I have been reading about tessellation representations of compact surfaces, such as how the square tiling the plane represents the torus. For surfaces of genus > 1 (the ones that interest me), we need to move to the hyperbolic plane in order to obtain a tessellation representation, which makes sense because of Gauss-Bonnet's restriction that the total curvature of a surface of genus > 1 be negative.

But, since we had to move to the hyperbolic plane in order to obtain a tessellation, my question is this: can we only find tessellation representations of compact surfaces of genus > 1 when we impose constant negative curvature on them? Or does the tessellation representation look the same when I change curvature on certain sets? Or maybe, is it slightly perturbed but still a tessellation? If my questions are completely misguided, could someone suggest some literature? Thank you!