Papers

Let $m$ denote the normalized Haar measure on the unit circle $\mathbb{T}$. For $0<r<\infty$, define $\lVert f\rVert_{r}:=\left(\int_{\mathbb{T}}|f|^{r}\,\mathrm{d} m\right)^{1/r}$, with $\lVert f\rVert_{0}$ and $\lVert f\rVert_{\infty}$ interpreted as the geometric mean and the supremum norm, respectively. Set $Q_n(z)=1+z^n$. We prove that, for every nonzero polynomial $p$ of degree $n$ whose zeros all lie on $\mathbb{T}$, $$ \frac{\lVert p\rVert_s}{\lVert Q_n\rVert_s} \le \frac{\lVert p\rVert_t}{\lVert Q_n\rVert_t}, \qquad 0\le s\le t\le\infty. $$ This settles Baernstein's quasi-norm monotonicity conjecture. As corollaries, we obtain an $L^r$ extension of Visser's coefficient inequality, the sharp O'Hara–Rodriguez inequality and its higher-power analogues, the Erdős–Szekeres product bound $ \left\lVert\prod_{j=1}^N(1-z^{s_j})\right\rVert_\infty\ge2\sqrt N $ for all positive integers $s_1,\ldots,s_N$ and Agler–McCarthy's entropy conjecture. We also provide a Lean 4 formalization of the main results.

We prove that every holomorphic expanding map on a compact connected complex manifold is biholomorphically conjugate to an affine expanding endomorphism of a complex infra-nilmanifold.

We develop a geometric way to uniformly carry out Hörmander's $L^2$-estimates for Hermitian-Yang-Mills connections with bounded energy on Kähler surfaces. We then show the Uhlenbeck convergence of smooth HYM connections on Kähler surfaces can be realized as algebraic convergence inside an analytic flat family. As applications, we prove a continuity theorem for smoothable families and a boundedness result for $\mu$-semistable sheaves with fixed topology. We also discuss potential applications for the moduli problem.

Given a $\mathbb{Q}$-Gorenstein normal projective Calabi-Yau variety $Y$ with ordinary double points as singularities and a small crepant resolution $\pi\colon X\to Y$ with $X$ in the Fujiki class $\mathcal{C}$, we establish uniform estimates away from the exceptional locus where the given metrics may degenerate. When $Y$ is a threefold, we prove that these metrics also converge in a Gromov-Hausdorff sense to the metric completion of $Y$ on the regular locus. In particular, our results extend the work of Song on a conjecture of Candelas and de la Ossa.

We study the Dirichlet problems with unbounded boundary data. Our main theorem proves interior continuity from a local biholomorphic deformation satisfying a one-sided compatibility condition near the infinite locus and barriers on the associated boundary cluster set. The deformed domains need not be nested, and the proof uses neither a Runge hypothesis nor a pseudoconvex ambient neighbourhood. We also prove nonmonotone Perron–Bremermann convergence outside an intrinsic neighbourhood pluripolar hull, a disk-fibre hull representation, stability under simultaneous nonmonotone variations of the domain and obstacle, and uniqueness in a fixed obstacle class.

We prove the continuity of the pluricomplex Green functions introduced by F. Lárusson and R. Sigurdsson on a class of bounded (B)-regular domains that includes all (B)-regular convexifiable domains.

We prove that the graph of a standard one-dimensional Brownian motion, $\Gamma_B=\{(t,B_t):t \in [0,1]\}$, is almost surely not conformally removable. This resolves the question left open by Doherty and Miller in their study of the Sobolev removability of the Brownian graph~\cite{DM26}. We also construct non-removable $\alpha$-Hölder graphs for every $\alpha \in (0,2/3)$, including the previously open range $\alpha \in [1/2, 2/3)$. Together with the known removability result for $\alpha>2/3$, this identifies $2/3$ as the threshold for conformal removability of Hölder graphs.

