arXiv subjects

David Kalaj

Publications and source records attributed to David Kalaj.

At least 19 recordsLinked to original sources

arXiv2026

Sharp mean-width and Jacobian bounds for Euclidean and hyperbolic harmonic maps

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 $k:[0,\pi]\to\mathbb R$ be continuous and nonincreasing, and let $T_k$ be the associated zonal integral operator, \[ (T_kF)(\xi)=\int_{\mathbb S^{n-1}} k\!\left(\arccos\langle\xi,\eta\rangle\right) F(\eta)\,d\sigma(\eta), \] where $\sigma$ is normalized surface measure. For every measurable $F:\mathbb S^{n-1}\to\overline{\mathbb B^n}$, we prove \[ w\!\left(\operatorname{co}T_kF(\mathbb S^{n-1})\right) \le2\lambda_1(k), \] where $w$ denotes mean width, normalized by $w(\overline{\mathbb B^n})=2$, and $\lambda_1(k)$ is the eigenvalue of $T_k$ on the space of spherical harmonics of degree one. For strictly decreasing kernels, equality holds exactly for $F(\eta)=Q\eta$ almost everywhere, with $Q\in O(n)$. For the ordinary Poisson kernel, the multiplier is $r$, yielding sharp mean-width, intrinsic-volume, and image-volume contraction. In particular, $|f(r\mathbb B^n)|\le\omega_n r^n$ without injectivity assumptions, answering the area and higher-dimensional volume question of Koh and Kovalev.

math.CV↗

arXiv2026

A Lewy-type theorem for pluriharmonic mappings in $\C^2$ and bi-Lipschitz quasiconformal harmonic maps

Lewy's theorem says that a one-to-one harmonic mapping between plane domains has nonvanishing Jacobian. In dimensions at least three this statement is false for general harmonic homeomorphisms. We prove a four-dimensional Lewy theorem under the additional complex-analytic assumption of pluriharmonicity. More precisely, if \[ F:\Omega\subset \mathbb C^2\to \mathbb C^2 \] is a $C^2$ pluriharmonic mapping which is one-to-one in a neighborhood of a point $p$, then the real Jacobian of $F$ is nonzero at $p$. The proof is local. If the Jacobian vanished, a nontrivial real linear projection of $F$ would be the real part of a holomorphic function $f$ with $df(p)=0$. The level hypersurface of this projection would therefore be a real analytic germ of the form \[ \{\operatorname{Re} f=0\}. \] We prove that such a germ cannot be locally flat at a critical point of $f$. The obstruction is topological: local flatness forces the local homology of the hypersurface germ to agree with that of a real hyperplane, and hence forces every sufficiently small admissible link to have the integral homology, in particular the Euler characteristic, of $S^2$. In the reduced case the Milnor open book of the plane curve singularity $f^{-1}(0)$ gives a contradictory Euler-characteristic formula, while in the nonreduced case the local normal form produces more than two local complementary components. We then prove a separate bi-Lipschitz criterion for harmonic quasiconformal mappings: a harmonic quasiconformal homeomorphism from the unit ball onto a bounded $C^1$-Dini domain is bi-Lipschitz, provided it is already a local $C^1$-diffeomorphism.

