14.26.9 Two forms with the same cubeDBEDC79E97CD

The cancellation identities now turn the scalar coefficients into two semipositive forms with the same cube. Their average is strictly positive except at the two ends of the central interval.

By the extensions described above, the form and the function

are smooth on $X$. The form is semipositive, and positive except on the hypersurfaces $M_- =\{ r=-1\} $ and $M_+=\{ r=1\} $. Put

For $|r|\geq 1$ we have $H=0$. On $|r|<1$,

All coefficients are nonnegative, and by (14.26.4) and (14.26.8), $A_0^3=B_0^3=0$ on this band. Outside it both cubes equal $\omega _0^3$. Thus $A_0,B_0$ are smooth semipositive forms on $X$ with equal cubes. This common top-degree form vanishes on the closed band $|r|\leq 1$ and is positive on its complement, including near $X\setminus U$.