We study non-autonomous basins of $F_j$ automorphisms of $\mathbb C^2$ satisfying $$a_j\|p\|\leq\|F_j(p)\|\leq b_j\|p\|\qquad (\forall p\in \mathbb B)$$ where $0<a_j<b_j<b<1$. We show that if $b_j^m<a_j$ for some integer $m$ and all $j>0$ then every bounded plurisubharmonic function on the non-autonomous basin is constant. Moreover, if in addition we have $\prod_{i=1}^{j-1}b_i\leq b_j^m$ then the basin is biholomorphic to $\mathbb C^2$. For the quadratic Hénon family $$F_n(z,w)=\bigl(z^2+c^{q_n}w,c^{q_n}z\bigr), \qquad 0<c<1,$$ we study how the asymptotic behavior of the sequence $q_n\to\infty$ influences the complex structure of the basin. In particular, we identify broad regimes in which the non-autonomous basin is either biholomorphic to $\mathbb C^2$ or is a Short $\mathbb C^2$, thereby isolating possible sequences for which other complex structures may occur.

A domain in the Riemann sphere is called a circle domain if every connected component of its boundary is either a round circle or a single point. A circle domain is conformally rigid if every conformal map onto another circle domain is the restriction of a Möbius transformation. Conformal rigidity is closely related to Koebe's conjecture, which asserts that every domain in the Riemann sphere is conformally equivalent to a circle domain. In this article, we survey recent progress on conformal rigidity, with particular emphasis on its relationship with conformal removability.

The rough response calculus for the rotationally marked SLE welding law gives a closed Weil–Petersson energy but does not by itself give a process on rough weldings. For $0<\kappa\leq4$, we prove that this form is quasi-regular in the natural topology of circle homeomorphisms and therefore generates a canonical diffusion reversible with respect to the SLE welding probability. The proof constructs a compact form nest from inverse arc masses. We identify the generator as $-D^*D$ and prove absolute $L^2$ convergence of its Fourier expansion on the response cylinder core, together with the Fukushima decomposition. A common Fourier noise and planar estimates preventing boundary collapse yield a second quasi-regular realization by normalized Jordan welding triples. The diffusion is irreducible and converges strongly in $L^2$ to equilibrium. For $0<\kappa<4$, the complement of the conformally removable welding locus has zero capacity.

We propose a pluripotential-theoretic framework for studying finiteness questions for polarized Calabi–Yau manifolds. The central idea is to use global \(m\)-Hessian capacities as a quantitative intermediate between complex potential theory and global geometric boundedness. We formulate a chain of uniform estimates connecting \(m\)-Hessian capacity, volume–capacity inequalities, capacity–perimeter estimates, Sobolev bounds, and non-collapsing of the associated Ricci-flat Kähler metrics. Under suitable regularity assumptions, these estimates lead to uniform diameter and curvature control and hence to metric compactness. We then describe the algebraic part of the argument. Uniform projective embeddings with bounded degree lead, through the Macaulay–Gotzmann theory, to only finitely many possible Hilbert polynomials and hence to a finite collection of Hilbert schemes. Ehresmann's fibration theorem then converts boundedness of the smooth algebraic families into finiteness of diffeomorphism and topological types. The resulting framework isolates the main analytic and algebraic bottlenecks in a potential-theoretic approach to Calabi–Yau finiteness. In particular, the capacity–perimeter estimate and the derivation of uniform algebraic embedding data from the preceding metric estimates constitute the essential steps requiring further development. Thus the paper provides a structured program linking complex Hessian potential theory with geometric and algebraic boundedness, rather than assuming that these implications are automatic.

We give a conformal mating description for the class $\Sigma_d$ of hyperbolic Schwarz reflections with connected and full filled Julia set; this parallels the recent development in several non-hyperbolic settings. We show that the escaping dynamics of $\Sigma_d$ can be characterized independently by the class of annular Schwarz reflections we introduce. Our main result is that each $\sigma\in\Sigma_d$ decomposes into a unique hyperbolic polynomial factor and a unique annular Schwarz reflection factor and, conversely, any two such factors may be mated to determine a unique such Schwarz reflection. We also show that annular Schwarz reflections converge to Nielsen maps in suitable settings, connecting the hyperbolic and non-hyperbolic mating descriptions.

