Papers

We propose k-Hessian equations as auxiliary equations for degenera- tions of Kähler classes to nef classes of numerical dimension k. As proof-of-concept results, we provide a uniform mean-value inequality and uniform à priori estimates for Hessian equa- tions.

It is known that there can be at most 3 disjoint nontrivial minimal graphs in R^3 with 0 boundary values, and conjectured that there can be at most 2. We explore properties that the graphs would have if there were 3.

We prove that every holomorphic isometric embedding, with respect to the Bergman metrics, from a complex unit ball of maximal dimension into irreducible classical bounded symmetric domain of rank at least two that is not biholomorphic to a Type IV domain is equivalent to Mok's embedding. This completely resolves a problem of Mok.

Let $F(z)=zf(z)$, where $f$ is a normalized scalar holomorphic function on the unit ball $\mathbb B$ of a complex Banach space $X$ or on the unit polydisk $\mathbb U^n$ of $\mathbb{C}^n$. We establish sharp bounds for two Toeplitz determinants involving the second, third and fourth Fréchet derivatives of $F$ when $F$ is starlike of order $\alpha$, $0\leq\alpha<1$. The first bound is sharp for the full range $0\leq\alpha<1$, while the second is sharp for $0\leq\alpha\leq 2/3$. The extremal mappings are explicitly identified, confirming the sharpness of the estimates.

Suppose $n\ge2$ and $K\subset\C^n$ is compact and polynomially convex. Write $q(z)=(e^{-iz_1},\ldots,e^{-iz_n})$, $X_K=(\C^*)^n\setminus(K\cap(\C^*)^n)$, and $Y_K=\C^n\setminus q^{-1}(K)$. We prove that $X_K$ and its logarithmic covering $Y_K$ are Oka. More strongly, $Y_K$ admits normalized holomorphic families of entire embeddings through prescribed holomorphic sections. The proof combines polynomial pair Hamiltonian transport in multiplicative coordinates, compatible with the coordinate hyperplanes, with a fixed Fatou–Bieberbach model. Applications include Oka slab crosses over arbitrary nonempty connected planar domains, complements of periodic equal-radius Euclidean balls centered at products of planar lattices, and complements of sufficiently small closed metric balls in products of elliptic curves.

We prove that the optimal constant in the Chang–Marshall inequality is $e$, as conjectured by Andreev and Matheson. More precisely, if $f$ is analytic in the unit disk, vanishes to order $m\geq1$ at the origin, and has normalized Dirichlet integral $d\leq1$, then the boundary mean of $e^{|f|^2}$ is at most $e^{d/m}$. Equality holds precisely for monomials. The proof uses an auxiliary closed planar curve and the isoperimetric inequality to establish monotonicity of the exponential functional along normalized radial dilations.

We develop a quaternionic formulation of linear elasticity in which the classical derivative is replaced by Yang's local fractional derivative and the classical Cauchy–Riemann operator by a fractal Cauchy–Riemann operator. We formulate fractal kinematics, a constitutive law with a material length scale and the equilibrium equation, and show that, for displacement fields with polynomial components, the resulting homogeneous fractal Lamé system is equivalent to the monogenicity of an explicit quaternionic field. On the polynomial sector, that is, for finite combinations of normalized fractal monomials, we prove a fractal Kolosov–Muskhelishvili representation of the displacement field in terms of two monogenic quaternionic potentials, together with its completeness, by a direct construction that uses neither the Galerkin nor the Papkovich–Neuber potentials. The position term of the classical formula has to be replaced by a corrected position operator, which satisfies the exact Weyl relation with the partial local fractional derivatives. Finally a fractal Papkovich-Neuber representation is derived.

We prove Krzyż's conjecture for the sixth coefficient. Every holomorphic function $f(z)=\sum a_kz^k$ from the unit disk to the punctured disk satisfies $|a_6| \le 2/e$. Equality holds precisely for rotations of $\exp(-(1-z^6)/(1+z^6))$. Every extremal has a Herglotz representation with at most six atoms. By the reduction of Agler and McCarthy and the entropy theorem of Lei and Zhang, it suffices to exclude extremals with fewer than six atoms. We do this by a rigorous computer-assisted branch-and-bound, leaving the equality case entirely to the analytic argument. The main ingredients are explicit first-order conditions for extremals with fewer than the maximal number of atoms, which reduce to Chebyshev relations when one atom is missing, and a bound for the total mass obtained by convexity of the Poisson integral. The same method independently establishes the full-degree property for the known cases $n=3,4,5$.

