Recent Results and Open Problems on Conformal Metrics on ℝⁿ with Constant Q-Curvature

Abstract

We consider solutions to the equation $\displaystyle{ (-\Delta )^{m}u = (2m - 1)!e^{2mu}\quad \text{in }\mathbb{R}^{2m}, }$ satisfying $\displaystyle{ V:=\int _{\mathbb{R}^{2m}}e^{2mu(x)}dx <+\infty. }$ Geometrically, if u solves (1)–(2), then the conformal metric g _u : = e ^2u | dx | ² has Q-curvature $Q_{g_{u}} \equiv (2m - 1)!$ and volume V (by | dx | ² we denote the Euclidean metric).

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1 Constant$\boldsymbol{Q}$-Curvature Metrics on $\mathbb{R}^{2m}$ and Their Volumes

We consider solutions to the equation

$$\displaystyle{ (-\Delta )^{m}u = (2m - 1)!e^{2mu}\quad \text{in }\mathbb{R}^{2m}, }$$

(1)

satisfying

$$\displaystyle{ V:=\int _{\mathbb{R}^{2m}}e^{2mu(x)}dx <+\infty. }$$

(2)

Geometrically, if u solves (1)–(2), then the conformal metric g _u : = e ^2u | dx | ² has Q-curvature $Q_{g_{u}} \equiv (2m - 1)!$ and volume V (by | dx | ² we denote the Euclidean metric). For the definition of Q-curvature and related remarks, we refer to [2, Chapter 4] or to [6].

Notice that, up to the transformation $\tilde{u}:= u + c$, the constant (2m − 1)! in (1) can be changed into any positive number, but it is natural to choose (2m − 1)! because it is the Q-curvature of the round sphere S ^2m. This implies that the function $u_{1}(x) =\log \frac{2} {1+\vert x\vert ^{2}}$, which satisfies $e^{2u_{1}}\vert dx\vert ^{2} = (\pi ^{-1})^{{\ast}}g_{S^{2m }}$, is a solution to (1)–(2) with V = vol(S ^2m) (here, $\pi: S^{2m} \rightarrow \mathbb{R}^{2m}$ is the stereographic projection). Translations and dilations of u ₁ (i.e., Möbius transformations) then produce a large family of solutions to (1)–(2) with V = vol(S ^2m), namely

$$\displaystyle{ u_{x_{0},\lambda }(x):= u_{1}(\lambda (x - x_{0}))+\log \lambda =\log \frac{2\lambda } {1 +\lambda ^{2}\vert x - x_{0}\vert ^{2}},\quad x_{0} \in \mathbb{R}^{2m},\quad \lambda> 0. }$$

(3)

We shall call the functions $u_{x_{0},\lambda }$ spherical solutions to (1)–(2).

The question whether the family of spherical solutions in (3) exhausts the set of solutions to (1)–(2) has raised a lot of interest and it is by now well understood. For instance, in dimension 2 we have the following result:

Theorem 1 (Chen-Li [5])

Every solution to (1)–(2) with m = 1 is spherical.

On the other hand, for every m > 1, i.e., in dimension 4 and higher, it was proven by Chang–Chen [3] that the Problem (1)–(2) admits solutions which are non spherical. More precisely:

Theorem 2 (Chang–Chen [3])

For every m > 1 and $V \in (0,\mathrm{vol}(S^{2m}))$ there exists a solution to (1)–(2) .

Several authors have given analytical and geometric conditions under which a solution to (1)–(2) is spherical (see [4, 14, 16]), and have studied properties of non-spherical solutions, such as asymptotic behavior, volume and symmetry (see [9, 11, 15]). In particular Lin proved:

Theorem 3 (Lin [9])

Let u solve (1)–(2) with m = 2. Then either u is spherical (i.e., as in (3) ) or $V <\mathrm{ vol}(S^{4}).$

Spherical solutions are radially symmetric (i.e., of the form u( | x − x ₀ | ) for some $x_{0} \in \mathbb{R}^{2m}$) and the solutions given by Theorem 2 might a priori all be spherically symmetric. The fact that this is not the case was proven by Wei–Ye in dimension 4:

Theorem 4 (Wei–Ye [15])

For every $V \in (0,\mathrm{vol}(S^{4}))$ there exist (several) non-radial solutions to (1)–(2) for m = 2.

