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Sławomir Dinew

Publications and source records attributed to Sławomir Dinew.

17 recordsLinked to original sources
arXiv2026

Hessian operators, null Lagrangians and the integral comparison principle

We investigate the conditions under which a Hessian operator satisfies the integral comparison principle. We establish a rather unexpected connection with the notion of a null Lagrangian from the calculus of variations. Under mild conditions we obtain a complete classification, showing that the only operators for which the integral comparison principle holds are linear combinations of $k$-Hessians and, in the homogeneous case, precisely the $k$-Hessians themselves. An analogous result is established for the complex Hessian operator, revealing in particular substantial obstructions to the development of a genuine pluripotential theory for the $\mathcal{J}$-equation.

math.AP↗
arXiv2026

Very weak solutions of quadratic Hessian equations

We extend the methods of Lewicka - Pakzad, Székelyhidi - Cao and Li - Qiu to study the notion of very weak solutions to the complex $σ_2$ equation in domains in $\mathbb C^n,\ n\geq 2$. As a by-product we sharpen the regularity threshold of the counterexamples obtained by Li and Qiu in the real case.

math.AP↗
arXiv2025

$m$-Pseudo-effectivity and a Monge-Ampère-Type Equation for Forms of Positive Degree

Given an $n$-dimensional compact Kähler manifold, we continue our study of $m$-positivity in two ways. We first propose generalisations of the notions of pseudo-effective and big Bott-Chern cohomology classes of bidegree $(1,\,1)$ by relaxing the standard positivity hypotheses to their $m$-counterparts after we have proved a Lamari-type duality lemma in bidegree $(m,\,m)$. Independently, we propose a Monge-Ampère-type non-linear pde whose distinctive feature is that its solutions, if any, are forms of positive degree rather than functions. We prove a form of uniqueness for the solutions and, under the assumption that a solution exists, we give a geometric application involving the $m$-bigness notion introduced in the first part.

math.DG↗
arXiv2025

$m$-Positivity and Regularisation

Starting from the notion of $m$-plurisubharmonic function introduced recently by Dieu and studied, in particular, by Harvey and Lawson, we consider $m$-(semi-)positive $(1,\,1)$-currents and Hermitian holomorphic line bundles on complex Hermitian manifolds and prove two kinds of results: vanishing theorems and $L^2$-estimates for the $\bar\partial$-equation in the context of $C^\infty$ $m$-positive Hermitian fibre metrics; global and local regularisation theorems for $m$-semi-positive $(1,\,1)$-currents whose proofs involve the use of viscosity subsolutions for a certain Monge-Ampère-type equation and the associated Dirichlet problem.

math.DG↗
arXiv2023

Liouville theorem for a class of Hessian equations

In this paper, we study a general class of Hessian elliptic equations, including the Monge-Ampère equation, the $k$-Hessian equation and $p$-Monge-Ampère equations. We propose new additional condition on the solution and prove Liouville theorem under this assumption. We show that our general condition covers as special cases numerous sets of assumptions known in the literature which were tailored for specific equations. Thus we obtain a significant generalization of multiple isolated Liouville theorems and conditional interior estimates.

math.AP↗
arXiv2023

On the Alesker-Verbitsky conjecture on hyperKähler manifolds

We solve the quaternionic Monge-Ampère equation on hyperKähler manifolds. In this way we prove the ansatz for the conjecture raised by Alesker and Verbitsky claiming that this equation should be solvable on any hyperKähler with torsion manifold, at least when the canonical bundle is trivial holomorphically. The novelty in our approach is that we do not assume any flatness of the underlying hypercomplex structure which was the case in all the approaches for the higher order a priori estimates so far. The resulting Calabi-Yau type theorem for HKT metrics is discussed.

math.DG↗
arXiv2023

A remark on Oka's lemma and a geometric property of pseudoconvex domains

A direct proof of Oka's lemma on the relation of holomorphic convexity and the properties of the distance to the boundary function is provided. Some related problems for subharmonicity properties of this function are also studied. A new geometric property of pseudoconvex domains is described.

