dg.differential geometry - What is the easiest way to classify all possible smooth orientable closed 2-manifolds?

Consider harmonic functions $f$ with exactly 2 log-singularities of weight $pm 1.$ (locally $f(z)=a logparallel zparallel+g,$ $g$ being smooth at $z=0$, $a$ being the weight) on your compact surface equipped with a Riemann metric. They exists by standard elliptic theory ( the two weights $a_1$ and $a_2$ have two add to zero). Consider $partial f,$ the complex linear part of the differential. This is a meromorphic section of your canonical bundle. Then $degpartial f=-frac{1}{2pi i}int KdA,$ as the Levi-Civita connection defines a complex linear connection on the canonical bundle.

This shows that if the total curvature is large enough $geq 4pi,$ $f$ will not have critical points (only two singularities). Moreover $e^f$ is the real part of a holomorphic bijection onto $CP^1.$

If $f$ has a critical point, then you can easily construct a non-sepreating loop, as in Morse theoretic proofs. You cut your surface, and add two disc (with the right orientation). One, can easily see, that this must increase the total curvature by $4pi,$ and you end up with the two-sphere after a finite number of steps.