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Slawomir Dinew

Publications and source records attributed to Slawomir Dinew.

At least 19 recordsLinked to original sources
arXiv2026

Three Bernstein type theorems for hypersurfaces with zero Gaussian curvature

In this paper, we prove Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in both Euclidean and Minkowski context. A supplementary example illustrates that zero Gaussian convex spacelike hypersurfaces are not necessary hyperplanes without additional conditions. We show that a zero Gaussian curvature convex hypersurface must be a hyperplane if the mean curvature goes to zero at infinity. In the Minkowski context, we prove similar results for hypersurface without timelike points.

math.DG↗
arXiv2024

Regularity of geodesics in the spaces of convex and plurisubharmonic functions II

In this note we continue our investigation of geodesics in the space of convex and plurisubharmonic functions. We show optimal regularity for geodesics joining two smooth strictly convex functions. We also investigate the regularity theory in the $C^{1,α}$ realm. Finally we discuss the regularity of geodesics joining two toric strictly plurisubharmonic functions.

math.CV↗
arXiv2023

Interior estimates for $p$-plurisubharmonic functions

We study a Monge-Ampère type equation in the class of $p$-plurisubharmonic functions and establish first and second order interior estimates. As an application of these we show that $p$-plurisubharmonic functions with constant operator and quadratic growth must be quadratic polynomials.

math.AP↗
arXiv2021

On a problem of CHirka

We observe that a slight adjustment of a method of Caffarelli, Li, and Nirenberg yields that plurisubharmonic functions extend across subharmonic singularities as long as the singularities form a closed set of measure zero. This solves a problem posed by Chirka.

math.CV↗
arXiv2020

m-subharmonic and m-plurisubharmonic functions -- on two problems of Sadullaev

We show that the spaces of $A$-$m$-subharmonic and $B$-$m$-subharmonic functions differ in sufficently high dimensions. We also prove that the Monge-Ampère type operator $\mathcal M_m$ associated to the space of $m$-plurisubharmonic functions does not allow an integral comparison principle except in the classical cases $m=1$ and $m=n$. These answer in the negative two problems posed by A. Sadullaev.

math.CV↗
arXiv2019

Regularity of geodesics in the spaces of convex and plurisubharmonic functions

In this note we investigate the regularity of geodesics in the space of convex and plurisubharmonic functions. In the real setting we prove (optimal) local C^{1,1} regularity. We construct examples which prove that the global C^{1,1} regularity fails both in the real and complex case in contrast to the Kähler manifold setting. Finally we show a necessary and sufficient conditions for existence of a smooth geodesic between two smooth strictly convex functions.

math.CV↗
arXiv2018

Regularity of Degenerate Hessian Equation

We show a second order a priori estimate for solutions to the complex $k$-Hessian equation on a compact Kähler manifold provided the $(k$-$1)$-st root of the right hand side is $\mathcal C^{1,1}$. This improves an estimate of Hou-Ma-Wu. An example is provided to show that the exponent is sharp.

math.AP↗
arXiv2017

A viscosity approach to the Dirichlet problem for degenerate complex Hessian type equations

A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in $\C^n$. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by Trudinger. As consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions.

math.CV↗
arXiv2016

The minimum sets and free boundaries of strictly plurisubharmonic functions

We study the minimum sets of plurisubharmonic functions with strictly positive Monge-Ampère densities. We investigate the relationship between their Hausdorff dimension and the regularity of the function. Under suitable assumptions we prove that the minimum set cannot contain analytic subvarieties of large dimension. In the planar case we analyze the influence on the regularity of the right hand side and consider the corresponding free boundary problem with irregular data. We provide sharp examples for the Hausdorff dimension of the minimum set and the related free boundary. We also draw several analogues with the corresponding real results.

math.CV↗
arXiv2015

Open problems in pluripotential theory

We propose a list of open problems in pluripotential theory partially motivated by their applications to complex differential geometry. The list includes both local questions as well as issues related to the compact complex manifold setting.

math.CV↗
arXiv2015

Liouville and Calabi-Yau type theorems for complex Hessian equations

We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex Hessian equation on a compact Kähler manifold. This terminates the program, initiated by Hou, Ma and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right hand side in some $L^p, $ with sharp bound on $p$.

math.CV↗
arXiv2015

Remarks on Mukai threefolds admitting $C^{*}$ action

We investigate geometric invariants of the one parameter family of Mukai threefolds that admit $\mathbb C^{*}$ action. In particular we find the invariant divisors in the anticanonical system, and thus establish a bound on the log canonical thresholds. Furthermore we find an explicit description of such threefolds in terms of the quartic associated to the variety-of-sum-of-powers construction. This yields that any such threefold admits an additional symmetry which anticommutes with the $\mathbb C^{*}$ action, a fact that was previously observed near the Mukai-Umemura threefold by Rollin, Simanca and Tipler. As a consequence the Kähler-Einstein manifolds in the class form an open subset in the standard topology.

math.DG↗
arXiv2011

A priori estimates for the complex Hessian equations

We prove some $L^{\infty}$ a priori estimates as well as existence and stability theorems for the weak solutions of the complex Hessian equations in domains of $C^n$ and on compact Kähler manifolds. We also show optimal $L^p$ integrability for m-subharmonic functions with compact singularities, thus partially confirming a conjecture of Blocki. Finally we obtain a local regularity result for $W^{2,p}$ solutions of the real and complex Hessian equations under suitable regularity assumptions on the right hand side. In the real case the method of this proof improves a result of Urbas.

math.CV↗
arXiv2011

Hölder continuous solutions to Monge-Ampère equations

Let $(X,ω)$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $\MAH(X,ω)$ of the complex Monge-Ampère operator acting on $ω$-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej's result that measures with $L^p$-density belong to $\MAH(X,ω)$ and proving that $\MAH(X,ω)$ has the "$L^p$-property", $p>1$. We also describe accurately the symmetric measures it contains.

math.CV↗
arXiv2010

A local regularity for the complex Monge-Ampère equation

We prove a local regularity (and a corresponding a priori estmate) for plurisubharmonic solutions of the nondegenerate complex Monge-Ampére equation assuming that their $W^{2,p}$-norm is under control for some $p>n(n-1)$. This condition is optimal. We use in particular some methods developed by Trudinger and an $L^q$-estimate for the complex Monge-Ampére equation due to Kołodziej.

math.CV↗