math.CV↗

arXiv2026

The Gromov--Ros conjecture for rank-one symmetric spaces of noncompact type

We prove the Gromov--Ros conjecture for every rank-one symmetric space of noncompact type: finite-perimeter isoperimetric regions are precisely the geodesic balls, up to ambient isometry and null sets. The proof is organized uniformly in the root multiplicities $(p,q)$. We introduce the volume radius \[ R(r)^n=n\int_0^r\sinh^{n-1}s\,\cosh^q s\,ds, \] which converts the polar volume form into $R^{n-1}dR\,d\sigma$. Hence radial--angular stretches $R\mapsto e^{t\varphi(\theta)}R$ have the exact Jacobian $e^{nt\varphi}$. Starting from a volume geometric median, a log-partition correction produces an exactly volume-preserving family of global bi-Lipschitz stretches. The reduced-boundary area formula and the ambient cofactor yield a universal pointwise trace identity depending only on $(p,q)$ and the horizontal and vertical components of the measure-theoretic normal. Its specializations for $q=1,3,7$ admit explicit strict sign certificates, forcing the normal of an isoperimetric region to be radial almost everywhere. Isotropy invariance in BV and a one-dimensional weighted endpoint comparison then force a single ball. For complex hyperbolic space we additionally prove that every smooth bounded fixed-volume stable critical domain is a geodesic ball. The resulting complex-hyperbolic isoperimetric theorem removes the geometric hypothesis in several sharp weighted-Bergman contraction, Faber--Krahn, and Lieb--Wehrl inequalities, and their high-weight scaling limit recovers holomorphic Gaussian hypercontractivity.

math.CV↗

arXiv2026

A Lewy theorem for harmonic quasiregular mappings in three-space

Lewy's classical theorem asserts that a one-to-one planar harmonic mapping has nonvanishing Jacobian. We prove a three-dimensional bounded-distortion analogue: if \[ f:\Omega\subset \mathbb R^3\to \mathbb R^3 \] is nonconstant, sense-preserving, quasiregular, and harmonic componentwise, then $J_f>0$ throughout $\Omega$. Thus harmonic quasiconformal mappings between domains in three-space are local harmonic diffeomorphisms. The new point is the Lewy-type differential conclusion $J_f\neq0$, not merely topological local invertibility, which is already known for sufficiently smooth quasiregular mappings. The proof is by blow-up. A hypothetical zero of $J_f$ produces a nonconstant homogeneous harmonic polynomial quasiregular mapping $P:\mathbb R^3\to\mathbb R^3$ of degree $m>1$. We exclude such homogeneous blow-ups by a second-order trace identity for $J_P|_{S^2}$: after normalizing the first jet at a positive minimum, the identity gives a negative spherical trace, contradicting the maximum principle. We also derive an affine Liouville theorem for entire harmonic quasiregular mappings in $\mathbb R^3$.

math.CV↗

arXiv2026

The Nitsche--Hopf conjecture for minimal graphs

We prove the Nitsche--Hopf conjecture for non-parametric minimal graphs over disks. If $S$ is a minimal graph over a disk of radius $R$, and if $\xi$ is the point above the center, then \[ W(\xi)^2 |K(\xi)|<\frac{\pi^2}{2R^2}. \] Here $K$ is the Gaussian curvature and \[ W=\sqrt{1+|\nabla u|^2}=\frac1{n_3} \] is the reciprocal of the vertical component of the upward unit normal. The constant is sharp, as shown by the horizontal tangent-plane extremal sequence of Finn and Osserman. The main difficulty is that the bicentric-quadrilateral comparison theorem for Gaussian curvature controls $|K|$, but it does not by itself control the normalized quantity $W^2|K|$: the slope factor $W$ can be arbitrarily large. We show that the missing information is recovered inside the Scherk-type comparison family from the zero equation for the horizontal harmonic projection. More precisely, in the fixed-arc normalization the point corresponding to the center of the physical disk is a distinguished zero $z_\circ$ of the harmonic projection. The equation $f(z_\circ)=0$, written in harmonic-measure coordinates, reduces the sharp Hopf estimate to a scalar derivative inequality at the admissible zero of a monotone function $G_{A,B}$. We prove this scalar inequality on the full admissible parameter domain by a barrier argument and two explicit Bernstein-polynomial positivity certificates. Combined with the bicentric-quadrilateral comparison theorem of the first author and Melentijevi\'c, the Scherk-family estimate gives the sharp normalized Hopf estimate for arbitrary minimal graphs over disks. As a byproduct, we obtain the two-sided bound \[ \frac{\pi^2}{4}\leq W^2|K|\leq \frac{\pi^2}{2} \] throughout the normalized Scherk-type comparison family, evaluated at the distinguished point corresponding to the center.

