The Sobolev inequality for Paneitz operator on three manifolds
type: Item
instance of: scholarly article
title: The Sobolev inequality for Paneitz operator on three manifolds
zbMATH Open document ID: 1066.53082
published in: Calculus of Variations and Partial Differential Equations
publication date: 2004-11-05T00:00:00Z
review text: Let \((M, g)\) be a smooth compact Riemannian manifold of dimension \(3\). The Paneitz operator is defined by \[ P \phi = \Delta^2 \phi + 4 \operatorname {div} (Rc(\nabla \phi, e_i)e_i) - \frac{5}{4} \operatorname {div} (R \nabla \phi) - \frac{1}{2} Q \phi,\quad \forall \phi \in C^{\infty}(M, \mathbb R), \] where \(Rc\) is the Ricci operator, \(\{e_1, e_2, e_3\}\) is an arbitrary local orthonormal frame, and \(Q\) is the curvature, \[ Q=-\frac{1}{4} \Delta R - 2 | Rc | ^2 + \frac{23}{32} R^2. \] One notes that \(P\) satisfies the conformal covariant property, \[ P_{{\rho}^{-4}g} \phi = {\rho}^7 P_g (\rho \phi),\quad \forall \rho \in C^{\infty} (M, \mathbb R),\;\rho >0,\quad \forall \phi \in C^{\infty} (M, \mathbb R). \] In particular, if \(\widetilde{g} = u^{-4}g\) for some \(u \in C^\infty (M), u >0\), one has \(\widetilde{Q}=-2u^7P_g u\), hence the problem of finding a conformal metric with constant \(Q\)-curvature becomes \[ P_gu={\text{constant}}\cdot u^{-7},\quad {\text{on}}\;M,\quad {\text{and}} \quad u \in C^{\infty} (M),\quad u>0. \] This problem has also a variational definition. Let \[ E(u,v)=\int_M \left( \Delta u \Delta v - 4Rc(\nabla u, \nabla v) + \frac{5}{4} R \nabla u \cdot \nabla v - \frac{1}{2} Q u v \right)\, d \mu, \quad \forall u, v \in H^2(M), \] and \[ I(u)=\frac{E(u,u)}{| u| ^2_{L^{-6}(M)}}= E(u,u) | u^{-1}| ^2_{L^6(M)},\quad \forall u \in H^2(M),\;u>0. \] Thus, the previous description is the Euler-Lagrange equation for \(I\). To find a critical point, it suffices to minimize the functional \(I\) over the class of positive functions in \(H^2(M)\). Its infimum, although a conformally invariant quantity, is no longer easily seen to satisfy \(Y_4^3(M, g):=\inf_{u \in H^2(M), u>0} I(u)> -\infty.\) This was trivial in the case of the analogous Yamabe constant where one simply applies the usual Sobolev inequality. The negative exponent of the present problem does not allow the same reasoning here. To deal with this, the following condition is intr…
reviewed by: Alina Stancu
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 35J35
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 2113925
zbMATH Keywords: Berger spheres
zbMATH Keywords: Paneitz operator
zbMATH Keywords: Yamabe problem
author: Fengbo Hang
author: Paul C. Yang
MaRDI profile type: Publication
full work available at URL: s00526-003-0247-4
OpenAlex ID: W2081798767
DOI: 10.1007/S00526-003-0247-4
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description: scientific article; zbMATH DE number 2113925