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Register a new userLectures on the fourth order Q curvature equation
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We discuss some open problems and recent progress related to the 4th order Paneitz operator and Q curvature in dimensions other than 4.
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In this note, we study Q-curvature flow on $S^4$ with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on $S^4$ has a solution provided the prescribed Q-curvature $f$ has its positive part, which possesses non-degenerate critical points such that $Delta_{S^4} f ot=0$ at the saddle points and an extra condition such as a nontrivial degree counting condition.
We first present a warped product manifold with boundary to show the non-uniqueness of the positive constant scalar curvature and positive constant boundary mean curvature equation. Next, we construct a smooth counterexample to show that the compactness of the set of lower energy solutions to the above equation fails when the dimension of the manifold is not less than $62$.
In this paper, we employ a nonlocal $Q$-curvature flow inspired by Gursky-Malchiodis work cite{gur_mal} to solve the prescribed $Q$-curvature problem on a class of closed manifolds: For $n geq 5$, let $(M^n,g_0)$ be a smooth closed manifold, which is not conformally diffeomorphic to the standard sphere, satisfying either Gursky-Malchiodis semipositivity hypotheses: scalar curvature $R_{g_0}>0$ and $Q_{g_0} geq 0$ not identically zero or Hang-Yangs: Yamabe constant $Y(g_0)>0$, Paneitz-Sobolev constant $q(g_0)>0$ and $Q_{g_0} geq 0$ not identically zero. Let $f$ be a smooth positive function on $M^n$ and $x_0$ be some maximum point of $f$. Suppose either (a) $n=5,6,7$ or $(M^n,g_0)$ is locally conformally flat; or (b) $n geq 8$, Weyl tensor at $x_0$ is nonzero. In addition, assume all partial derivatives of $f$ vanish at $x_0$ up to order $n-4$, then there exists a conformal metric $g$ of $g_0$ with its $Q$-curvature $Q_g$ equal to $f$. This result generalizes Escobar-Schoens work [Invent. Math. 1986] on prescribed scalar curvature problem on any locally conformally flat manifolds of positive scalar curvature.
In this paper, we establish that: Suppose a closed Riemannian manifold $(M^n,g_0)$ of dimension $geq 8$ is not locally conformally flat, then the Paneitz-Sobolev constant of $M^n$ has the property that $q(g_0)<q(S^n)$. The analogy of this result was obtained by T. Aubin in 1976 and had been used to solve the Yamabe problem on closed manifolds. As an application, the above result can be used to recover the sequential convergence of the nonlocal Q-curvature flow on closed manifolds recently introduced by Gursky-Malchiodi.
We apply the theory of Lie symmetries in order to study a fourth-order $1+2$ evolutionary partial differential equation which has been proposed for the image processing noise reduction. In particular we determine the Lie point symmetries for the specific 1+2 partial differential equations and we apply the invariant functions to determine similarity solutions. For the static solutions we observe that the reduced fourth-order ordinary differential equations are reduced to second-order ordinary differential equations which are maximally symmetric. Finally, nonstatic closed-form solutions are also determined.
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