math.CV↗

arXiv2026

On an extremal problem for harmonic maps conformal at a point

Let $\mathbb D$ denote the unit disc in $\mathbb C$. For a domain $D\subset\mathbb C$ and a point $p\in D$, let $M_D(p)$ denote the supremum of $\|df_0\|$ over all harmonic maps $f:\mathbb D\to D$ with $f(0)=p$ whose differential $df_0$ at $0\in \mathbb D$ is conformal. If $f:\mathbb D\to D$ is a conformal diffeomorphism onto $D$ with $f(0)=p$, then $\|df_0\|\le M_D(p)$. In a recent paper, the authors proved that equality holds when $D=\mathbb D$, and they asked whether equality can hold only when $D$ is a round disc. We give a negative answer by proving that, among bounded convex pointed domains $p\in D\subset\mathbb C$ and up to translations, rotations, and reflections, equality holds if and only if, after moving $p$ to the origin, $D=F(\mathbb D)$ where $F:\mathbb D\to\mathbb C$ is a holomorphic map with $F(0)=0$ and $F'(z)=\frac{c}{1+az+\lambda z^2}$, where $c>0$, $|\lambda|<1$, and $|a-\bar a\lambda|<1-|\lambda|^2$. This family contains strongly convex examples which are not round discs.

math.CV↗

arXiv2026

A bounded globally univalent quasiconformal harmonic map whose analytic part is unbounded

We construct, for every $0<k<1$, a bounded globally univalent harmonic mapping \[ f=h+\overline g \colon \D\to\C \] such that \[ |g'(z)|\le k|h'(z)|,\qquad z\in\D, \] while the analytic part $h$ is unbounded. The construction is based on a bounded logarithmic spiral map on a far right horizontal strip, together with a smaller logarithmic perturbation.

math.CV↗

arXiv2026

Contraction properties for holomorphic functions via isoperimetric stability on the Bergman ball

We prove a local contraction property for holomorphic functions that are nearly constant, relating weighted Bergman spaces $A^p_\alpha(\B_n)$ and $A^q_\beta(\B_n)$. Our approach converts geometric information on weighted superlevel sets into analytic deficit inequalities and rests crucially on a quantitative stability result (of Fuglede type) for the isoperimetric inequality in the Bergman ball. As an application, along the contractive line $q/p=\beta/\alpha$, we obtain a deficit contraction near the extremizer $f\equiv 1$: if $f=1+\phi$ with $\phi$ small and its weighted level sets are nearly spherical (after recentering), then the $A^q_\beta$-deficit is controlled by the $A^p_\alpha$-deficit, and the same deficit quantitatively controls the deviation of the level sets from spheres.

math.CV↗

arXiv2026

The $L^2$-Norm of the Cauchy transform on circular annuli

We compute the exact $L^2$ operator norm of the Cauchy transform \[ (C_\Omega f)(z)=\frac1\pi\int_\Omega \frac{f(w)}{z-w}\,dA(w) \] on a circular annulus $\Omega=A(r,R)=\{r<|z|<R\}$. Exploiting rotational symmetry and a Fourier mode decomposition, we reduce the problem to a one--dimensional weighted Hardy operator and obtain \[ \|C_{A(r,R)}\|_{L^2\to L^2} = \frac{2}{\sqrt{\mu_1^{ND}(r,R)}}, \] where $\mu_1^{ND}(r,R)$ is the first eigenvalue of the Laplacian on $A(r,R)$ with Neumann condition on the inner boundary and Dirichlet condition on the outer boundary. The extremizers are explicitly described in terms of Bessel functions.

math.CV↗

arXiv2026

The $L^2$ Norm of the Interior Cauchy Transform: Beyond the First Dirichlet Eigenvalue

