also obtain a disk analogue of Rosenbloom's composition estimate, with an explicit boundary prefactor. The key step combines a boundary change of variable with a monotone auxiliary function whose derivative is exactly the variance of a rescaled member of the Khinchin family. A classical example shows that the leading logarithmic exponent $1/2$ cannot be decreased.

We give an algebraic proof in characteristic zero of the Miyaoka–Mori criterion: every point of a curve of negative canonical degree on a smooth projective variety lies on a rational curve. Our jet technique gives, in addition, an effective numerical decomposition of the original curve class. For each prescribed point, the decomposition contains a rational curve through that point, with a positive coefficient independent of the point and with anticanonical degree at most $\dim X+1$. Together with BDPP cone duality, this recovers the projective uniruledness criterion over $\mathbb C$. The main result of this paper was obtained using the Pharos system. A detailed report on the use of Pharos and on the Lean~4 formalization of the main arguments is given in the Appendix B, written by Bin Dong, Guoxiong Gao, Zeming Sun, and Bin Wu.