Searcharxiv⌕ Search

arXiv subjects

Xingwang Xu

Publications and source records attributed to Xingwang Xu.

8 recordsLinked to original sources
arXiv2026

A sharp isoperimetric inequality and the top order $Q$-curvature

For a smooth, complete and normal metric $g = e^{2u}|dx|^2$ with finite total $n$-th order $Q$-curvature on $\mathbb{R}^n$ with dimension $n \geq 2$, we first show that everywhere non-negativity (resp. non-positivity) $n$-th order $Q$-curvature $Q_g^{(n)}$ implies everywhere non-negativity (resp. non-positivity) of the sectional curvature. Based on this fact, we secondly show that, once $Q_g^{(n)}$ is non-negative, then for any compact domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial\Omega$, the following sharp isoperimetric inequality holds: $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}} \left(1 - \frac{2}{(n-1)!\,|\mathbb{S}^n|} \int_{\mathbb{R}^n} Q_g^{(n)} \, d\mu_g\right) |\Omega|_g.$$ The third claim in this article is that, if the $n$-th order $Q$-curvature, $Q_g^{(n)}$, is non-positive and under the main assumption that Cartan-Hadamard conjecture holds true, then we have the sharp inequality $$|\partial\Omega|_g^{\frac{n}{n-1}} \geq n^{\frac{n}{n-1}} |\mathbb{B}^n|^{\frac{1}{n-1}}|\Omega|_g.$$

math.DG↗
arXiv2025

On geometry of $Q^{(2k)}_g$-curvature

The main purpose of current article is to study the geometry of $Q$-curvature. For simplicity, we start with a simple model: a complete and conformal metric $g=e^{2u}|dx|^2$ on $\mathbb{R}^n$. Assuming that the metric $g$ has non-negative $nth$-order $Q$-curvature and non-negative scalar curvature, we show that the Ricci curvature is non-negative. If we further assume that the isoperimetric ratio near the end is positive, we show that the growth rate of $kth$ elementary symmetric function $\sigma_k(g)$ of Ricci curvature over geodesic ball of radius $r$ is at most polynomial in $r$ with order $n-2k$ for all $1 \leq k \leq \frac{n-2}{2}$. Similarly, we are able to show that the same growth control holds for $2kth$-order $Q$-curvature. Finally, we show that for $k=1$ or $2$, the gap theorems for $Q^{(2k)}_g$ hold true.

math.DG↗
arXiv2024

On positivity of the Q-curvatures of conformal metrics

We mainly show that for a conformal metric $g=u^{\frac{4}{n-2m}}|dx|^2$ on $\mathbb{R}^n$ with $n\geq 2m+1$, if the higher order Q-curvature $Q^{(2m)}_g$ is positive and has slow decay barrier near infinity, the lower order Q-curvature $Q^{(2)}_g$ and $Q^{(4)}_g$ are both positive if $m$ is at least two.

math.DG↗
arXiv2015

A pointwise inequality for the fourth order Lane-Emden equation

We prove that the following pointwise inequality holds \begin{equation*} -Δu \ge \sqrt\frac{2}{(p+1)-c_n} |x|^{\frac{a}{2}} u^{\frac{p+1}{2}} + \frac{2}{n-4} \frac{|\nabla u|^2}{u} \ \ \text{in}\ \ \mathbb{R}^n \end{equation*} where $c_n:=\frac{8}{n(n-4)}$, for positive bounded solutions of the fourth order Hénon equation that is \begin{equation*} Δ^2 u = |x|^a u^p \ \ \ \ \text {in }\ \ \mathbb{R}^n \end{equation*} for some $a\ge0$ and $p>1$. Motivated by the Moser's proof of the Harnack's inequality as well as Moser iteration type arguments in the regularity theory, we develop an iteration argument to prove the above pointwise inequality. As far as we know this is the first time that such an argument is applied towards constructing pointwise inequalities for partial differential equations. An interesting point is that the coefficient $\frac{2}{n-4}$ also appears in the fourth order $Q$-curvature and the Paneitz operator. This in particular implies that the scalar curvature of the conformal metric with conformal factor $u^\frac{4}{n-4}$ is positive.

math.AP↗
arXiv2013

Remarks on the $\mathbf Q$-curvature flow

The main purpose of this short note is to point out that the negative gradient flow for the prescribed $\mathbf Q$-curvature problem on $S^n$ can be extended to handle the case that the $\mathbf Q$-curvature candidate $f$ may change signs.

math.DG↗
arXiv2007

Ricci Flow with hyperbolic warped product metrics

In this short note, we show that the negative curvature is preserved in the deformation of hyperbolic warped product metrics under Ricci flow. It is also showed that the flow converges to a flat metric as time going to infinity.

math.DG↗