We study sharp $L^2$ bounds for the interior Cauchy transform $C_D$ on a bounded planar domain $D$ and clarify its connection with the Dirichlet spectrum. We analyze an approach that replaces fractional Dirichlet powers on $D$ by Euclidean Fourier multipliers after extension by zero, and show that this substitution can change the optimal constants. In particular, we construct an explicit endpoint counterexample on the unit disk to a Fourier-weighted inequality appearing in \cite{Dostanic1996}. This identifies a gap in the derivation of the conjectured identity $\|C_D\|_{L^2\to L^2}=2/\sqrt{\lambda_1(D)}$. We then identify the correct sharp constant for the endpoint Fourier weight $|\xi|^{-1}$ in terms of the top eigenvalue of a natural positive potential-type operator on $D$. Finally, we show that testing $C_D$ on the first Dirichlet eigenfunction already exceeds the spectral threshold, so $\|C_D\|_{L^2\to L^2}>2/\sqrt{\lambda_1(D)}$ for simply connected domains and also for annuli $A(r,R)$, and we prove a rigidity result: equality with the spectral value occurs if and only if $D$ is a disk.

math.CV↗

arXiv2026

Lipschitz regularity of harmonic quasiconformal maps between Lyapunov domains in $\mathbb{R}^n$

We prove that every sense-preserving harmonic $K$--quasiconformal homeomorphism $f\colon D\to\Omega$ between Lyapunov domains (equivalently, bounded $C^{1,\alpha}$ domains) in $\mathbb{R}^n$, $\alpha\in(0,1]$, is globally Lipschitz on $\overline D$. The argument is based on a boundary iteration scheme: an initial H\"older modulus for the boundary trace (coming from quasiconformality) is improved via the $C^{1,\alpha}$ graph representation of $\partial\Omega$, yielding higher H\"older regularity for the normal component. This boundary gain is converted into a near-boundary gradient bound for harmonic functions through a basepoint boundary H\"older-to-gradient estimate obtained by flattening the boundary and using local harmonic-measure bounds. Quasiconformality then propagates the resulting control from one component to the full differential, and iteration gives boundedness of $|Df|$ up to the boundary. Along the way we briefly survey several standard tools from the theory of quasiconformal harmonic mappings (QCH), including boundary H\"older continuity, distortion of derivatives, and boundary-to-interior propagation principles that enter the iteration.

math.AP↗

arXiv2025

Minimal surfaces over the Pitot quadrilaterals

We develop a fully explicit framework for constructing Scherk-type minimal graphs over the Pitot quadrilaterals (i.e. such that the two pairs of opposite sides have the same total length). For any Pitot quadrilateral $Q$, we first produce a harmonic diffeomorphism of the unit disk onto $Q$, whose dilatation is the square of a M\"obius automorphism determined directly by the vertices of $Q$. Using this map as the Weierstrass data, we obtain a minimal graph $\Sigma$ whose Gauss map is a univalent M\"obius transformation and whose height function exhibits alternating blow-up behavior along opposite sides of $Q$, mirroring the classical Scherk surfaces. We further construct an associated canonical surface $\Sigma^\diamond$, with the same boundary asymptotics, and prove a sharp curvature comparison theorem: at the harmonic center of $Q$, among all bounded minimal graphs with matching normal direction and mixed derivative, $\Sigma^\diamond$ uniquely maximizes the absolute Gaussian curvature. This provides a complete and constructive description of Scherk-type minimal graphs over all, both convex or concave, Pitot quadrilaterals.

math.DG↗

arXiv2025

On generalized M. Riesz conjugate function theorem for harmonic mappings

Let $L^p(\mathbf{T})$ be the Lesbegue space of complex-valued functions defined in the unit circle $\mathbf{T}=\{z: |z|=1\}\subseteq \mathbb{C}$. In this paper, we address the problem of finding the best constant in the inequality of the form: $$\|(|P_+ f|^2+c| P_{-} f|^2)^{1/2}\|_{L^p(\mathbf{T})}\le A_{p,c} \|f\|_{L^p(\mathbf{T})}.$$ Here $2\le p<\infty$, $c>0$, and by $P_{-} f$ and $ P_+ f$ are denoted co-analytic and analytic projection of a function $f\in L^p(\mathbf{T})$. The sharpness of the constant $A_{p,c}$ follows by taking a family quasiconformal harmonic mapping $f_\gamma$ and letting $\gamma\to 1/p$. The result extends a sharp version of M. Riesz conjugate function theorem of Pichorides and Verbitsky and some well-known estimates for holomorphic functions.

