Suppose I have a cusp in a Riemann surface with a shrinking geodesic which is degenerating towards a surface with a cusp, and I consider a neighborhood of the shrinking geodesic, $C_[a,b] = [a,b]\times \mathbb{T}^1$ and a harmonic function $g$ on $C_[a,b]$, and I want a bound on $|g|$ which is independent of the radius of the cusp. "cusp". $g$ here is determined up to an additive constant. I thought I could do this using Harnack, since by adding a constant we can assume $\inf_{C[-\log{2},1]} {g} = 1$ and then Harnack bounds $|g|$ on a smaller $C$. It was pointed out to me that Harnack won't work in this case since I only have one degree of freedom so I can choose to make $g$ positive or make the inf = 1 but not necessarily both at once. I'm still quite sure this should be true, but any ideas about how to prove it greatly appreciated.
Suppose I have a cusp in a Riemann surface, $C_[a,b] = [a,b]\times \mathbb{T}^1$ and a harmonic function $g$ on $C_[a,b]$, and I want a bound on $|g|$ which is independent of the radius of the cusp. $g$ here is determined up to an additive constant. I thought I could do this using Harnack, since by adding a constant we can assume $\inf_{C[-\log{2},1]} {g} = 1$ and then Harnack bounds $|g|$ on a smaller $C$. It was pointed out to me that Harnack won't work in this case since I only have one degree of freedom so I can choose to make $g$ positive or make the inf = 1 but not necessarily both at once. I'm still quite sure this should be true, but any ideas about how to prove it greatly appreciated.