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 $d_ B(0,c)=\sqrt{2n}|c|$.
17.16.23 Negative directions of the raw action124D4C83F5E6
On this Hopf background, constant Perron translations are stationary. Radial variations detect the failure of minimization. Every smooth two-periodic function of $r$ defines a smooth function on $X$, and for a radial path $u(t,r)$,
\[ \omega +\mathrm{dd}^{\mathrm{c}}u=(1+u_ r)\alpha _ n+(1+u_{rr})\beta _ n. \]
For $\langle F\rangle =\frac12\int _0^2F(r)\, \mathrm dr$, fiber integration therefore gives
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17.16.24
\begin{equation} \mathscr S(u)=n\int _0^1 \left\langle u_ t^2(1+u_ r)^ d(1+u_{rr})\right\rangle \mathrm dt. \end{equation}
Changing the local base section shifts the start of the full period and leaves the average unchanged.
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
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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 $\sqrt{2n}|c|$. $\square$
These competing paths need not be subgeodesics. The strict length inequality does not establish definiteness or completeness of the unrestricted length infimum. For $n=3$ and $c=1$, a concrete path is
\[ u_\varepsilon (t,r)=t+\varepsilon \sin (\pi t)\cos \bigl (\pi (r-2t)\bigr ). \]
Its spatial coefficients are at least $1-\pi |\varepsilon |$ and $1-\pi ^2|\varepsilon |$. Its exact action is
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17.16.27
\begin{equation} \mathscr S(u_\varepsilon ) =3-\frac{3\pi ^2}2\varepsilon ^2+\frac{111\pi ^4}{64}\varepsilon ^4. \end{equation}
Indeed, put $f=\sin (\pi t)$, $p=f'$, and $\vartheta =\pi (r-2t)$. Substitute $u_ t=1+\varepsilon (p\cos \vartheta +2\pi f\sin \vartheta )$, $u_ r=-\varepsilon \pi f\sin \vartheta $, and $u_{rr}=-\varepsilon \pi ^2f\cos \vartheta $ into (17.16.24). Odd powers have zero periodic average. The averaged quadratic coefficient is $(p^2-3\pi ^2f^2-2\pi ^2fp)/2$, and the quartic coefficient is $\pi ^2f^2(p^2+6\pi ^2fp+12\pi ^2f^2)/8$. The identities
\[ \begin{gathered} \int _0^1fp\, \mathrm dt=\int _0^1f^3p\, \mathrm dt=0, \\ \int _0^1f^2\, \mathrm dt=\frac12, \quad \int _0^1p^2\, \mathrm dt=\frac{\pi ^2}2,\\ \int _0^1f^4\, \mathrm dt=\frac38, \quad \int _0^1f^2p^2\, \mathrm dt=\frac{\pi ^2}8 \end{gathered} \]
give (17.16.27). Taking $\varepsilon =1/100$ gives a positive path with action $2.9985212487\ldots <3$.
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