Q-curvature Flow Research Articles

The prescribed Q-curvature flow for arbitrary even dimension in a critical case

In this paper, we study the prescribed Q-curvature flow equation on a arbitrary even dimensional closed Riemannian manifold (M,g), which was introduced by S. Brendle in [3], where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and ∫MQdμ<(n−1)!Vol(Sn). In this paper we study the critical case that ∫MQdμ=(n−1)!Vol(Sn), we will prove the convergence of the flow under some geometric hypothesis. In particular, this gives a new proof of Li-Li-Liu's existence result in [21] in dimension 4 and extend the work of Li-Zhu [22] in dimension 2 to general even dimensions. In the proof, we give a explicit expression of the limit of the corresponding energy functional when the blow up occurs.

Prescribed Q-curvature flow on closed manifolds of even dimension

On a closed Riemannian manifold $(M,g_0)$ of even dimension $n \geqslant 4$, the well-known prescribed $Q$-curvature problem asks whether or not there is a metric $g$ comformal to $g_0$ such that its $Q$-curvature, associated with the GJMS operator $\mathbf P_g$, is equal to a given function $f$. Letting $g = e^{2u}g_0$, this problem is equivalent to solving \[ \mathbf P_{g_0} u+Q_{g_0} = f e^{nu}, \] where $Q_{g_0}$ denotes the $Q$-curvature of $g_0$. The primary objective of the paper is to introduce the following negative gradient flow of the time dependent metric $g(t)$ conformal to $g_0$, \[ \frac{\partial g (t)}{\partial t}= -2\Big(Q_{g (t)} - \frac{\int_M f Q_{g(t)} d\mu_{g(t)} }{\int_M f^2 d\mu_{g(t)} }f \Big)g(t) \quad \text{ for } t >0, \] to study the problem of prescribing $Q$-curvature. Since $\int_M Q_g d\mu_g$ is conformally invariant, our analysis depends on the size of $\int_M Q_{g_0} d\mu_{g_0}$, which is assumed to satisfy \[ \int_M Q_0 d\mu_{g_0} \ne k (n-1)! \, {\rm vol}(\mathbb S^n) \quad \text{ for all } \; k = 2,3,... \] The paper is twofold. First, we identify suitable conditions on $f$ such that the gradient flow defined as above is defined to all time and convergent, as time goes to infinity, sequentially or uniformly. Second, we show that various existence theorems for prescribed $Q$-curvature problem can be derived from the convergence of the flow.

Compactness properties for geometric fourth order elliptic equations with application to the Q-curvature flow

Abstract We prove the compactness of solutions of general fourth order elliptic equations which are L 1 L^{1} -perturbations of the Q-curvature equation on compact Riemannian 4-manifolds. Consequently, we prove the global existence and convergence of the Q-curvature flow on a generic class of Riemannian 4-manifolds. As a by-product, we give a positive answer to an open question by A. Malchiodi [J. reine angew. Math. 594 (2006), 137–174] on the existence of bounded Palais–Smale sequences for the Q-curvature problem when the Paneitz operator is positive with trivial kernel.

Formula omitted]-curvature flow for GJMS operators with non-trivial kernel

We investigate the prescribed Q-curvature flow for GJMS operators with non-trivial kernel on a compact manifold of even dimension. When the total Q-curvature is negative, we identify a conformally invariant condition on the nodal domains of functions in the kernel of the GJMS operator, allowing us to prove the global existence of the flow and its convergence at infinity to a metric which is conformal to the initial one, and having a prescribed Q-curvature. If the total Q-curvature is positive, we show that the flow blows up in finite time.

Prescribed [formula omitted]-curvature flow on [formula omitted

In this paper, we study the problem of prescribing Q-curvature on the even-dimensional standard sphere Sn as a given function f. Using the prescribed Q-curvature flow, which was initially studied by Malchiodi and Struwe on S4, we prove an existence result under suitable assumptions on f.

Q-curvature flow on the standard sphere of even dimension

Using a gradient flow approach initiated by S. Brendle, we generalize the existence theorem for the prescribing Q-curvature equation on S 2 (Gauss curvature) by M. Struwe (2005) [14] and on S 4 by Malchiodi and Struwe (2006) [12] to S n for all even n with the similar assumption on the prescribed curvature candidate f.

