Weyl Curvature, Del Pezzo Surfaces, and Almost-Kähler Geometry

Abstract

If a smooth compact $4$-manifold $M$ admits a Kähler–Einstein metric $g$ of positive scalar curvature, Gursky (Ann Math 148:315–337, 1998) showed that its conformal class $[g]$ is an absolute minimizer of the Weyl functional among all conformal classes with positive Yamabe constant. Here we prove that, with the same hypotheses, $[g]$ also minimizes of the Weyl functional on a different open set of conformal classes, most of which have negative Yamabe constant. An analogous minimization result is then proved for Einstein metrics $g$ which are Hermitian, but not Kähler.

Appendix The Yamabe Invariant

In this appendix, we contrast Gursky’s theorem [24] with Theorem B by showing that “most” symplectic conformal classes on any $4$-manifold have negative Yamabe constant. For related results, see [27, 28].

Proposition 4

Let $(M^4 ,\omega )$ be any compact symplectic $4$-manifold. Then there are sequences of conformal classes $[g_k]$ on $M$ which are compatible with $\omega$, and have Yamabe constants $Y([g_k]) \rightarrow -\infty$. Moreover, among all $\omega$-compatible conformal classes, those with negative Yamabe constant are dense in the $C^0$ topology.

Proof

The Yamabe constant of a conformal class $[g]$ on a compact $4$-manifold $M$ is given by

$$\begin{aligned} Y([g]) = \inf _{g^\prime \in [g]} \frac{\int _M s_{g^\prime }d\mu _{g^\prime }}{\sqrt{\int _M \,d\mu _{g^\prime }}} , \end{aligned}$$

so it suffices to produce a sequence of almost-Kähler metrics $g_k$ adapted to $\omega$ such that $\int s_{g_k}\,d\mu _{g_k}\rightarrow -\infty$; here we are using the fact that all such metrics have the same volume form $d\mu =\omega ^2/2$, and hence the same total volume. On the other hand, for any almost-Kähler metric $g$,

$$\begin{aligned} 4\pi c_1\cdot [\omega ] = \int \limits _M \frac{s+s^*}{2}d\mu = \int \limits _M \left( s+ \frac{|\nabla \omega |^2}{2}\right) d\mu , \end{aligned}$$

so it suffices to show that we can choose $g_k$ adapted to $\omega$ so that

$$\begin{aligned} \Vert \nabla \omega \Vert _{L^2}\rightarrow +\infty . \end{aligned}$$

Of course, a choice of almost-Kähler metric $g$ compatible with $\omega$ is equivalent to the choice of an almost-complex structure $J$ such that $\omega$ is $J$-invariant; the metric is then given by

$$\begin{aligned} g = \omega (\cdot , J \cdot ), \end{aligned}$$

and, with respect to this metric, $J$ then just becomes $\omega$ with an index raised. The Nijenhuis tensor $N_J$, given by

$$\begin{aligned} N_J (X,Y)=( \nabla _X J)(Y)-(\nabla _Y J)(X)-( \nabla _{JX} J)(JY)+(\nabla _{JY} J)(JX), \end{aligned}$$

is then just four times the projection of $\nabla J$ to $(\Lambda ^{0,2}_J\oplus \Lambda ^{2,0}_J)\otimes TM$, and represents the O’Neill tensor of $T^{0,1}_J$ in that

$$\begin{aligned}{}[X+iJX,Y+iJY]^{1,0}=J N_J(X,Y)+iN_J(X,Y). \end{aligned}$$

It thus suffices to produce a sequence $J_k$ of $\omega$-compatible almost-complex structures which satisfy

$$\begin{aligned} \Vert N_{J_k}\Vert _{L^2}\rightarrow \infty , \end{aligned}$$

where the norms are to be computed with respect to the associated sequence of metrics

$$\begin{aligned} g_k = \omega (\cdot , J_k \cdot ). \end{aligned}$$

This can be done via an entirely local construction. First, choose an arbitrary background almost-Kähler metric $g$ adapted to $\omega$, which amounts to choosing a background $\omega$-compatible almost-complex structure $J$. Now take a Darboux chart $(x,y,u,v)$ centered on an arbitrary point $p\in M$, so that

$$\begin{aligned} \omega = dx\wedge dy + du\wedge dv \end{aligned}$$

(14)

and such that $J$ coincides with the standard Euclidean almost-complex structure at the origin, which represents $p$ in these coordinates. Next, by freezing the almost-complex structure near the origin, introduce a perturbed back-ground almost-Kähler metric $g_0=g_{0,\varepsilon }$ which agrees with the standard Euclidean metric on, say, the coordinate ball of radius $3\varepsilon$ about the origin, but agrees with $g$ outside the ball of radius $4\varepsilon$; and notice that we can do this so that $g_{0,\varepsilon }\rightarrow g$ in the $C^0$ topology as $\varepsilon \rightarrow 0$. Next, for any fixed $\varepsilon$, notice that we can construct a new almost-Kähler metric $g_f$ associated with an almost-complex structure given by