For endomorphisms of the complex projective plane of a given degree, we prove that the existence of a special curve of bounded degree, in the sense of the dynamical Manin–Mumford conjecture, is confined to a proper algebraic subvariety of the parameter space. This confirms a conjecture of DeMarco and Mavraki in dimension two. For general regular polynomial endomorphisms of $\mathbb A^k$, we obtain a uniform dynamical Manin–Mumford bound outside an exceptional hypersurface. These results arise from a common strategy, valid in every dimension and codimension, which reduces such questions to the geometry of exceptional loci in the space of pairs formed by subvarieties and maps. We control these loci by combining arithmetic equidistribution, blender constructions, dynamical directions, monodromy arguments, and degenerations at the boundary of parameter space.

Motivated by three-dimensional wave diffraction by conical shapes, we consider the two-dimensional problem of wave scattering on the real sphere $\mathcal{S}^2$ resulting from a point-source incidence on a scatterer $\mathbb{S}$. Using the recently introduced notion of plane waves on $\mathcal{S}^2$ as well as recent results on the Laplace–Beltrami Green's function $G$ on $\mathcal{S}^2$, we derive three plane-wave representations for the scattered field $u$. We then explain how $u$ can be analytically continued to the complexified sphere $\mathcal{S}^2_{\mathbb{c}}$. This allows us to formulate a complexified version of Green's third identity, and to provide a characterisation of complex sources. All our results are then used to draw strong analogies between planar and spherical scattering problems, notably through the introduction of directivities.

Let $f$ be a nonzero, non-monomial entire function, and let $\nu_f(r)$ count the points at which $f$ attains its maximum modulus on $|z|=r$. Write $f(z)=z^m(a_0+a_kz^k+O(z^{k+1}))$, where $a_0a_k\ne0$. We prove that $\nu_f(r)\le 2k$ outside a locally finite subset of $(0,\infty)$. The constant $2k$ is sharp. In particular, $\liminf_{r\to\infty}\nu_f(r)<\infty$, so the number of maximum-modulus points of a fixed non-monomial entire function cannot tend to infinity on all sufficiently large circles. The proof associates the logarithmic derivative $q=zf'/f$ with its reflection $q^*(w)=\overline{q(\bar w)}$ through the correspondence $q(z)=q^*(w)$ and the map $(z,w)\mapsto(zw,q(z))$. Local valence at the origin bounds the number of inverse pairs that can escape. A local elimination argument then bounds the generic degree of each nonvertical image component. The residue structure of $f'/f$ excludes vertical components.

Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\infty$, and let $\phi=b/a$, where $a$ is the Pythagorean mate of $b$. In an earlier article, the author showed that, if $\phi\in H^2$, then \[ f\in\mathcal{H}(b) \iff \Bigl(f\in H^2 \text{ and }\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2<\infty\Bigr), \] in which case we have \[ \|f\|_{\mathcal{H}(b)}^2=\sum_{m\ge0}|\hat{f}(m)|^2 +\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2. \] It was also shown that there exists a space $\mathcal{H}(b)$ for which $\phi\in\cap_{p<2}H^p$, yet \[ f\in\mathcal{H}(b)\not\Rightarrow \sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2<\infty. \] Here we complete this circle of ideas by showing that there exists a space $\mathcal{H}(b)$ for which $\phi\in\cap_{p<1}H^p$, yet \[ \Bigl(f\in H^2\text{ and } \sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2<\infty\Bigr) \not\Rightarrow f\in\mathcal{H}(b). \]

We study the exponential Green energies associated with a finite point configuration on a Riemann surface. We focus on studying optimal or close-to-optimal configurations, and show that they must be well-spread out on the surface. We study in particular the round sphere and obtain values for the minimal Green energy of a configuration. The first version of this paper was written with the help of ChatGPT.

We extend the packing results of Etayo, Hedenmalm, and Ortega-Cerda for complex tori to arbitrary compact Riemann surfaces. The construction uses two different approaches, one based on complex geometry of line bundles, and the other on Laplace designs. Both approaches give the same basic existence of close-to-optimal point configurations on the given surface. This version was written with the help of ChatGPT.

