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Xi-Nan Ma

Publications and source records attributed to Xi-Nan Ma.

At least 19 recordsLinked to original sources
arXiv2026

$C^{1,1}$ Regularity Global Estimates for Homogeneous Complex Hessian Equations on Punctured Domains

In this paper, we prove the $C^{1,1}$ regularity of Green functions associated with the complex $k$-Hessian operator for $1\leq k<n$, and give a new proof of the corresponding regularity of pluricomplex Green functions. We establish real Hessian estimates for approximating problems associated with homogeneous complex Hessian equations on punctured domains. The main idea is to introduce new auxiliary functions generated by complex linear vector fields to reduce the global real Hessian estimate to the boundary. We also treat the complex Monge-Ampère case. Similar real Hessian estimates also works for the exterior problem.

math.CV↗
arXiv2026

A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive $σ_k$-curvature Metrics

Let $k\in\mathbb N$ satisfy $1 0$ must obey \[ p\leq p_k(n), \] where $p_k$ is the model threshold determined by $\Hh^{p+1}\times\Sn^{n-p-1}$. When $k=2$ and $n=m^2$, we construct a smooth complete equality example on $\Sn^n\setminus\Sn^{(m^2-m-2)/2}$. We also prove that the strict inequality $p<p_k(n)$ holds under a finite positive linear-contact hypothesis.

math.DG↗
arXiv2026

Sharp Gradient Stability for the Sobolev Trace Inequality

Let \(n\ge3\) and \(1<p<n\). We prove a quantitative stability estimate for the critical Sobolev trace inequality on the upper half-space. More precisely, the Sobolev trace deficit controls the \(\max\{2,p\}\)-th power of the gradient distance to the manifold of trace bubbles. A central part of the proof is the spectral nondegeneracy of the trace bubbles: the first two eigenspaces of the linearized weighted Steklov problem are exactly the amplitude, dilation, and tangential translation modes.

math.AP↗
arXiv2026

Liouville Rigidity and Universal Spacelikeness Estimates for a Lorentzian Prescribed Mean Curvature Equation

We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p<\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key independent ingredient is a universal bound, valid for every $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. Thus pointwise strict spacelikeness automatically improves to a uniform spacelike gap, including in the critical and supercritical regimes. The proof combines a geometric Bernstein estimate, comparison with an explicit hyperbolic cap, and weighted trace-free tensor identities. The result extends the known radial nonexistence theorem to arbitrary entire solutions and yields a geometric half-space rigidity theorem for complete spacelike hypersurfaces.

math.AP↗
arXiv2026

The Higher-Dimensional Nitsche Conjecture: Sharp Bounds and Rigidity

Let $n\ge3$ and let $h:\A(r,1)\to\A(R,1)\subset\mathbb R^n$ be an onto homeomorphism with harmonic coordinate functions. We prove the sharp Nitsche bound \[ R\le R_{n,+}(r):=\frac{nr}{n-1+r^n}, \] and, when $h$ interchanges the two ends, the strictly stronger sharp bound \[ R\le R_{n,-}(r):=\frac{nr^{n-1}}{1+(n-1)r^n}. \] Both critical cases are rigid: equality forces, up to an orthogonal transformation, the corresponding end-preserving or end-reversing radial harmonic homeomorphism. No continuous extension to the closed annulus, boundary homeomorphism, boundary Jacobian, or sign condition on the Jacobian is assumed. The proof converts the nonzero degree of each interior direction map into a probability coupling and establishes a sharp contraction principle for vector measures under positive zonal kernels, using the strict concavity of spherical-cap barycenters. At either critical value, a second-order endpoint defect forces equality for a limiting transfer kernel, whose equality classification yields an orthogonal coupling graph. The remaining trace is locked by a Dirichlet-to-Neumann spectral gap in the end-preserving case and by endpoint Hölder regularity and uniform convergence of the direction maps in the end-reversing case.

math.AP↗
arXiv2026

Global Minimality and Rigidity of the Constraint Map Vortex

We consider the minimization problem \[ \min\left\{\int_{B_1}|Du|^2:\ u\in W^{1,2}(B_1;\mathbb R^n),\quad u=x\ \text{on }\partial B_1,\quad |u|\ge a\right\}, \quad 0<a<1. \] Figalli, Guerra, Kim, and Shahgholian proved that the canonical radial vortex is the unique global minimizer for $n\ge7$, and asked whether the same holds in dimensions \(3\le n \le6\). We answer this question affirmatively, thereby completing the global minimality and rigidity of the constraint map vortex in every dimension \(n\ge3\).

math.AP↗
arXiv2026

Liouville theorem for a class of p-Laplace type equations on manifolds

We study a class of $p$-Laplace equations $$Δ_p u-λu^{p-1}+ u^{q-1}=0$$ on a closed $n$-dimensional Riemannian manifold $(M,g)$ with $\operatorname{Ric}\geqslant(n-1)g$. For $1 2$ and $p 0$; aside from the constant solution, the equation admits a positive nonconstant solution. This answers Véron's problem raised in \cite{Ver92}.