$$\begin{aligned} J_f= e^{2f} dx \otimes \frac{\partial }{\partial y}- e^{-2f} dy \otimes \frac{\partial }{\partial x} + du \otimes \frac{\partial }{\partial v}- dv \otimes \frac{\partial }{\partial u} \end{aligned}$$

on the $(3\varepsilon )$-ball, where $f$ is a smooth function supported in the ball of radius $2\varepsilon$; we then extend $J$ to all of $M$ by taking it to coincide with the almost-complex structure $J_0$ of $g_0$ outside the $(2\varepsilon )$-ball. The corresponding metric $g_f$ is given by

$$\begin{aligned} g_f= e^{2f}dx^2 + e^{-2f}dy^2 + du^2 + dv^2 \end{aligned}$$

(15)

on the region in question, and

$$\begin{aligned} (e^fdx+ie^{-f}dy) \left( \left[ e^{-f} \frac{\partial }{\partial x} + i e^f \frac{\partial }{\partial y}, \frac{\partial }{\partial u} +i \frac{\partial }{\partial v}\right] \right) = 2\left( \frac{\partial }{\partial u} +i \frac{\partial }{\partial v}\right) f \end{aligned}$$

is a component of $JN_J+iN_J$ in an orthonormal frame. Now let $g_k=g_{k,\varepsilon }$ be the sequence of almost-Kähler metrics $g_{f_k}$ associated with the sequence of functions $f_k = \frac{1}{k} \sin (\frac{2\pi k^2}{\varepsilon } v) \cdot \phi$, where the cut-off function $\phi : M\rightarrow [0,1]$ is supported in the $(2\varepsilon )$-ball and $\equiv 1$ on the $\varepsilon$-ball. The Nijenhuis tensors of the corresponding almost-complex structures then satisfy

$$\begin{aligned} \Vert N_{J_k}\Vert _{L^2}^2 > \int \limits _{[-\frac{\varepsilon }{2}, \frac{\varepsilon }{2}]^4} |\frac{\partial f}{\partial v}|^2\, \frac{\omega ^2}{2} = 2\pi ^2 k^2 \varepsilon ^2 \rightarrow + \infty \end{aligned}$$

with respect to the associated metrics.

In particular, for fixed $\varepsilon$, the Yamabe constant of $[g_k] = [g_{k,\varepsilon }]$ is negative for all large $k$. However, the above choice of $f_k$ converges to $0$ in $C^0$, so that $g_{k,\varepsilon }\rightarrow g_{0,\varepsilon }$ in the $C^0$ topology as $k\rightarrow \infty$. On the other hand, we also have $g_{0,\varepsilon }\rightarrow g$ in the $C^0$ topology as $\varepsilon \rightarrow 0$. By choosing a suitably large $k(j)$ for each $\varepsilon = 2^{-j}$, we therefore obtain a sequence of $\omega$-compatible metrics converging to the given $\omega$-compatible metric $g$ in $C^0$, even though their conformal classes have negative Yamabe constants.$\square$

In fact, Jongsu Kim [27, 28] has proved much stronger and more difficult results in this direction. Indeed, he shows that one can always construct $\omega$-compatible almost-Kähler metrics $g$ for which the scalar curvature is everywhere negative, even without conformal rescaling.

Finally, we point out that this phenomenon persists in the toric setting:

Proposition 5

Let $(M^4,\omega )$ be a Hamiltonian $T$-space. Then there are $\omega$-compatible, $T^2$-invariant almost-Kähler metrics $g_k$ such that $Y([g_k]) \rightarrow -\infty$.

Proof

On a neighborhood of a $T^2$ orbit, we can again take coordinates $(x,y,u,v)$ so that (14) holds, with $y,v$ coordinates on the base, and with $(x,u)$ now $({\mathbb {R}}/{\mathbb {Z}})$-valued fiber coordinates adapted to the action. We again consider metrics $g_f$ as in (15), but with $f$ now a function of $(y,v)$ only. Choosing a sequence $f_k$ of such functions which are highly oscillatory in $v$ in a small region makes the $L^2$ norm of the Nijenhuis tensor tend to infinity, so that $\int s_{g_k}\,d\mu _{g_k} \rightarrow -\infty$, while the volume $[\omega ]^2/2$ remains fixed. Hence $Y([g_{f_k}])\rightarrow -\infty$, as claimed.$\square$