We study weighted Bergman spaces of quasiregular mappings on the unit ball in $\mathbb{R}^n$, with weight $(1-|x|)^\alpha$, $\alpha>-1$. For every fixed distortion bound $K\ge1$ and every $p>0$, we characterize Bergman integrability by averaged derivatives. We also give geometric conditions under which integration over a subset, or against another measure, controls the full Bergman integral. When $K$ is sufficiently large, the dominating sets are precisely the relatively dense sets. For comparison, we determine the dominating sets for the Möbius class $K=1$ in dimensions $n\ge3$. At the critical exponent $p=n+\alpha$, its domination criterion differs from that for every fixed $K>1$. Our results extend aspects of the Bergman theory developed by Nolder~\cite{Nolder90} and by Oleinik and Vuorinen~\cite{OV}, and complement the recent work of Adamowicz and González~\cite{AG21,AG25} and Adamowicz and Caamaño~\cite{AC} on quasiregular Hardy spaces.

In this work, we focus on deriving the absolute moments of the general ized hyperbolic (GH) distribution via its mean-variance mixture structure. Conditioning on the GIG mixing variable reduces the problem to Gaussian moment calculations and known GIG power moments, avoiding direct integration with Bessel functions. The variance gamma (VG), normal inverse Gaussian (NIG), hyperbolic, Normal, and Student t distributions emerge as natural special or limiting cases of the generalized hyperbolic family. This approach offers a substantially simpler and more transparent derivation, facilitating both implementation and verification.

We classify graphs of complex polynomials in one variable up to isometry, both for the inner (length) distance and for the outer (Euclidean) distance of $\mathbb{C}^2\cong\mathbb{R}^4$. For polynomials $p,q$ with $\deg p\ge 2$ we show that the graphs of $p$ and $q$ are inner isometric if and only if they are outer isometric, if and only if $p(z)=\alpha\,q(\beta z+h)+k$ or $p(z)=\alpha\,\bar q(\beta z+h)+k$ for some $\alpha,\beta,h,k\in\mathbb{C}$ with $|\alpha|=|\beta|=1$, where $\bar q$ denotes the polynomial with conjugate coefficients. Moreover, every inner isometry between such graphs is the restriction of an isometry of $\mathbb{C}^2$; in this sense graphs of complex polynomials are isometrically rigid. We contrast this with the real case, where the two distances behave very differently: the graphs of any two real polynomials are inner isometric, whereas, for degree at least $2$, outer isometries are generated by translations, reflections in horizontal and vertical lines and half-turns.

We establish characterizations of planar quasiconformal mappings in weighted Bergman spaces $A_\alpha^p(\mathbb D)$. For $p>0$ and $\alpha>-1$, our characterization involves the convergence of certain integrals involving superlevel set areas, the Hausdorff measure of boundary superlevel sets, and the averaged derivative. These include quasiconformal analogues of characterizations established by Karafyllia and Karamanlis for conformal mappings. Moreover, for $-1<\alpha\leq 0$, we obtain an additional equivalent characterization in terms of the Hausdorff measure of level sets. The power $\alpha$ is sharp: for every $\alpha>0$, we construct a bounded $K_\alpha$-quasiconformal mapping for which the weighted level-set length integral with exponent $\alpha+2$ diverges.

Motivated by the role of piecewise linear functions in ReLU neural networks, we study systems of dilated functions in the Lebesgue spaces $L_q([0,1]^d)$, $q\in(1,\infty)$. For a sequence $\tau\in\ell_1$ we consider the operator $T_\tau g(x)=\sum_{k=1}^\infty \tau_k g(kx)$ and the function system $\mathcal Z^\tau$ obtained by applying it to the trigonometric system. We show that $\tau\mapsto T_\tau$ is an isomorphism of algebras from $\ell_1$ with the Dirichlet convolution as multiplication onto an algebra of operators and connect the inverse of $T_\tau$ to the inverse of $\tau$ with respect to the Dirichlet convolution. We give explicit conditions on $\tau$, under which the resulting system $\mathcal Z^\tau$ is a Schauder basis of $L_q([0,1]^d)$ and, for $q=2$, a Riesz basis for which we estimate the Riesz constants in terms of $\tau$. We characterise the Riesz bases of this type in terms of a sequence space of Dirichlet convolution multipliers, and we treat mixed systems of cosine and sine functions with different sequences. Finally, we apply these results to the system generated by two piecewise linear functions inspired by ReLU networks and improve the known lower Riesz constant to $2/3$, independently of the dimension.