math.AP↗
arXiv2026

A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains

We prove the strict log-concavity of the positive first eigenfunction \(-u\) of the \(2\)-Hessian equation and the strict $1/2$-convexity of the solution for the corresponding torsion problem in smooth bounded uniformly convex domains in $\mathbb{R}^{n}$. As applications, we establish the associated Brunn--Minkowski inequalities. We also show that this transformed-convexity phenomenon fails for \(3\)-Hessian equations by constructing, in dimension four, a smooth uniformly convex domain whose admissible zero-boundary solution has a nonconvex sublevel set.

math.AP↗
arXiv2026

Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation

Translating solitons arise as models for type~II singularities of mean-convex mean curvature flow. We construct a smooth bounded uniformly convex domain \(\Om\Subset\R^2\) such that the zero-Dirichlet solution of the planar translating mean curvature equation has a nonconvex sublevel set. The construction is based on a corrected near-critical grim-reaper profile and explicit barriers on a long convex channel.

math.AP↗
arXiv2026

Brunn--Minkowski Inequality for the First Complex $σ_{2}$-Hessian Eigenvalue

There are relatively few results on the convexity of solutions to complex equations. In this paper, We prove a strict real log-concavity theorem for the first eigenfunction of the complex $σ_{2}$-Hessian operator on smooth, bounded, real uniformly strictly convex domains in $\mathbb{C}^{n}$. As an application, we obtain a Brunn--Minkowski inequality for the first complex $σ_{2}$-Hessian eigenvalue. The proof combines a Bian--Guan constant-rank argument, a new inverse-convexity lemma for the compressed real Hessian, and Salani's viscosity admissible-test-function method.

math.AP↗
arXiv2026

A Brunn--Minkowski inequality for the Hessian eigenvalue in convex domain

We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in $\mathbb{R}^{n}$, as an application we get the Brunn--Minkowski inequality for the Hessian eigenvalue and characterize the equality case in bounded strictly convex domain in $\mathbb{R}^{n}$.

math.AP↗
arXiv2025

The Liouville-type equation and an Onofri-type inequality on closed 4-manifolds

In this paper, we study the Liouville-type equation \[Δ^2 u-λ_1κΔu+λ_2κ^2(1-\mathrm e^{4u})=0\] on a closed Riemannian manifold \((M^4,g)\) with \(\operatorname{Ric}\geqslant 3κg\) and \(κ>0\). Using the method of invariant tensors, we derive a differential identity to classify solutions within certain ranges of the parameters \(λ_1,λ_2\). A key step in our proof is a second-order derivative estimate, which is established via the continuity method. As an application of the classification results, we derive an Onofri-type inequality on the 4-sphere and prove its rigidity.

math.AP↗
arXiv2025

Liouville theorem of the subcritical biharmonic equation on complete manifolds

In this paper, we study the subcritical biharmonic equation \[Δ^2 u=u^α\] on a complete, connected, and non-compact Riemannian manifold $(M^n,g)$ with nonnegative Ricci curvature. Using the method of invariant tensors, we derive a differential identity to obtain a Liouville theorem, i.e., there is no positive $C^4$ solution if $n\geqslant5$ and $1<α<\frac{n+4}{n-4}$. We establish a crucial second-order derivative estimate, which is established via Bernstein's technique and the continuity method.

math.AP↗
arXiv2024

Liouville theorem for elliptic equations involving the sum of the function and its gradient in $\mathbb R^n$

We prove Liouville theorem for the equation $Δv + N v^p + M |\nabla v|^{q}= 0$ in $\mathbb R^n$, with $M, N > 0, q = \frac{2p}{p + 1}$ in the critical and subcritical case. The proof is based on a differential identity and Young inequality. We remark that this is the second version for the paper. And we thank Prof. Bidaut-Véron and Véron for their very useful comments on this paper. Compared with the first one, in this version we correct some errors and adjust the arrangement of the proof so that it can be understood easily.

math.AP↗
arXiv2024

Liouville theorem for elliptic equations with a source reaction term involving the product of the function and its gradient in $\mathbb R^n$

We improve the Liouville theorem for the equation $-Δv = v^p |\nabla v|^q$ in $\mathbb R^n$, which was studied by Bidaut-Véron, García-Huidobro, and Véron. The proof is based on a differential identity and Young inequality. We remark that this is the second version for this paper and the first one was submitted one year ago. We thank Prof. Bidaut-Véron and Véron for their very useful comments on this paper. Compared with the first version, we correct some errors and provide more details for the proof.

math.AP↗
arXiv2024

Best constant and extremal functions for a class Hardy-Sobolev-Maz'ya inequalities

We derive an integral identity for a class $p$-Laplace equation, and then classify all positive finite energy cylindrically symmetric solutions of the equation (\ref{1.2}) for $3\leq k\leq n-1,$ with the help of some a prior estimates. Combining this with the result of Secchi-Smets-Willem{\cite{SSW03}}, as a consequence, we obtain the best constant and extremal functions for the related Hardy-Sobolev-Maz'ya inequalities.

math.AP↗
arXiv2024

$σ_k$-Yamabe measure

We found a special divergence structure for the $σ_k$-Yamabe operator and use it to get a monotonicity formula. We also get an interior $L^{\infty}$ estimate via its $L^1$ norm for the $σ_k$-Yamabe operator when $1\le k \le \frac{n}{2}$. Combining these two tools, we prove the weak continuity of the $σ_k$-Yamabe measure with respect to convergence in measure.

math.AP↗