PomeTheorem 17.16.25.944944D6FF68 For every nonzero constant $c$, the full Perron segment $u_ t=ct$ is stationary for the raw action, whose Hessian has an infinite-dimensional negative subspace. The infimum of raw lengths over all smooth strictly positive paths from $0$ to $c$ is strictly smaller than $d_ B(0,c)=\sqrt{2n}|c|$.
Theorem 17.16.25944944D6FF68
Proof. The constant path is basic, equals the chord, and is the full Perron segment by corollary 17.16.14. Its stationarity also follows from proposition 17.16.22. Let $w(t,r)$ be smooth, two-periodic in $r$, and zero at $t=0,1$. The paths $ct+\varepsilon w$ have positive spatial forms for sufficiently small $|\varepsilon |$. Expanding (17.16.24) to second order gives
\[ 2n\int _0^1\left\langle w_ t^2+2cw_ t(dw_ r+w_{rr}) +c^2\left(\frac{d(d-1)}2w_ r^2+dw_ rw_{rr}\right) \right\rangle \mathrm dt. \]
The integral of $w_ rw_{rr}$ over a period vanishes, while $\int _0^1\langle w_ tw_{rr}\rangle \mathrm dt =-\frac12[\langle w_ r^2\rangle ]_0^1=0$. Completing the square gives the Hessian
F20F3B72C244
17.16.26
\begin{equation} 2n\int _0^1\left\langle (w_ t+dcw_ r)^2-\frac{d(d+1)}2c^2w_ r^2 \right\rangle \mathrm dt. \end{equation}
For a positive integer $k$, take
\[ w_ k(t,r)=\sin (\pi t)\cos \bigl (\pi k(r-dct)\bigr ). \]
Then $w_{k,t}+dcw_{k,r}=\pi \cos (\pi t)\cos (\pi k(r-dct))$, and (17.16.26) is
\[ \frac{n\pi ^2}2\left(1-\frac{d(d+1)}2c^2k^2\right). \]
It is negative for all sufficiently large $k$. Distinct modes and their derivatives are orthogonal under the $r$ average, so the quadratic form is negative on every nonzero finite combination of these modes. Each combination gives a positive path for sufficiently small variation parameter and has action strictly below $nc^2$. Its raw length satisfies $L^2\leq 2\mathscr S$ on the unit interval, so it is strictly below $\sqrt{2n}|c|$. $\square$
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