arXiv subjects

Yulun Xu

Publications and source records attributed to Yulun Xu.

12 recordsLinked to original sources

arXiv2026

Regularization and H\"older continuity for complex Hessian equations on Hermitian manifolds

Let $(M,\omega)$ be a compact Hermitian manifold, and let $\Gamma$ be a symmetric convex cone. We develop a quantitative regularization method for $\Gamma$-admissible functions. As an application, we prove H\"older continuity for every pluripotential solution of complex $m$-Hessian equations whose right-hand sides belong to $L^p$, for $p>\frac{n}{m}$. These results extend to a more general class of complex Hessian equations satisfying the structural condition used by Guo--Phong--Tong.

math.AP↗

arXiv2026

A numerical criterion for complex Hessian type equations on projective manifolds

We prove a Nakai-Moishezon-type criterion for complex Hessian-type equations on projective manifolds whose associated degree-$n$ polynomials are strongly strictly right-Noetherian. For strictly right-Noetherian polynomials of arbitrary degree, we prove a uniform Nakai-Moishezon-type criterion. This class includes the complex Hessian and Hessian quotient equations.

math.DG↗

arXiv2026

Poincar\'e type J-equation

We introduce a two-parameter continuity path for the J-equation and use it to characterize the solvability of the J-equation for K\"ahler metrics with Poincar\'e type singularities along a divisor $D$, allowing simple normal crossings and self-intersections. On K\"ahler surfaces, we show that the classical subsolution condition in the smooth setting implies solvability in the Poincar\'e type setting for any smooth divisor $D$. As a consequence, if $X$ contains no curves of negative self-intersections and $K_X[D]$ is ample, then the K-energy is bounded from below on any Poincar\'e type K\"ahler class. In the smooth divisor case, we further analyze the asymptotic behavior of solutions near $D$, and show that existence of a Poincar\'e type solution implies existence of a solution to the J-equation on $D$.

math.DG↗

arXiv2025

The uniqueness of Poincar\'e type extremal K\"ahler metric

Let $D$ be a smooth divisor on a closed K\"ahler manifold $X$. Suppose that $Aut_0(D)=\{Id\}$. We prove that the Poincar\'e type extremal K\"ahler metric with a cusp singularity at $D$ is unique up to a holomorphic transformation on $X$ that preserves $D$. This generalizes Berman-Berndtson's work on the uniqueness of extremal K\"ahler metrics from closed manifolds to some complete and noncompact manifolds.

math.DG↗

arXiv2025

Investigation on the Spreading Behaviour of Sand Powder Used in Binder Jet 3D Printing

The spreading behaviour of cohesive sand powder is modelled by Discrete Element Method, and the spreadability and the mechanical jamming are focused. The empty patches and total particle volume of the spread layer are examined, followed by the analysis of the geometry force and jamming structure. The results show that several empty patches with different size and shapes could be observed within the spread layer along the spreading direction even when the gap height increases to 3.0D90. Large particles are more difficult to be spread onto the base due to jamming, although their size is smaller than the gap height. Size segregation of particles occurs before particles entering the gap between the blade and base. There are almost no particles on the smooth base when the gap height is small, due to the full-slip flow of particles. The difference of the spread layer and spreadability between the cases with rough and smooth base is reduced by the increase of the gap height. An interesting correlation between jamming effect and local defects (empty spaces) in the powder layer is identified. The resistance to particle rolling is important for the mechanical jamming reported in this work. The jammed particles with a larger size ratio tend to be more stable.

physics.flu-dyn↗

arXiv2025

Viscosity solution to complex Hessian quotient equations

In this paper, we prove the existence of viscosity solutions to complex Hessian equations on compact Hermitian manifolds, assuming the existence of a strict subsolution in the viscosity sense. The results cover the complex Hessian quotient equations. This generalized our previous results where the equation needs to satisfy a determinant domination condition.

math.AP↗

arXiv2024

The uniqueness of Poincar\'e type constant scalar curvature K\"ahler metric

Let $D$ be a smooth divisor on a closed K\"ahler manifold $X$. First, we prove that Poincar\'e type constant scalar curvature K\"ahler (cscK) metric with a singularity at $D$ is unique up to a holomorphic transformation on $X$ that preserves $D$, if there are no nontrivial holomorphic vector fields on $D$. For the general case, we propose a conjecture relating the uniqueness of Poincar\'e type cscK metric to its asymptotic behavior near $D$. We give an affirmative answer to this conjecture for those Poincar\'e type cscK metrics whose asymptotic behavior is invariant under any holomorphic transformation of $X$ that preserve $D$. We also show that this conjecture can be reduced to a fixed point problem.

math.DG↗

arXiv2024

Viscosity solution to complex Hessian equations on compact Hermitian manifolds

We prove the existence of viscosity solutions to complex Hessian equations on a compact Hermitian manifold that satisfy a determinant domination condition. This viscosity solution is shown to be unique when the right hand is strictly monotone increasing in terms of the solution. When the right hand side does not depend on the solution, we reduces it to the strict monotonicity of the solvability constant.

math.AP↗

arXiv2023

Interior Hölder estimate for the linearized complex Monge-Ampere equation

Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$. Suppose that we have a function $ϕ$ in this neighborhood with $1-ε\le MA(u)\le 1+ε$ and there exists a function $u$ solving the linearized complex Monge-Ampere equation: $det(ϕ_{k\bar{l}})ϕ^{I\bar{j}}u_{I\bar{j}}=0$. Then one has an estimate on $|u|_{C^α(B_{\frac{1}{2}})}$ for some $α>0$ depending on $n$, as long as $ε$ is small depending on $n$. This partially generalizes Caffarelli's estimate for linearized real Monge-Ampere equation to the complex version.

math.CV↗

arXiv2023

Interior $W^{2,p}$ estimate for small perturbations to the complex Monge-Ampere equation

Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$ with the following property: for any continuous, plurisubharmonic function $u$ in this neighborhood solving $1-\eps \le MA(u)\le 1+\eps$, one has $u\in W^{2,p}(B_{\frac{1}{2}})$, as long as $\eps>0$ is small enough depending only on $n$ and $p$. This partially generalizes Caffarelli's interior $W^{2,p}$ estimates for real Monge-Ampere to the complex version.

math.AP↗

arXiv2022

Regularization Of $m$-subharmonic Functions And HÖlder Continuity

We use sup-convolution to find upper approximations of a bounded $m$-subharmonic function on a compact Kähler manifold with nonnegative holomorphic bisectional curvature. As an application, we show the Hölder continuity of solutions to $σ_m$ equation when the right hand side is in $L^p$, $p>\frac{n}{m}$. All these results generalize to more general complex Hessian equations.

math.AP↗

arXiv2019

The k-Power Domination Number in Some Self-Similar Graphs

The $k$-power domination problem is a problem in graph theory, which has applications in many areas. However, it is hard to calculate the exact $k$-power domination number since determining k-power domination number of a generic graph is a NP-complete problem. We determine the exact $k$-power domination number in two graphs which have the same number of vertices and edges: pseudofractal scale-free web and Sierpiński gasket. The $k$-power domination number becomes 1 for $k\ge2$ in the Sierpiński gasket, while the $k$-power domination number increases at an exponential rate with regard to the number of vertices in the pseudofractal scale-free web. The scale-free property may account for the difference in the behavior of two graphs.

math.CO↗