math.CV↗
arXiv2023

A Variational Approach to SKT and Balanced Metrics

We investigate compact complex manifolds endowed with SKT or balanced metrics. In each case we define a new functional whose critical points are proved to be precisely the Kähler metrics, if any, on the manifold. As general manifolds of either type need not admit Kähler metrics, this provides an approach to new obstructions to Kählerianity within these two families of metrics.

math.DG↗
arXiv2021

A Generalised Volume Invariant for Aeppli Cohomology Classes of Hermitian-Symplectic Metrics

We investigate the class of compact complex Hermitian-symplectic manifolds $X$. For each Hermitian-symplectic metric $ω$ on $X$, we introduce a functional acting on the metrics in the Aeppli cohomology class of $ω$ and prove that its critical points (if any) must be Kähler when $X$ is $3$-dimensional. We go on to exhibit these critical points as maximisers of the volume of the metric in its Aeppli class and propose a Monge-Ampère-type equation to study their existence. Our functional is further utilised to define a numerical invariant for any Aeppli cohomology class of Hermitian-symplectic metrics that generalises the volume of a Kähler class. We obtain two cohomological interpretations of this invariant. Meanwhile, we construct an invariant in the form of an $E_2$-cohomology class, that we call the $E_2$-torsion class, associated with every Aeppli class of Hermitian-symplectic metrics and show that its vanishing is a necessary condition for the existence of a Kähler metric in the given Hermitian-symplectic Aeppli class.

math.DG↗
arXiv2021

Radó- type theorem for subharmonic and plurisubharmonic functions

We observe that a recent result by Gardiner and Sjödin, solving a problem of Král on subharmonic functions, can be easily generalized to yield a somewhat stronger result. This can be combined with a viscosity technique of ours, which we slightly improve, to obtain Radó- type theorems for plurisubharmonic functions. Finally, we study the Borel complexity of the critical set and the set where the gradient does not exist finitely of subharmonic functions and general real valued functions.

math.CV↗
arXiv2020

Uniform estimates for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Ampère equation

We prove sharp uniform estimates for strong supersolutions of a large class of fully nonlinear degenerate elliptic complex equations. Our findings rely on ideas of Kuo and Trudinger who dealt with degenerate linear equations in the real setting. We also exploit the pluripotential theory for the complex Monge-Ampère operator as well as suitably tailored theory of $L^p$-viscosity subsolutions.

math.AP↗
arXiv2016

Differential tests for plurisubharmonic functions and Koch curves

We study minimum sets of singular plurisubharmonic functions and their relation to upper contact sets. In particular we develop an algorithm checking when a naturally parametrized curve is such a minimum set. The case of Koch curves is studied in detail. We also study the size of the set of upper non-contact points. We show that this set is always of Lebesgue measure zero thus answering an open problem in the viscosity approach to the complex Monge-Ampère equation. Finally, we prove that similarly to the case of convex functions, strictly plurisubharmonic lower tests yield existence of upper tests with a control on the opening.

math.CV↗
arXiv2014

Mixed Hessian inequalities and uniqueness in the class $\mathcal{E}(X,ω,m)$

We prove a general inequality for mixed Hessian measures by global arguments. Our method also yields a simplification for the case of complex Monge-Ampère equation. Exploiting this and using Kołodziej's mass concentration technique we also prove the uniqueness of the solutions to the complex Hessian equation on compact Kähler manifolds in the case of probability measures vanishing on $m$-polar sets.

math.CV↗
arXiv2008

Uniqueness and Stability in $\mathcal E(X,ω)$

We prove uniqueness for the Dirichlet problem for the complex Monge-Ampère equation on compact Kähler manifolds in the case of measures vanishing on pluripolar sets. As a by-product we generalize Xing's stability theorem.

math.CV↗