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Ni Xiang

Publications and source records attributed to Ni Xiang.

At least 19 recordsLinked to original sources
arXiv2025

Second order estimates for $χ$-semi convex solutions of Hessian equations on Hermitian manifolds

In this paper, we establish the modified concavity inequality for complex Hessian equations under the semi-convexity assumption inspired by Lu \cite{Lu23} and Zhang \cite{Z24} for real case. Then second order estimates for admissible solutions of complex Hessian equations on compact Hermitian manifolds with both sides of equations depending on gradient terms are obtained by taking advantage of the crucial inequality.

math.AP↗
arXiv2025

The $L_p$-dual Christoffel-Minkowski problem for the case $p\geq q$

In this paper, we consider a class of Hessian equations associated to the $L_p$-dual Christoffel-Minkowski problem for the case $p\geq q$. By combining the tools of constant rank theorem, the a priori estimates and the continuity method, we obtain the existence and uniqueness for strictly spherical convex solutions to the $L_p$-dual Christoffel-Minkowski problem.

math.AP↗
arXiv2024

The oblique boundary value problem for degenerate Hessian quotient type equations

In this paper, we investigate the oblique boundary value problem for degenerate Hessian quotient type equations in a smooth bounded domain. Without imposing any geometric restrictions on the domain, we establish the a priori estimates and derive the existence and uniqueness of admissible $C^{1,1}$ solutions under the condition $f^{\frac{1}{k-l}}\in C^{1,1}(\overline{\Omega}\times\mathbb{R})$.

math.AP↗
arXiv2024

$k$-convex hypersurfaces with prescribed Weingarten curvature in warped product manifolds

In this paper, we consider Weingarten curvature equations for $k$-convex hypersurfaces with $n<2k$ in a warped product manifold $\overline{M}=I\times_λM$. Based on the conjecture proposed by Ren-Wang in \cite{Ren2}, which is valid for $k\geq n-2$, we derive curvature estimates for equation $σ_k(κ)= ψ(V, ν(V))$ through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface $Σ$ satisfying the above equation by the degree theory under some sufficient conditions.

math.AP↗
arXiv2024

Dirichlet problem for degenerate Hessian quotient type curvature equations

In the paper, we prove the existence and uniqueness results of the $C^{1,1}$ regular graphic hypersurface for Dirichlet problem of a class of degenerate Hessian quotient type curvature equations under the condition $ψ^{\frac{1}{k-l}}\in C^{1,1}(\overlineΩ\times\mathbb{R}\times\mathbb{S}^n)$. Specially, we also consider the second order derivative estimates for the corresponding degenerate Hessian type curvature equations under the optimal condition $ψ^{\frac{1}{k-1}}\in C^{1,1}(\overlineΩ\times\mathbb{R}\times\mathbb{S}^n)$.

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arXiv2022

The Dirichlet problem for mixed Hessian equations on Hermitian manifolds

In this paper we study the Dirichlet problem for a class of Hessian type equation with its structure as a combination of elementary symmetric functions on Hermitian manifolds. Under some conditions with the initial data on manifolds and admissible subsolutions, we derive a priori estimates for this complex mixed Hessian equation and solvability of the corresponding Dirichlet problem.

math.AP↗
arXiv2021

A class of Hessian quotient equations in the warped product manifold

In this paper, we consider a class of Hessian quotient equations in the warped product manifold $\overline{M}=I\times_λM$. Under some sufficient conditions, we obtain an existence result for the star-shaped compact hypersurface $Σ$ in $\overline{M}$ using standard degree theory based on a priori estimates for solutions to the Hessian quotient equations.

math.DG↗
arXiv2021

The Dirichlet problem for a class of Hessian quotient equations on Riemannian manifolds

In this paper, we consider the Dirichlet problem for a class of Hessian quotient equations on Riemannian manifolds. Under the assumption of an admissible subsolution, we solve the existence and the uniquness for the Dirichlet problem in a domain without any geometric restrictions on the boundary, based on the a priori estimates for the solutions to the Hessian quotient type equations.

math.AP↗
arXiv2021

The $L_p$ Gauss image problem

In this paper we study the $L_p$ Gauss image problem, which is a generalization of the $L_p$ Aleksandrov problem and the Gauss image problem in convex geometry. We obtain the existence result for the $L_p$ Gauss image problem in two cases (i) $p>0$ or (ii) $p<0$ with the given even measures.

math.AP↗
arXiv2020

Smooth solutions to the Gauss image problem

In this paper we study the the Gauss image problem, which is a generalization of the Aleksandrov problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain the existence of smooth solutions to this problem.

math.AP↗
arXiv2020

Inverse Gauss curvature flows with free boundaries in a cone

We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of round sphere after rescaling.

math.DG↗
arXiv2020

Existence of smooth even solutions to the dual Orlicz-Minkowski problem

In this paper we study the dual Orlicz-Minkowski problem, which is a generalization of the dual Minkowski problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain a new existence result of smooth even solutions to this problem for smooth even measures.

math.AP↗
arXiv2020

A Class of Hessian quotient equations in Euclidean space

In this paper, we consider a Class of Hessian quotient equations in Euclidean space. Under some sufficient condition, we obtain an existence result by the standard degree theory based on the a prior estimates for the solutions to the Hessian quotient equations.

math.AP↗
arXiv2020

The Neumann problem for a class of mixed complex Hessian equations

In this paper, we consider the Neumann problem of a class of mixed complex Hessian equations, and establish the global C^1 estimates a nd reduce the global second derivative estimate to the estimate of double normal second derivatives on the boundary. In particular, we can prove the global C^2 estimates and the existence theorems when k=n.

math.AP↗
arXiv2020

A curvature flow approach to Lp-Christoffel-Minkowski problem for p>1

We study the motion of smooth, closed, strictly convex hypersurfaces in Rn+1 expanding in the direction of their normal vector field with speed depending on the k-th elementary symmetric polynomial of the principal radii of curvature. As an application, we gives a unifed fow approach to Lp-Christoffel-Minkowski problem for p > 1.

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