In this paper, we study biholomorphic mappings $F: \mathbb{B} \to X$ of the form $F(z) = f(z)z$ for $z \in \mathbb{B}$, where $f: \mathbb{B} \to \mathbb{C}$ is a holomorphic function satisfying $f(0) = 1$. Our main contributions are as follows: We establish sharp estimates for the coefficients of starlike functions of order $\alpha$ as well as quasi-convex functions of type $B$ and order $\alpha$ by employing the notion of majorization. The results established here directly extend the results due to Robertson \cite{Robertson-1936} to the broader setting of complex Banach spaces.

Based on earlier work by Wiegerinck and Huang-Li, Hayashida-Kamimoto constructed unbounded non-smooth Reinhardt domains in \({\mathbb C}^n\), \(n\geq 3\), whose Bergman spaces are finite-dimensional and whose Bergman metrics have constant positive holomorphic sectional curvature \(2\). In this note, we prove a Sandwich Principle for the Hayashida-Kamimoto domains that allows us to construct, inside every prescribed sandwich, uncountably many pairwise biholomorphically inequivalent smooth non-Reinhardt domains whose Bergman metrics also have constant positive holomorphic sectional curvature \(2\). We also classify biholomorphisms between domains sandwiched by the domains of Hayashida-Kamimoto with the help of an affine rigidity theorem in the spirit of Cartan rigidity, which should be of independent interest. For explicit angular perturbation families, we determine the full automorphism group and obtain, in particular, an abundance of examples with trivial automorphism group.

We show that the cotangent bundle of a holomorphically flexible Stein manifold has the symplectic density property. As application we deduce holomorphic approximation for Hamiltonian diffeomorphisms of real cotangent bundles by complex symplectic automorphisms of the complexified cotangent bundles, which preserve the real bundle, in the Carleman sense, i.e. in the Whitney topology.

We study the de Rham cohomology and Deligne splitting for a class of Eisenstein-Kronecker forms by introducing a modular weight grading. As an application, we compute a particular cohomology class that gives rise to generating series of Gromov-Witten invariants of elliptic curves.

A Hsiang cubic is a cubic form on $\mathbb{R}^{n}$ whose zero set is a minimal cone; equivalently, it is the cubic form of a commutative nonassociative metrized algebra satisfying a single identity of degree four. We show that the intrinsic geometry of such an algebra, encoded in its idempotents and their Peirce decompositions, is the geometry of a Chebyshev-type extremal problem on the sphere. A Hsiang cubic equioscillates on $S^{n-1}$: its maximum set is the set of normalized idempotents, and on suitable great circles through an idempotent it restricts to the Chebyshev polynomial $T_3$. For the spherical analogue of the problem of least deviation, the minimization of $\|p\|_\infty^2/\|p\|_2^2$ over harmonic cubics, a sharp lower bound based on Kellogg's inequality shows that Cartan's isoparametric cubics are the global minimizers in dimensions $5$, $8$, $14$ and $26$. In general, we compute the ratio of every Hsiang cubic in terms of the Killing form and characterize stationarity by a Kolmogorov criterion: it amounts to a positive measure on the idempotent variety. Such measures are given explicitly in several cases which together cover every Hsiang cubic known to exist; in particular, all of them are stationary. These measures give quadrature identities on the idempotent variety whose moments are invariants of the algebra; they are spherical designs exactly in the Einstein (Killing metrized) case, i.e.\ when the Killing form is proportional to the inner product. Distinguished low-dimensional Hsiang cubics are strict local minimizers, whereas a polar Hsiang cubic is a stationary non-minimizer.

Let $\Omega$ ___ C n be a bounded hyperconvex domain and $\Phi$\,: $\Omega$ $\rightarrow$ R a maximal plurisubharmonic function in $\Omega$ (possibly unbounded). We introduce a new class E 1 $\Phi$ ($\Omega$) of plurisubharmonic functions with boundary values equal to $\Phi$ in the sense of Cegrell and finite $\Phi$-energy on $\Omega$. Then given a Borel measure $\mu$ on $\Omega$, we apply a variational approach to prove that $\mu$ is the Monge-Ampère measure of a function u $\in$ E 1 $\Phi$ ($\Omega$) if and only if it satisfies the integrability condition We also establish a quantitative version of this condition extending previous results of Cegrell and others. Using a logarithmic transformation, we obtain analogous results for the real Monge-Ampère equations with (unbounded) boundary data.

