A geodesic segment before its first conjugate point is locally energy minimizing

The full $H^1$ energy bound. [F1, F2, F3, F4, given] Take $\epsilon$ from [F2] and an admissible $\alpha$. Applying [F4] to $s=\mathrm{\mid }\dot{\alpha }{\mathrm{\mid }}_g\in L^2$ and the constant function $1$, using [F1], gives $L_g(\alpha {)}^2\le T{\int }_a^b{s}^2=2TE(\alpha )$. Both lengths are nonnegative and [F2] gives $L_g(\alpha )\ge L_g(\gamma )$. By [F3], $E(\alpha )\ge \frac{L_g(\alpha {)}^2}{2T}\ge \frac{L_g(\gamma {)}^2}{2T}=E(\gamma )\mathrm{.}$ This argument uses the $L^2$ speed of every $H^1$ competitor, not a smooth approximation of the equality case.

Equality forces constant speed and the radial reparametrization. [F2, F3, F4, step 1.1, given] If $E(\alpha )=E(\gamma )$, both inequalities in step 1.1 are equalities. Thus $L_g(\alpha )=L_g(\gamma )$. Put $s=\mathrm{\mid }\dot{\alpha }{\mathrm{\mid }}_g$ and $m=L_g(\alpha )\mathrm{/}T$. Expanding the square using $\int s=Tm$ gives ${\int }_a^b(s-m{)}^2={\int }_a^b{s}^2-T{m}^2=2E(\alpha )-L_g(\alpha {)}^2\mathrm{/}T=0$. By the zero-integral clause of [F4], $s=m$ a.e., and by [F3] $m=\mathrm{\mid }v{\mathrm{\mid }}_g$. If $v=0$, [F2] already gives $\alpha =\gamma$. Suppose $v\mathrm{\ne }0$. The equality clause of [F2] produces an absolutely continuous nondecreasing $H^1$ surjection $\tau$ fixing $a,b$ and satisfying $\alpha =\gamma \circ \tau$.

The only equal-energy reparametrization is the identity. [F1, F3, step 2.1, given] The a.e. chain rule [F1] applied to the smooth geodesic and $H^1$ function $\tau$ gives $\dot{\alpha }(t)=\dot{\gamma }(\tau (t)){\tau }^{\mathrm{\prime }}(t)$ a.e. Since $\tau$ is nondecreasing and a primitive, ${\tau }^{\mathrm{\prime }}\ge 0$ a.e.; [F3] then gives $\mathrm{\mid }\dot{\alpha }{\mathrm{\mid }}_g=\mathrm{\mid }v{\mathrm{\mid }}_g{\tau }^{\mathrm{\prime }}$ a.e. By step 2.1 the left side equals $\mathrm{\mid }v{\mathrm{\mid }}_g>0$ a.e., so ${\tau }^{\mathrm{\prime }}=1$ a.e. The primitive identity for $\tau$ and $\tau (a)=a$ yield $\tau (t)=a+{\int }_a^{t}1ds=t$ for every $t$. Hence $\alpha =\gamma$. Conversely $\gamma$ is admissible and has its own energy, proving the equality clause.

Locality and boundaries. [A1, F1, step 1.1, step 3.1, given] The radius in [F2] is positive, and [F1] makes its uniform neighborhood an open neighborhood in the fixed-endpoint $H^1$ path topology. The preceding steps prove strictness there. For a constant geodesic, step 2.1 handles equality, including dimension zero. Only [A1] and the stated local suppliers are used; no global compactness or geodesic completeness enters.