In this paper, we study a problem posed by Matsumura on the extension of pluricanonical forms in Kähler families with a relatively nef canonical bundle. We give an affirmative answer in the smooth case, and for one-parameter degenerations under the additional assumption that the relative canonical bundle restricts to a big line bundle on an irreducible component of the central fiber. In particular, we confirm Siu's conjecture on the invariance of plurigenera for a smooth Kähler family with one fiber admitting the nef canonical bundle. The key step in the smooth case is to construct a semipositively curved singular metric on the relative canonical bundle with the integrability required for Cao's $L^2$ extension theorem, using Păun's twisted form of Schumacher's curvature formula and a uniform Monge–Ampère estimate obtained by adapting the capacity method of Boucksom–Eyssidieux–Guedj–Zeriahi. For one-parameter degenerations, we propagate bigness from an irreducible component of the central fiber to nearby smooth fibers and then apply Matsumura's relative Kawamata–Viehweg vanishing theorem on a projective modification to obtain the desired extension.

We disprove a conjecture of Liu, Luo, and Luo (2020) concerning the Landau radius for bounded planar harmonic mappings. We construct an explicit harmonic mapping $F_M$ satisfying the standard normalization conditions, which fails to be locally univalent strictly within the classical holomorphic Landau disk for all $M > M^* \approx 4.451$. By combining this counterexample with the lower bound established by Ponnusamy, Kalaj, and Vuorinen (2014), we demonstrate that the exact asymptotics of the harmonic Landau radius is $\frac{\pi}{8M}$.

Korenblum's maximum principle states that if $|f| \le |g|$ near the boundary, then $\| f \| \le \| g \|$. The optimal size of the region of domination has been studied in Bergman spaces since 1991 and in Fock spaces since 2006, but only bounds have been obtained. We determine the optimal size for the Fock space $F^2$ of entire functions that are square integrable with respect to $e^{-|z|^2}\,dA(z)$: if $f, g\in F^2$ and $|f(z) \le |g(z)|$ for all $|z|>1$, then $\| f \| \le \| g \|$. The radius $1$ is optimal and we determine all pairs for which equality holds. To our knowledge, this is the first space of Bergman or Fock type in which the optimal radius in Korenblum's principle has been determined. The proof combines a Schwarz-Pick estimate centred at infinity with a moment-duality argument, and requires no numerical computation. The same method gives a moment criterion for weighted Fock-type spaces. For every non-increasing radial weight, the optimal radius equals the upper bound given by the pairs $f\equiv c$, $g(z)=z$. In particular, the upper bounds of Wee and Le for such weighted Fock spaces are sharp in the Hilbert space case. We also show that the optimal radius is $\sqrt{(\beta+1)/\alpha}$ for the weight $|z|^{2\beta}e^{-\alpha |z|^2}$, whenever $-1<\beta\le5.44$.

Let $f$ be a nonzero holomorphic germ at $0 \in \mathbb C^n$ with $f(0)=0$, and let $\chi$ be a $C^2$ non-decreasing convex function on the left half-line. We establish sharp necessary and sufficient conditions for the local Sobolev regularity of the plurisubharmonic function $v=\chi(\log|f|).$ The criteria for the $L^p$-integrability of the classical Laplacian and for $W^{1,p}$-regularity are given by a weighted integral involving $\chi''$ and $\chi'$, respectively, and depend on $f$ only through the smallest multiplicity of $\operatorname{Div}(f)$. We also obtain the $W^{2,p}_{\mathrm{loc}}$ criterion for $1<p<\infty$. At the endpoint $p=1$, we prove that \[ v\in W^{2,1}_{\mathrm{loc}} \quad\Longleftrightarrow\quad \chi'\in L^1((-\infty,A)), \] equivalently, $v$ is locally bounded. As applications, we obtain counterexamples to Calderón-Zygmund theory and characterize a class of functions in the local Monge–Ampère domain. % and exhibit a natural family in $W^{2,1}_{\mathrm{loc}}$ whose limit fails to belong to $W^{2,1}_{\mathrm{loc}}$.

