arXiv subjects

Huihuang Zhou

Publications and source records attributed to Huihuang Zhou.

4 recordsLinked to original sources

arXiv2026

Comparison Theorems for Fractional GJMS Operators

In this paper, we mainly focus on the fractional GJMS operators $P_{2γ}$ which are defined on the conformal infinity of a Poincaré-Einstein manifold. We derive two comparison inequalities of the fractional Yamabe constants associated to the fractional GJMS operators. One is between $P_{1}$ and $P_{2γ}$ for $γ\in (1/2,1)$, and the other is between $P_{2}$ and $P_{2γ}$ for $γ\in (1,2)$. They both imply the rigidity theorems by characterizing the equalities. Together with the result in \cite{WZ1}, we partially provide some evidence for the monotonicity of the fractional Yamabe constants.

math.DG↗

arXiv2024

Heintze-Karcher Type Inequalities for Fractional GJMS Operators

In this paper, we genelize the Heintze-Karcher type inequalities for fractional Q-curvature $Q_{2γ}$ on conformally compact Einstein manifolds. Such inequality holds for all $γ\in (0,1]$. In particular, for $γ=\frac{1}{2}$ and $γ=1$, we obtain some rigidity theorems by characterising the equalities.

math.DG↗

arXiv2021

Lower Bounds for the Relative Volume of Poincare-Einstein Manifolds

In this paper, we show that for a Poincaré-Einstein manifold $(X^{n+1},g_+)$ with conformal infinity $(M,[\hat{g}])$ of nonnegative Yamabe type, the fractional Yamabe constants of the boundary provide lower bounds for the relative volume. More explicitly, for any $γ\in (0,1)$, $$ \left(\frac{Y_{2γ} (M,[\hat{g}])}{Y_{2γ} (\mathbb{S}^n , [g_{\mathbb{S}}])}\right)^{\frac{n}{2γ}} \leq \frac{V(Γ_t(p),g_+)}{V(Γ_t(0),g_{\mathbb{H}})} \leq \frac{V( B_t(p),g_+)}{V( B_t(0), g_{\mathbb{H}})} \leq 1,\quad 0<t<\infty, $$ where $B_t(p)$, $Γ_t(p)$ are the the geodesic ball and geodesic sphere of radius $t$ in $(X,g_+)$ with center at $p\in X^{n+1}$; and $B_t(0)$, $Γ_t(0)$ are the the geodesic ball and geodesic sphere in $\mathbb{H}^{n+1}$ with center at $0\in\mathbb{H}^{n+1}$.

math.DG↗

arXiv2021

A note on the Compactness of Poincare-Einstein manifolds

For a conformally compact Poincaré-Einstein manifold $(X,g_+)$, we consider two types of compactifications for it. One is $\bar{g}=ρ^2g_+$, where $ρ$ is a fixed smooth defining function; the other is the adapted (including Fefferman-Graham) compactification $\bar{g}_s=ρ^2_sg_+$ with a continuous parameter $s>\frac{n}{2}$. In this paper, we mainly prove that for a set of conformally compact Poincaré-Einstein manifolds $\{(X, g_{+}^{(i)})\}$ with conformal infinity of positive Yamabe type, $\{\bar{g}^{(i)}\}$ is compact in $C^{k,α}(\overline{X})$ topology if and only if $\{\bar{g}_s^{(i)}\}$ is compact in some $C^{l,β}(\overline{X})$ topology, provided that $\bar{g}^{(i)}|_{TM}=\bar{g}_s^{(i)}|_{TM}=\hat{g}^{(i)}$ and $\hat{g}^{(i)}$ has positive scalar curvature for each $i$. See Theorem 1.1 and Corollary 1.1 for the exact relation of $(k,α)$ and $(l,β)$.

math.DG↗