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Zhenan Sui

Publications and source records attributed to Zhenan Sui.

12 recordsLinked to original sources
arXiv2026

Interior Hessian estimates for Hessian quotient equations in dimension three

In this paper, we establish the interior Hessian estimates for $2$-convex solutions to $\frac{\sigma_2}{\sigma_1} (D^2 u) = \psi (x,u)$ in dimension three. In higher dimensions ($n \geq 4$), we prove the interior Hessian estimates for semi-convex solutions. We provide a new method to prove the doubling inequality for smooth solutions in dimensions three and four. In higher dimensions ($n\geq 5$) the doubling inequality is proved under an additional dynamic semi-convexity condition which is the same to that in \cite{SY2025}. The method also applies to the equation $\sigma_2 (D^2 u) = \psi (x, u, \nabla u)$.

math.AP↗
arXiv2025

Asymptotic Plateau problem for $3$-convex hypersurface in $\mathbb{H}^5$

We prove the existence of a smooth complete $3$-convex hypersurface which satisfies prescribed curvature equation $\prod\limits_{i = 1}^n (H - \kappa_i) = \big( (n - 1) \sigma \big)^n$ for $n = 4$ and has prescribed asymptotic boundary $\Gamma$ at the infinity of hyperbolic space of dimension 5, where $\sigma \in (0, 1)$ is a constant and $\Gamma$ is assumed to have nonnegative mean curvature. We introduce Lagrange multiplier method to compute the extreme value of the concavity of $f (\kappa) = \frac{1}{n - 1} \Big( \prod\limits_{i = 1}^n (H - \kappa_i) \Big)^{\frac{1}{n}}$ during uniform global curvature estimate.

math.DG↗
arXiv2024

On Fully Nonlinear Loewner-Nirenberg Problem of Ricci curvature

We prove the existence of a smooth complete conformal metric with prescribed kth elementary symmetric function of negative Ricci curvature under certain condition on general domain in Euclidean space. We then formulate this problem for more general equations.

math.DG↗
arXiv2024

Existence of solution to modified Gursky-Streets equation

We solve the modified Gursky-Streets equation, which is a fully nonlinear equation arising in conformal geometry with uniform $C^{1, 1}$ estimates when (i) $\gamma > 0$ and $1 \leq k \leq n$ or (ii) $r > 0$ and $2 s k \leq r n$. We also prove the existence of a Lipschitz continuous viscosity solution when $r \neq 0$.

math.AP↗
arXiv2022

Smooth Solutions to Asymptotic Plateau Type Problem in Hyperbolic Space

We investigate on the existence of smooth complete hypersurface with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under the assumption that there exists an asymptotic subsolution. We give an affirmative answer for the case $k = n$ when the asymptotic boundary $Γ$ bounds a uniformly convex domain, and for $k < n$ when $Γ$ bounds a disk, utilizing Pogorelov type interior second order estimate. Our result complements our previous work \cite{Sui2019, Sui-Sun}, and generalizes the asymptotic Plateau type problem to non-constant prescribed curvature case.

math.DG↗
arXiv2019

Strictly Locally Convex Hypersurfaces with Prescribed Curvature and Boundary in Space Forms

This paper is devoted to $C^2$ a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature equation under the assumption of a strictly locally convex subsolution.

math.AP↗
arXiv2019

Strictly Locally Convex Radial Graphs of Prescribed Curvature and Boundary in Space Forms

We obtain $C^2$ a priori estimates for solutions of the nonlinear second-order elliptic equation related to the geometric problem of finding a strictly locally convex hypersurface with prescribed curvature and boundary in a space form. Under the assumption of a strictly locally convex subsolution, we establish existence results by using degree theory arguments.

math.DG↗