Let $p$ be a complex polynomial of degree $n\ge2$, and let $\sigma_2(p)$ be the root mean square distance of its zeros from their centroid, including multiplicities. We prove that every zero of $p$ lies within distance $\sigma_2(p)$ of a critical point. This proves Borcea's quadratic conjecture and, in particular, gives a new proof of Sendov's conjecture. Two polynomial integration identities prove the result analytically for $n\ge2048$. For the remaining degrees, we derive explicit necessary inequalities, including a variable integration endpoint and an exact centered coefficient identity. Directed integer arithmetic certifies a finite covering of the resulting parameter domains. Degrees $4$–$26$ use $150\,845$ certified three-dimensional regions; degrees $27$–$2047$ use a further $27\,429$ regions, with degree intervals allowed. The cases of degree 2 and 3 were already covered by Khavinson–Pereira–Putinar–Saff–Shimorin.

We study the zero distribution of the sum of the first $n$ Gegenbauer polynomials \[ S_n^{(\alpha)}(z)=\sum_{k=0}^n C_{k}^{(\alpha)}(z). \] We prove that for $\alpha>1$, the zeros of $S_n^{(\alpha)}(z)$ lie on the interval $(-1,1)$ for all large $n$.

We consider the multi-spinor Seiberg-Witten equations on a 4-manifold in a general setting, where we allow the auxiliary bundle to be non-trivial. We show that when the metric is Kähler and the auxiliary connection is unobstructed anti-self-dual Yang-Mills, the moduli space of solutions is compact for all parameters in some open neighborhood of the given metric and auxiliary connection. We use this fact to compute a count of solutions on some examples of Kähler surfaces. These computations have a surprising purely algebro-geometric corollary which may be of independent interest.

We study the local geometry, singularity classification, and wavefront dynamics of complex evolutoids for regular holomorphic curves in $\mathbb{C}^2$ equipped with the standard complex bilinear metric. By integrating the generating family of rotated complex affine lines, we derive an explicit parametrization for the 1-parameter family of complex wavefronts $\Gamma^{\theta, w}$ and establish that their singular cusp locus sweeps out the complex evolutoid $E^\theta$. Using a square-root-free formulation, we classify the local behavior of $E^\theta$, proving that evolutoids remain regular and tangent to the curve at isotropic points ($q=0$) while escaping to infinity at complex inflections ($\kappa=0$). Finally, we analyze the three-dimensional complex discriminant surface $\mathcal{D}_F \subset \mathbb{C}^3$ formed by the evolving family $E^\theta$. We prove that $A_2$ cusps are versally unfolded by spatial parameters, whereas $A_3$ swallowtail singularities are holomorphically versally unfolded by $(x_1, x_2, \theta)$ if and only if the complex non-degeneracy condition $\kappa^4 + \kappa_s^2 \neq 0$ holds. Crucially, we demonstrate that simple isotropic points drive the breakdown of this versality condition, inducing degenerate, non-transverse sections of the complex swallowtail surface.

A recent manuscript produced by Claude under the direction of Levent Alpöge constructs a 1-parameter family of compact complex threefolds diffeomorphic to $S^6$. Our goal is to describe the geometric ideas behind the construction, give a self-contained proof, and remark on possible future research directions.

We prove a conjecture proposed by the first-named author in 1998. Let $f\colon\mathbb{C}\to\mathbb{P}^n$ be a transcendental linearly non-degenerate holomorphic curve of finite lower order. If the counting function $N_1(r,f)$ of its Wronskian zeros satisfies $N_1(r,f)=o(T(r,f))$, then its order and lower order coincide and belong to $\{1+k/q:k\in\mathbb{Z}_{\geq 0},\ 2\le q\le n+1\}$, and its characteristic is regularly varying. Every order in this set occurs. We also prove the sharp inequality $\limsup_{r\to\infty}N_1(r,f)/T(r,f)\geq 1$ for transcendental linearly non-degenerate curves of order zero.

We construct an automorphism of a smooth projective K3 surface with positive topological entropy and a non-empty Fatou set. The Fatou set contains invariant domains biholomorphic to products of two annuli. On each domain, the automorphism is holomorphically conjugate to a fixed-point-free irrational rotation. These domains are two-dimensional analogues of Herman rings. The construction answers a question posed by McMullen in 2002 and several questions posed by Cantat in his 2018 ICM address.

