# Geodesic curvature on hyperbolic manifold with boundary

Let $$\Sigma$$ be a compact oriented surface of genus $$1$$ having a single boundary component (i.e. $$T^2$$ minus an open disk) and let $$g$$ be a Riemannian metric on $$\Sigma$$ with constant Gaussian curvature $$K=-1$$. Is it necessarily the case that $$\int_{\partial \Sigma} k_gds \leq 0$$? The Gauss-Bonnet theorem gives a lower bound $$\int_{\partial \Sigma} k_gds = \text{Vol}(\Sigma)-2\pi \geq -2\pi$$ but I am wondering about an upper bound.

• The boundary can even be strictly convex. Jan 5 at 0:13
• For examples, think about circles in the hyperbolic plane. (You can always build a compact hyperbolic surface by gluing the sides of an appropriate geodesic $4g$-gon in the hyperbolic plane.) Circles can have any geodesic curvature $>1$. Jan 5 at 0:58

Given a topological surface $$S$$ which is the torus with closed disk removed, there is are two types of complete hyperbolic structures $$h$$ on $$S$$:
1. The ones of finite (hyperbolic area) area. In this case, the only "end" of $$(S,h)$$ is a "cusp."
2. The ones of infinite area, where the the only "end" of $$(S,h)$$ is a funnel.
I will be using the 2nd type. The hyperbolic surface $$(S,h)$$ in this case has the following decomposition: There exists a unique simple closed geodesic $$c$$ on $$(S,h)$$ such that cutting $$S$$ open along $$c$$ results in a compact hyperbolic surface with geodesic boundary and an annulus $$S$$. Then annulus $$S$$ is foliated by simple loops $$c_t$$ equidistant from $$c$$ (the distance from each point of $$c_t$$ to $$c$$ equals $$t>0$$). These curves are convex in $$(S,h)$$ and have strictly positive curvature: If I cut $$S$$ open along $$c_t$$, the result is a compact surface $$\Sigma_t$$ with convex boundary and an annulus $$A_t$$.
Now, take one of the surfaces $$\Sigma_t$$ as your surface $$\Sigma$$.
To get a better idea about the nature of the curves $$c_t$$, lift the closed geodesic $$c$$ to a biinfinite geodesic $$\tilde{c}$$ in the hyperbolic plane (the universal cover of $$(S,h)$$). I will be using the upper half-plane model of the hyperbolic plane. Then one can arrange it so that $$\tilde{c}$$ is given by $$\{(x,y): x> 0, y=0\},$$ the positive $$y$$-semiaxis. Then each $$c_t$$ lifts to a straight line $$y=m_t x, x>0$$; WLOG, $$m_t>0$$ for all $$t$$.