Remark 5

As recently shown by A. Hyder [7], the proof of Theorem 4 can be extended to higher dimension 2m ≥ 4, yielding several non-symmetric solutions to (1)–(2) for every V ∈ (0, vol(S ^2m)), but failing to produce solutions for V ≥ vol(S ^2m). As in the proof of Theorem 2, the condition V < vol(S ^2m) plays a crucial role.

Theorems 2–4 and Remark 5 strongly suggest that, also in dimension 6 and higher, all non-spherical solutions to (1)–(2) satisfy V < vol(S ^2m), i.e., (1)–(2) has no solution for V > vol(S ^2m) and the only solutions with V = vol(S ^2m) are the spherical ones. Quite surprisingly it turns out that this is not at all the case. In fact, in dimension 6 there are solutions to (1)–(2) with arbitrarily large V:

Theorem 6 (Martinazzi [13])

For m = 3 there exists V ^∗ > 0 such that for every V ≥ V ^∗ there is a solution u to (1)–(2) , i.e., there exists a metric on $\mathbb{R}^{6}$ of the form g _u = e ^2u |dx| ² satisfying $Q_{g_{u}} \equiv 5!$ and $\mathrm{vol}(g_{u}) = V.$

The proof of Theorem 6 is based on a ODE argument: one considers radial solutions to (1)–(2), so that (1) reduces to an ODE. Precisely, given $a \in \mathbb{R}^{}$ let u = u _a (r) be the solution of

$$\displaystyle{ \left \{\begin{array}{l} \Delta ^{3}u = -120e^{6u}\quad \text{in }\mathbb{R}^{6} \\ u(0) = u^{{\prime}}(0) = u^{{\prime\prime\prime}}(0) = u^{{\prime\prime\prime}{\prime\prime}}(0) = 0,\quad u^{{\prime\prime}}(0) = -a,\quad u^{{\prime\prime\prime}{\prime}}(0) = 1. \end{array} \right. }$$

(4)

Then one shows that

$$\displaystyle{\int _{\mathbb{R}^{6}}e^{6u_{a} }dx <\infty \mbox{ for $a$ large,}\quad \lim _{a\rightarrow \infty }\int _{\mathbb{R}^{6}}e^{6u_{a} }dx = \infty.}$$

In particular the conformal metric $g_{u_{a}} = e^{2u_{a}}\vert dx\vert ^{2}$ of constant Q-curvature $Q_{g_{u_{a}}} \equiv 5!$ satisfies

$$\displaystyle{ \mathrm{vol}(g_{u_{a}}) <\infty \mbox{ for $a$ large,}\quad \lim _{a\rightarrow \infty }\mathrm{vol}(g_{u_{a}}) = \infty. }$$

(5)

Theorem 6 then follows from (5) and the remark that the quantity $\mathrm{vol}(g_{u_{a}})$ is a continuous function of a when a is sufficiently large (this seems to be false in general if a > 0 is not large enough).

The proof of Theorem 2, which is variational and based on the sharpness of Beckner’s inequality [1], does not extend to the case V > vol(S ^2m). On the other hand with the previous ODE approach one can prove that, at least when m ≥ 3 is odd, Theorem 2 extends as follows.

