Example 11.26.6.8C86B880454A On a sufficiently small ball in $\mathbb C^2$, equip the trivial bundle of rank two with the metric
\[ H(z)=\operatorname {Id}-zz^*+\tfrac 12|z|^2\operatorname {Id}. \]
At the origin this metric has zero mean curvature for the Euclidean Hermitian form and is RC-positive, but it is not uniformly RC-positive.
Proof.
The two eigenvalues of $H(z)$ are $1-\tfrac 12|z|^2$ and $1+\tfrac 12|z|^2$, so the metric is positive on a small ball. Its first derivatives vanish at the origin, where
\[ F_{i\bar j}=e_ ie_ j^*-\tfrac 12\delta _{ij}\operatorname {Id},\qquad h(R_ h(u,\bar u)v,v)=|v^*u|^2-\tfrac 12|u|^2|v|^2. \]
The sum $F_{1\bar1}+F_{2\bar2}$ is zero. For each nonzero $v$, taking $u=v$ gives a positive evaluation. For each nonzero $u$, taking $v\perp u$ gives a negative evaluation, so no single $u$ works for every $v$.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.