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Wangzhe Wu

Publications and source records attributed to Wangzhe Wu.

12 recordsLinked to original sources
arXiv2026

A sharp threshold for mixed $Q$-curvature rigidity

We resolve Case's question (Crelle's Journal, 2024) on the sharp lower threshold for Obata-type rigidity of $I_a=Q+aσ_2(A)$ in positive Einstein conformal classes of dimension $n\ge4$. Using a new reference-curvature identity, we prove that, on a closed connected manifold, every smooth metric in such a class with nonnegative scalar curvature and constant $I_a$ is Einstein whenever $a\ge-4$. The scalar-curvature sign assumption can be removed for $-4\le a\le-2(n-2)/(n-1)$. We also extend the quotient rigidity theorem of Ge--Wang--Wei to $I_a=ΛR^θ$ for $R>0$, $a\ge-4$, and $θ\le1$. Sharpness is established in every dimension by a shooting construction. For each sufficiently small $η>0$, we construct a smooth non-Einstein metric in the round conformal class on $\mathbb S^n$ with positive scalar curvature and the same constant $I_{-4-η}$ as the unit round metric. The construction smoothly matches a perturbed Schwarzschild neck near the equator with perturbed round caps at both poles.

math.DG↗
arXiv2026

Beyond the $L^{9/5}$ Vorticity Criterion in the Stationary Navier--Stokes Liouville Problem

Let $(v,p)$ be a smooth stationary Navier--Stokes solution in $\mathbb R^3$ with $v(x)\to0$ as $|x|\to\infty$. We prove that $v\equiv0$ if its vorticity $ω:=\nabla \times v$ belongs to $L^{s,\infty}(\mathbb R^3)$ for some $9/5\le s\le1.87$. This gives a global weak-Lorentz vorticity criterion beyond $9/5$ requiring neither smallness nor an additional Fubini-type hypothesis. The finite Dirichlet integral is recovered from the vorticity hypothesis rather than assumed. In particular, $|ω(x)|=O(|x|^{-α})$ implies triviality for $α\ge300/187$, crossing the $|x|^{-5/3}$ vorticity-decay scale. The proof combines this automatic finite-energy upgrade with a new variable-denominator Bernoulli--vorticity quotient identity. Its completed-square form gives nonnegative defect terms, and localizing the concavity of the denominator creates a positive measure on a Bernoulli level surface. A pressure-weighted convex extension of this identity yields a quantitative normalized-vorticity trace and quotient-gradient control. These estimates are then coupled to the Bernoulli-gradient identity through a nonlinear feedback estimate.

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 $1\leqslant p\leqslant\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. This resolves, in the classical strictly spacelike setting, the nonexistence conjecture of Byeon, Ikoma, Malchiodi, and Mari, including the critical endpoint. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key ingredient is a universal bound, valid for every $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. Then a weighted trace-free tensor identity from the invariant-tensor approach, combined with a common cutoff estimate, a core-counting argument and Souplet-type feedback inequality, yields a unified proof in the subcritical and critical ranges. The upper endpoint is sharp for $n\geqslant3$, as supercritical radial solutions exist. The theorem also gives half-space rigidity for complete spacelike hypersurfaces, including at the critical exponent.

math.AP↗
arXiv2026

Rigidity, sharp inequalities, and stability for $σ_2$-curvature

Using classical conformal divergence identities with a variable reference curvature, we prove four main results. First, we establish a general mean-curvature estimate for conformal metrics on the round hemisphere with $A_g\in\overline{Γ_2^+}$ and positive prescribed $H_2$ data. When $σ_2(A_g)=0$ and the boundary data are nonincreasing, the estimate yields rigidity, removing Case-Wang's pinching condition $\sup_ΣH_g\le 3\inf_ΣH_g$. For $n\ge 5$, the constant-data case also classifies the smooth critical metrics associated with their sharp $σ_2$ Sobolev trace conjecture. Second, within positive Einstein conformal classes, we extend the constant-$σ_2$ rigidity results of Viaclovsky and Gursky-Streets to nonincreasing prescribed data, including backgrounds with nonzero Weyl curvature for $n\ge 5$. Third, we extend Li-Li's spherical $σ_2/σ_1$ rigidity and Guan-Wang's sharp integral inequality to positive Einstein backgrounds, with the latter holding for $n\ge 5$ under positive scalar curvature. Fourth, we extend Frank-Peteranderl's spherical $σ_2$ stability to fixed nonround positive Einstein backgrounds in dimensions $n\ge 5$, retaining $H^1$ and $W^{1,4}$ control under positive scalar curvature.

math.DG↗
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↗
arXiv2025

A remark for characterizing blowup introduced by Giga and Kohn

Giga and Kohn studied the blowup solutions for the equation $v_{t} - Δv - |v|^{p - 1} v = 0 $ and characterized the asymptotic behavior of $v$ near a singularity. In the proof, they reduced the problem to a Liouville theorem for the equation $Δu - \frac{1}{2} x \cdot \nabla u + |u|^{p - 1} u - βu = 0$ where $β= \frac{1}{p - 1}$ and $|u|$ is bounded. This article is a remark for their work and we will show when $u \geq 0$, the boundedness condition for $|u|$ can be removed.

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

Liouville theorem for one kind of elliptic equations on complete Riemannian manifold

We use maximum principle to prove the Liouville theorem of the equation $ΔU + b\cdot \nabla U + h U^α = 0, U \geq 0, 0 < α< \frac{n + 2}{n - 2}$ on the complete Riemannian manifold with non-negative Ricci tensor, which improve the result of Gidas-Spruck and Catino-Monticelli. We remark that this is the second version and all of the results come from the first version. Two months after we posted version 1 of this preprint on arXiv, we found Zhihao Lu has already posted a paper arXiv:2308.14764 before us and part of his result coincides with ours. So after deleting these parts and adding more reference and details, we post this second version on arXiv.

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↗
arXiv2023

Liouville theorem for quasilinear elliptic equations in $\mathbb R^N$

We prove Liouville theorem for the equation $Δ_m v + v^p + M |\nabla v|^{q}= 0$ in a domain $Ω\subset\mathbb R^n$, with $M\in \mathbb{R}$ in the critical and subcritical case. As a natural extension of our recent work \cite{MWZ}, the proof is based on an integral identity and Young's inequality.

math.AP↗