Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-06-26 , DOI: 10.1007/s00526-020-01780-y Quốc Anh Ngô , Hong Zhang
On a closed Riemannian manifold \((M,g_0)\) of even dimension \(n \ge 4\), the well-known prescribed Q-curvature problem asks whether 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
$$\begin{aligned} \mathbf {P}_{g_0} u+Q_{g_0} = f e^{nu}, \end{aligned}$$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 \(g_0\),
$$\begin{aligned} \frac{\partial g (t)}{\partial t}= -2\left( Q_{g (t)} - \frac{\int _M f Q_{g(t)} d\mu _{g(t)} }{\int _M f^2 d\mu _{g(t)} }f \right) g(t) \quad \text { for } t >0, \end{aligned}$$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
$$\begin{aligned} \int _M Q_0 \, d\mu _{g_0}\ne k (n-1)! \, \mathrm{vol}(\mathbb {S}^n) \quad \text {for all } \; k = 2,3,\ldots \end{aligned}$$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 results for prescribed Q-curvature problem can be derived from the convergence of the flow.
中文翻译：
尺寸均匀的闭合歧管上的规定Q曲率流
在关闭的黎曼流形\（（M，G_0）\）甚至尺寸的\（N \ GE 4 \） ，公知的规定的Q -curvature问题询问是否有一个度量克适形到\（G_0 \）这样其Q -curvature，与GJMS运营商相关联\（\ mathbf {P} _g \） ，是等于给定函数˚F。令\（g = e ^ {2u} g_0 \），此问题等效于解决
$$ \ begin {aligned} \ mathbf {P} _ {g_0} u + Q_ {g_0} = fe ^ {nu}，\ end {aligned} $$其中\（Q_ {g_0} \）表示\（g_0 \）的Q曲率。本文的主要目的是引入以下随时间变化的度量g（t）与\（g_0 \）保形的负梯度流，
$$ \ begin {aligned} \ frac {\ partial g（t）} {\ partial t} = -2 \ left（Q_ {g（t）}-\ frac {\ int _M f Q_ {g（t）} d \ mu _ {g（t）}} {\ int _M f ^ 2 d \ mu _ {g（t）}} f \ right）g（t）\ quad \ text {for} t> 0，\ end {aligned} $$研究规定Q曲率的问题。由于\（\ INT _M Q_G d \亩_g \）是共形不变，我们的分析依赖于的大小\（\ INT _M Q_ {G_0} \，d \亩_ {G_0} \） ，其被假定为满足
$$ \ begin {aligned} \ int _M Q_0 \，d \ mu _ {g_0} \ ne k（n-1）！\，\ mathrm {vol}（\ mathbb {S} ^ n）\ quad \ text {for all} \; k = 2,3，\ ldots \ end {aligned} $$本文是双重的。首先，我们在f上确定合适的条件，以使上述定义的梯度流在所有时间均被定义，并且随着时间流向无穷大而逐渐或均匀地收敛。其次，我们表明可以从流的收敛中得出规定的Q曲率问题的各种存在结果。
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