2.II.12A

Let $U$ be an open subset of $\mathbf{R}^{2}$ equipped with a Riemannian metric. For $\gamma:[0,1] \rightarrow U$ a smooth curve, define what is meant by its length and energy. Prove that length $(\gamma)^{2} \leq \operatorname{energy}(\gamma)$, with equality if and only if $\dot{\gamma}$ has constant norm with respect to the metric.

Suppose now $U$ is the upper half plane model of the hyperbolic plane, and $P, Q$ are points on the positive imaginary axis. Show that a smooth curve $\gamma$ joining $P$ and $Q$ represents an absolute minimum of the length of such curves if and only if $\gamma(t)=i v(t)$, with $v$ a smooth monotonic real function.

Suppose that a smooth curve $\gamma$ joining the above points $P$ and $Q$ represents a stationary point for the energy under proper variations; deduce from an appropriate form of the Euler-Lagrange equations that $\gamma$ must be of the above form, with $\dot{v} / v$ constant.

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