Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0611894

On the higher order conformal covariant operators on the sphere

Fengbo Hang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We will show that in the conformal class of the standard metric $g_{S^n}$ on $S^n$, the scaling invariant functional $(μ_g(S^n))^{\frac{2m-n}{n}}\int_{S^n}Q_{2m,g}dμ_g$ maximizes at $g_{S^n}$ when $n$ is odd and $m=\frac{n+1}{2}$ or $\frac{n+3}{2}$. For $n$ odd and $m\geq\frac{n+5}{2}$, $g_{S^n}$ is not stable and the functional has no local maximizer. Here $Q_{2m,g}$ is the $2m$th order $Q $-curvature.

Explore related subjects

math.DGmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Fengbo Hang. 2006-11-28. On the higher order conformal covariant operators on the sphere. https://arxiv.org/abs/math/0611894

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two

In this article, we investigate when a left-invariant Riemannian metric on a Lie group is a Ricci soliton, under the assumption that the derived algebra has dimension at most two. We establish computable necessary and sufficient conditions for a given left-invariant Riemannian metric to be a Ricci soliton. As applications, we obtain several examples of Ricci nilsolitons and apply our results to indecomposable Lie groups of dimension at least five with two-dimensional derived algebra.

math.DG↗
arXiv2026

Cohn--Vossen-Type Inequalities for Three-Manifolds and Locally Conformally Flat Manifolds

We prove Cohn--Vossen-type scalar curvature inequalities on complete noncompact Riemannian manifolds with nonnegative Ricci curvature, motivated by Yau's higher-dimensional problem. For $n\ge3$, we obtain an $O(r^{n-2})$ normalized growth estimate under a $\mathbb Z^{n-2}$ subgroup condition on $π_1(M)$. For locally conformally flat manifolds, we prove the corresponding normalized estimate in the non-$\mathbb R^n$ case and derive polynomial or exponential upper bounds in the conformally Euclidean case. In dimension three, we prove the sharp asymptotic scalar-curvature flux estimate $8π$ under quadratic scalar-curvature decay, confirming the Munteanu--Wang conjectural $8π$ bound in this setting; we also prove finiteness of the flux for manifolds with a foliated end. Finally, under the Cohn--Vossen-scale scalar-growth hypothesis, we prove weighted analogues for the weighted scalar curvature $\mathrm{Sc}_{α,β}$ on weighted Riemannian manifolds with nonnegative Bakry--Émery Ricci curvature, including sharp distinctions between the finite-dimensional and infinite-dimensional Bakry--Émery regimes.

math.DG↗
arXiv2026

A sharp relative comparison inequality for conformal fillings of Poincaré--Einstein manifolds

Let $(X^{n+1},g_+)$ be a Poincaré--Einstein manifold with conformal infinity $(M^n,[h])$ of positive Yamabe constant $Y(M,[h])$. Then a relative comparison inequality \[ \frac{Y_1(X,M,[\bar g])}{Y_1(\mathbb{S}^{n+1}_+,\mathbb{S}^n,[g_{\mathbb{S}_+^{n+1}}])} \geq \left(\frac{Y(M,[h])}{Y(\mathbb{S}^n,[g_{\mathbb{S}^n}])}\right)^{\frac{n}{n+1}} \] holds for the \textbf{type-I} Escobar--Yamabe compactification $Y_1(X,M,[\bar g]$, with equality if and only if $\left(X, g_{+}\right)$ is isometric to the hyperbolic space $\left(\mathbb{H}^{n+1}, g_{\mathbb{H}^{n+1}}\right)$. This confirms a conjecture raised by Sun-Yung A. Chang.

math.DG↗