Theorem 7 (Martinazzi [13])

Set $V _{m}:= \frac{(2m)!} {4(m!)^{2}} \mathrm{vol}(S^{2m})>\mathrm{ vol}(S^{2m})$ . Then, for m ≥ 3 odd and for every V ∈ (0,V _m ], there is a non-spherical (but radially symmetric) solution u to (1)–(2) , i.e., there exists a metric on $\mathbb{R}^{2m}$ of the form g _u = e ^2u |dx| ² satisfying $Q_{g_{u}} \equiv (2m - 1)!$ and $\mathrm{vol}(g_{u}) = V.$

The condition m ≥ 3 odd is (at least in part) necessary in view of Theorems 1 and 3, but the case m ≥ 4 even is open. Notice also that, when m = 3, Theorems 6 and 7 guarantee the existence of solutions to (1)–(2) for

$$\displaystyle{V \in (0,V _{m}] \cup [V ^{{\ast}},\infty ),}$$

but do not rule out that V _m  < V ^∗ and the existence of solutions to (1)–(2) is unknown for V ∈ (V _m , V ^∗). Could there be a gap phenomenon?

We remark that the case m even is more difficult to treat since the ODE corresponding to (1), in analogy with (4), becomes

$$\displaystyle{\Delta ^{m}u(r) = (2m - 1)!e^{2mu(r)},\quad r> 0,}$$

whose solutions can blow up in finite time (i.e., for finite r) if the initial data are not chosen carefully (contrary to what happens when m is odd).

2 Negative Curvature and Odd Dimension

It is natural to investigate how large the volume of a metric g _u  = e ^2u | dx | ² on $\mathbb{R}^{2m}$ can be, also with constant and negative Q-curvature $Q_{g_{u}} <0$. Again with no loss of generality we assume $Q_{g_{u}} \equiv -1$. In other words, consider the problem

$$\displaystyle{ (-\Delta )^{m}u = -e^{2mu}\quad \text{on }\mathbb{R}^{2m}, }$$

(6)

subject to condition (2). Although for m = 1 it is easy to see that Problem (6)–(2) admits no solution for any V > 0, when m ≥ 2 we have

Theorem 8 (Martinazzi [10])

For any m ≥ 2 Problem (6)–(2) has solutions for some V > 0.

Using the fixed point argument from [15] and a compactness result from [12], Hyder–Martinazzi recently proved:

Theorem 9 (Hyder–Martinazzi [7])

For any m ≥ 2 and any V > 0 Problem (6)–(2) has solutions.

Also the odd-dimensional case is interesting, but more delicate since (1) becomes a non-local equation for $m = (k + 1)/2$, $k \in \mathbb{N}$. Building upon previous results from [3, 9, 16], we recently proved the following existence result:

Theorem 10 (Jin–Maalaoui–Martinazzi–Xiong [8])

Fix $m = 3/2$ . For every V ∈ (0,2π ² ], Problem (1)–(2) has a solution (where $(-\Delta )^{\frac{3} {2} }$ needs to be suitably defined). Moreover, if u is a non-spherical solution to (1)–(2) , then $V <2\pi ^{2} =\mathrm{ vol}(S^{3})$ .

It is interesting to compare the volume restrictions of Theorems 3 and 10 for dimension 3 and 4, with the results of Theorems 6 and 7 for dimension 6 and higher. It is then natural to ask what does the situation look like in dimension 5, i.e., when $m = 5/2$.

Conjecture 11

Problem (1)–(2) for $m = 5/2$ admits solutions for some values of V > vol(S ⁵).

In other words, we conjecture that dimension 5 is similar to dimension 6 more than to dimension 4. The intuition behind this is that the kernel of $(-\Delta )^{5/2}$ contains polynomials of degree 4, just as the kernel of $(-\Delta )^{3}$, while the kernels of $(-\Delta )^{3/2}$ and $(-\Delta )^{2}$ contain polynomials of degree 2 but not of degree 4, which is crucial in the proofs of Theorems 3 and 10. On the other hand, we remark that there seems to be no chance to extend the proofs of Theorem 6 to dimension 5, since ODE techniques do not fit well in a non-local framework.