We construct closed Levi non-degenerate CR manifolds with non-vanishing Chern monomials. For every $n \geq 2$ and $\lceil n / 4 \rceil \leq m \leq \lfloor n / 2 \rfloor$, there exists a $(2 n + 1)$-dimensional example of signature $(\lfloor n / 2 \rfloor, \lceil n / 2 \rceil)$ on which all Chern monomials with positive indices at most $m$ and total weight at most $n - m$ are non-zero in cohomology with real coefficients. Taking $m = \lceil n / 4 \rceil$ gives examples with $c_{1}^{\lfloor 3 n / 4 \rfloor} \neq 0$. For $n = 4$ or $n \geq 6$, the latter examples show that the Chern-class vanishing theorem for strictly pseudoconvex CR manifolds does not extend to the Levi non-degenerate case. In these dimensions, their underlying manifolds admit no strictly pseudoconvex CR structures. The construction uses circle bundles over products of integral pseudo-Kähler manifolds.

In this note we show that if a contractible Stein manifold $X$ with the density property admits a proper holomorphic embedding into $\mathbb{C}^k$, then $X \times \mathbb{C}^k$ is biholomorphic to $\mathbb{C}^{n+k}$ with $n=\dim X$. Combined with an announced result of Auroux, Bai and Pomerleano, this shows that for some $4 \le n \le 8$ there is a complex affine variety $X$ such that $X \times \mathbb{C}$ is biholomorphic to $\mathbb{C}^{n+1}$ and $X$ is not biholomorphic to $\mathbb{C}^n$. This settles the Holomorphic Cancellation Problem in the negative.

We prove that every weakly pseudoconvex Oka manifold admitting a positive line bundle is elliptic in the sense of Gromov, thereby proving Gromov's ellipticity conjecture. The proof establishes a cohomological construction of local sprays. Given a bundle morphism $\varphi\colon E\to T_X$, we introduce quadratic $\varphi$-vector fields on $E$ whose flows yield local sprays with fibre derivative $\varphi$. Their existence is characterized by the vanishing of the symmetrized $\varphi$-Atiyah class of $E$. This gives a local dominating spray on every weakly pseudoconvex manifold admitting a positive line bundle, without any Oka assumption. The positivity hypothesis is essential even for local existence: blow-ups at a single point of complex tori of algebraic dimension zero, and Kummer surfaces of algebraic dimension zero, admit no local dominating spray. The torus examples give compact Kähler Oka manifolds that are not elliptic and show that ellipticity is not preserved under blowing up, and that it is neither open nor closed in holomorphic families of compact Oka manifolds. Combined with the work of Xie and Zhao, the Kummer examples yield Oka K3 surfaces that are not elliptic. Under the additional Oka assumption, we globalize the local dominating sprays constructed above. The proof combines Oka approximation with a gluing argument based on weighted $L^2$ estimates for the $\bar\partial$-equation.

It was known that the canonical bundle of a smooth toroidal compactification of a noncompact finite-volume ball quotient is nef in dimension $n\geq3$ and ample in dimension $n\geq6$. We prove that the canonical bundle is ample in dimension $n\geq4$ and show that this dimension bound is sharp. We construct infinitely many pairwise noncommensurable noncompact Picard modular threefolds whose smooth projective toroidal compactifications have non-ample canonical bundle. Each compactification contains a smooth rational curve meeting the interior and having canonical degree zero. More generally, in every dimension $n\geq2$, we construct infinitely many pairwise noncommensurable noncompact Picard modular varieties whose smooth toroidal compactifications contain smooth rational curves meeting the interior. In particular, such curves persist in arbitrarily high dimension, even when the canonical bundle is ample.

We establish first-order variational formulas for the driving function and the half-plane capacity of a chordal Loewner chain under quasiconformal deformations. Under the standard normalization fixing $0,1$ and $\infty$, the formulas hold for every bounded Beltrami differential. If the curve has positive area, the formulas contain additional integrals over the curve; these terms vanish when the curve has zero area or when the differential is extended by zero on the curve. The proof uses the normalized Ahlfors–Bers variation together with a Stoilow factorization adapted to the slit domain. We then consider the asymptotic normalization for differentials whose symmetric extensions to $\mathbb{C}$ have the form $\nu(z)=a(\bar z/z)^{n-1}+\nu_0(z)$, where $a\in(-1,1)$, $n\ge1$ is an integer, and $\nu_0\in L^r(\mathbb{C})$ for $0<r<2$. The asymptotically normalized map differs from the standardly normalized map by a positive dilation. Using this relation, the variational formulas in the second normalization follow directly from the standardly normalized case.