Q-curvature flow on 4-manifolds with boundary

Given a compact four-dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero. Using integral methods, we prove global existence and convergence for the Q-curvature flow to a smooth metric conformal to g of prescribed Q-curvature, zero T-curvature and vanishing mean curvature under conformally invariant assumptions.

Q-curvature flow with indefinite nonlinearity

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 non-negative Q-curvature f has its positive part, which possesses non-degenerate critical points such that Δ S 4 f ≠ 0 at the saddle points and an extra condition such as a nontrivial degree counting condition.

STEADY STATES FOR ONE-DIMENSIONAL CURVATURE FLOWS

We define two conformal structures on S1which give rise to a different view of the affine curvature flow and a new curvature flow, the "Q-curvature flow". The steady states of these flows are studied. More specifically, we prove four sharp inequalities, which state the existence of the corresponding extremal metrics.

A fourth order curvature flow on a CR 3-manifold

Let (M 3 , J, θ 0 ) be a closed pseudohermitian 3-manifbld. Suppose the associated torsion vanishes and the associated Q-curvature has no kernel part with respect to the associated Paneitz operator. On such a background pseudohermitian 3-manifold, we study the change of the contact form according to a certain version of normalized Q-curvature flow. This is a fourth order evolution equation. We prove that the solution exists for all time and converges smoothly to a contact form of zero Q-curvature. We also consider other background conditions and obtain a priori bounds on high-order norms on the solutions.

Q-curvature flow on S4

We study a natural counterpart of the Nirenberg problem, namely to prescribe the Q-curvature of a conformal metric on the standard S4 as a given function f. Our approach uses a geometric flow within the conformal class, which either leads to a solution of our problem as, in particular, in the case when f ≡ const, or otherwise induces a blow-up of the metric near some point of S4. Under suitable assumptions on f, also in the latter case the asymptotic behavior of the flow gives rise to existence results via Morse theory.

Q-curvature flow on 4-manifolds

We formulate an appropriate gradient flow in order to study the evolution of the Q-curvature to a prescribed function on a 4-manifold. For a class of prescribed functions, we show convergence and describe the asymptotic behaviour at infinity.

Nonlinear Evolution Problems

The workshop ‘Nonlinear Evolution Problems’ focussed on three types of nonlinear evolution equations: (1) Geometric evolution equations (essentially of parabolic type) (2) Nonlinear hyperbolic equations (3) Dispersive equations The programme consisted of 22 talks presented by international specialists from Australia, France, Germany, Italy, Sweden, Switzerland and the United States.Generally, three lectures were delivered in the morning sessions and two in the late afternoon which left ample time for individual discussions. In the first group of equations, in particular the Ricci flow and conformal flows such as the Yamabe flow and the Q-curvature flow were considered. A further focus was on curvature flows for hypersurfaces such as the Gauss and harmonic mean curvature flow and on geometric flows of higher order. Julie Clutterbuck presented a direct approach to certain fundamental estimates for a general class of parabolic equations. Mete Soner pointed out important relations between fully nonlinear parabolic equations and backward stochastic differential equations. The class of hyperbolic equations was represented by wave maps, Einstein's equations of gravitation and nonlinear wave equations in waveguides and cones. Among the dispersive equations the nonlinear Schroedinger equation and the KdV equation were considered. The individual discussions covered connections between approaches to the different equations in each group but also on common techniques used across the three classes of equations. Among such techniques are for instance the careful examination of the algebraic and geometric structure of the nonlinear terms. Another common theme appears to be the phenomenon of blow-up (singularity formation) for solutions and methods to describe the solution near these. Selfsimilar solutions and blow-up rates play an important role here. The organizers decided to particularly encourage the young researchers among theparticipants to present their work, including a PhD student and several recent post-doctoral fellows.

Convergence of the Q-curvature flow on [formula omitted

In this paper, we study the Q-curvature flow on the standard sphere S 4 . We show that the flow converges exponentially for all initial data.