math.CV↗

arXiv2025

Conjugate type properties of harmonic $(K,K')$-quasiregular mappings

The main purpose of this paper is to investigate conjugate type properties for harmonic $(K,K')$-quasiregular mappings, where $K \geq 1$ and $K' \geq 0$ are constants. We first establish a Riesz type conjugate function theorem for such mappings, which generalizes and refines several existing results. Additionally, we derive an asymptotically sharp constant for a Riesz type theorem pertaining to a specific class of $K$-quasiregular mappings. Furthermore, we obtain Kolmogorov type and Zygmund type theorems for harmonic $(K,K')$-quasiregular mappings.

math.CV↗

arXiv2025

The Schwarz lemma for holomorphic and minimal disks at the boundary

We first prove a Boundary Schwarz lemma for holomorphic disks on the unit ball in $\mathbb{C}^n$. Further by using a Schwarz lemma for minimal conformal disks of Forstneri\v c and Kalaj (F.~Forstneri{\v{c}} and D.~Kalaj. \newblock Schwarz-pick lemma for harmonic maps which are conformal at a point. \newblock {\em Anal. PDE}, 17(3):981--1003, 2024.) we prove the boundary Schwarz lemma for such minimal disks.

math.CV↗

arXiv2025

On a sharp form of curvature conjecture for minimal graphs

Recently, the author and Melentijevi\'c resolved the longstanding Gaussian curvature problem by proving the sharp inequality \[ |\mathcal{K}| < c_0 = \frac{\pi^2}{2} \] for minimal graphs over the unit disk, evaluated at the point of the graph lying directly above the origin. The constant $ c_0 $ is known as the \emph{Heinz constant}. Building on this result, we obtain an improved estimate for the Hopf constant $ c_1 $. In addition, we show that for any prescribed unit normal vector $ \mathbf{n} $, there exists a minimal graph over the unit disk -- bending in the coordinate directions -- whose Gaussian curvature at the point above the origin is strictly smaller, yet arbitrarily close to, the curvature of the associated Scherk-type surface with the same normal, situated above a bicentric quadrilateral. This sharp inequality strengthens the classical result of Finn and Osserman, which applies in the special case when the unit normal is $ (0,0,1) $.

math.DG↗

arXiv2025

Isoperimetric inequality for nearly spherical domains in the Bergman ball

We prove a quantitative isoperimetric inequality for the nearly spherical subset of the Bergman ball in $\mathbb{C}^n$. We prove the Fuglede theorem for such sets. This result is a counterpart of a similar result obtained for the hyperbolic unit ball and it makes the first result on the isoperimetric phenomenon in the Bergman ball.

math.CV↗

arXiv2025

Zygmund theorem for harmonic quasiregular mappings

Let $K\ge 1$. We prove Zygmund theorem for $K-$quasiregular harmonic mappings in the unit disk $\mathbb{D}$ in the complex plane by providing a constant $C(K)$ in the inequality $$\|f\|_{1}\le C(K)(1+\|\mathrm{Re}\,(f)\log^+ |\mathrm{Re}\, f|\|_1),$$ provided that $\mathrm{Im}\,f(0)=0$. Moreover for a quasiregular harmonic mapping $f=(f_1,\dots, f_n)$ defined in the unit ball $\mathbb{B}\subset \mathbb{R}^n$, we prove the asymptotically sharp inequality $$\|f\|_{1}-|f(0)|\le (n-1)K^2(\|f_1\log f_1\|_1- f_1(0)\log f_1(0)),$$ when $K\to 1$, provided that $f_1$ is positive.

math.CV↗