A Zoll counterexample to a geodesic length conjecture.

*(English)*Zbl 1201.53050Consider a Riemannian metric on the 2-sphere and denote by \(D\) its diameter and by \(L\) the length of its shortest non trivial geodesic. It is known that \(L\leq 4D\), and the inequality \(L\leq 2D\) has been conjectured by A. Nabutovsky and R. Rotman. Among the known surfaces satisfying \(L= 2D\) there are the rotationally symmetric Zoll surfaces. A Zoll surface is a surface all of whose geodesics are closed and whose prime geodesics all have length \(2\pi\).

The authors give a counterexample proving that there exist other families of Zoll surfaces obtained as smooth variations of the round metric and satisfying \(L> 2D\). They use the existence theorem of Zoll surfaces due to Guillemin stating that for every \(f\in C^\infty_{\text{odd}}(S^2,\mathbb{R})\) there exists a smooth 1-parameter family \(g_t= \Psi^f_g g_0\) of smooth Zoll metrics such that \(\Psi^f_0= 1\), \({d\Phi^f_t\over dt}\biggl|_{t= 0}= f\) and all prime geodesics of \((S^2,g_t)\) have length \(2\pi\). The key notions are those of \(Y\)-like subset and amply negative function. More precisely a subset of the unit circle is called \(Y\)-like if it contains a triple of vectors \((u,v,w)\) such that there exist positive real numbers \(a\), \(b\), \(c\) satisfying \(au+ bv+ cw= 0\). A subset of the unit tangent bundle of \(S^2\) is said to be \(Y\)-like if its intersection with the unit tangent vectors at \(p\) is \(Y\)-like for each \(p\in S^2\). An odd function \(f\) is said to be amply negative if the set of unit tangent directions to great half-circles \(\tau\) satisfying \(\int_\tau f\,ds_0< 0\) is a \(Y\)-like subset of the unit tangent bundle of \(S^2\). The authors prove the existence of amply negative functions.

Finally, the main result reads: If \(f\) is an amply negative function then the smooth variation \(g_t= \Psi^f_t g_0\) of the round metric \(g_0\) by smooth Zoll metrics satisfies \(L(g_t)> 2D(g_t)\) for sufficiently small \(t> 0\).

The authors give a counterexample proving that there exist other families of Zoll surfaces obtained as smooth variations of the round metric and satisfying \(L> 2D\). They use the existence theorem of Zoll surfaces due to Guillemin stating that for every \(f\in C^\infty_{\text{odd}}(S^2,\mathbb{R})\) there exists a smooth 1-parameter family \(g_t= \Psi^f_g g_0\) of smooth Zoll metrics such that \(\Psi^f_0= 1\), \({d\Phi^f_t\over dt}\biggl|_{t= 0}= f\) and all prime geodesics of \((S^2,g_t)\) have length \(2\pi\). The key notions are those of \(Y\)-like subset and amply negative function. More precisely a subset of the unit circle is called \(Y\)-like if it contains a triple of vectors \((u,v,w)\) such that there exist positive real numbers \(a\), \(b\), \(c\) satisfying \(au+ bv+ cw= 0\). A subset of the unit tangent bundle of \(S^2\) is said to be \(Y\)-like if its intersection with the unit tangent vectors at \(p\) is \(Y\)-like for each \(p\in S^2\). An odd function \(f\) is said to be amply negative if the set of unit tangent directions to great half-circles \(\tau\) satisfying \(\int_\tau f\,ds_0< 0\) is a \(Y\)-like subset of the unit tangent bundle of \(S^2\). The authors prove the existence of amply negative functions.

Finally, the main result reads: If \(f\) is an amply negative function then the smooth variation \(g_t= \Psi^f_t g_0\) of the round metric \(g_0\) by smooth Zoll metrics satisfies \(L(g_t)> 2D(g_t)\) for sufficiently small \(t> 0\).

Reviewer: Anna Maria Pastore (Bari)