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A Zoll counterexample to a geodesic length conjecture. (English) Zbl 1201.53050
Consider 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 $$t> 0$$.

##### MSC:
 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C22 Geodesics in global differential geometry
##### Keywords:
closed geodesic; sphere Guillemin deformation; Zoll surface
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