Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 14E / c

If $K<0$ everywhere, compactness makes the integral negative, so $\chi (S)<0$. For every triangulation, $3F=2E$ again yields the general identitySince $V\ge 1$ and $\chi (S)<0$,This supplies bounds depending only on the surface.