We compute the Dolbeault cohomology of minimally stable polytopal LVM manifolds using a finite curvature complex. Let $B$ be the indispensable weight block. A canonical exact sequence identifies the kernel of the curvature map with $(\operatorname{coker}B)^\vee$ and its cokernel with $(\ker B)^\vee$. At full rank, the Dolbeault groups are direct sums of reduced cohomology groups of induced subcomplexes of the visible simplicial sphere, tensored with the exterior algebra on the complex dual of the deck lattice. One proof uses the curvature model and Stanley–Reisner Tor; a second gives an additive comparison through completed Laurent expansions and holomorphic descent. Full curvature rank is equivalent to Frolicher degeneration at $E_1$. Every manifold in this class fails the $\partial\bar\partial$ lemma. For polygons, the number of facets and the curvature rank determine the entire Hodge diamond. We construct a proper holomorphic family over a disk with fixed visible square whose curvature rank drops at the origin. The total dimension of Dolbeault cohomology increases there by sixteen. At full rank, holomorphic descent also computes tangent-sheaf cohomology, including resonant contributions in positive Cech degree. We determine the Kodaira–Spencer map of the normalized weight family and give a cohomological criterion for semiuniversality with smooth base. For a resonant example over the square, the global vector fields have a polynomial basis of nine elements and $h^1(\Theta)=19$. The primary obstruction map is nonzero. The additional Cech class integrates in a holomorphic family.

Let $p:X\rightarrow\Delta$ be a proper holomorphic submersion from a Kähler manifold $X$. We prove that, if $K_{X_t}$ is nef for every $t\in\Delta$, then $\dim_{\mathbb C}H^0(X_t,mK_{X_t})$ is independent of $t$ for every integer $m\geq1$. This gives a partial answer to Siu's conjecture on the invariance of plurigenera for Kähler families.

At the critical density, the classical density theory of the Bargmann–Fock space no longer determines exact zero sets. We show that this failure is substantial: the complete radial counting function together with any prescribed finite collection of angular moments does not determine exact realizability, and the finite-dimensional relation between the dimension of the vanishing space and the zero-insertion capacity breaks down in infinite dimension. For arbitrary locally finite multisets, we establish a variational characterization of exact zero sets in terms of a renormalized logarithmic potential and relative entropy. This characterization yields sharp criteria for several classes of critical configurations.

We determine the cotangent operator space structure at the origin of the tetrablock (denoted by $\mathbb E$). In particular we prove that $\operatorname{COT}_0(\overline{\mathbb E}) =\operatorname{MIN}(\ell_1^2\oplus_\infty\mathbb C)$ completely isometrically. This shows the existence of commuting tuple of operators having $\overline{\mathbb E}$ as a spectral set but not as a complete spectral set.

We prove that every codimension-one holomorphic Anosov diffeomorphism of a compact connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. This verifies a conjecture of Ghys in the codimension-one case and, in particular, in complex dimension three.

For orientation-preserving planar harmonic mappings, postcomposition by real-affine mappings induces automorphisms on the space of complex dilatations. Motivated by this fact and the lack of natural affine-invariant geometric quantities in classical distortion theory, we introduce the affine circumradius and affine diameter associated with the image of the complex dilatation. The exponentiated affine circumradius gives a characterization of minimal quasiconformal distortion under affine normalization. Optimal affine balancing is unique up to similarity and yields canonical harmonic mappings with centrally symmetric dilatation. Using a three-point support principle and a sharp hyperbolic Jung theorem, we establish universal sharp two-sided bounds for these invariants and provide their pseudohyperbolic reformulations. For canonically balanced mappings, we prove sharp second-order estimates for pre-Schwarzian derivatives, showing that affine normalization cancels leading-order conformal discrepancies. We develop an affine-stable hierarchical classification for families of harmonic quasiconformal mappings, including normality and an invariant ordering. This framework unifies extremal distortion problems and connects classical analytic function theory with the distortion theory of harmonic quasiconformal mappings.

Rempe and Rippon gave a general result which enables the construction of a transcendental entire function with a Baker domain, in which the iterates tend to infinity at a prescribed arbitrarily slow rate. Here we construct a transcendental entire function with a Baker domain, in which the iterates are controlled from both above and below in a prescribed arbitrarily slow but regular manner. The proof uses distortion theorems for conformal mappings due to Ahlfors and Warschawski, and an approximation theorem due to Rempe and Rippon.