Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2

The paper proves Theorem 1.2 with a clean split into the cases e<0 and e>0, using the S1-moment map K̃=ιΘα (constant along the Reeb flow) and a careful reduction to invariant surfaces of section. For e<0, the scaling argument α=K̃·α0 gives Vol(α)=∫(K̃)²α0∧dα0 and sys(α)≤min|K̃|, hence sys(α)²≤Vol(α)/|e| with equality iff α is Zoll, exactly as stated (Proposition 2.4) . For e>0, the paper constructs an S1-invariant union of cylinders Σ as a surface of section (Proposition 3.1) and decomposes α=Kσ+π*β on M\Z, leading to a family of potentials Ji on intervals of regular values. The key lower bound Vol(α)≥2∫_{-Kmin}^{Kmin}J(k)dk (from Lemma 3.8 plus positivity, Lemma 3.11) together with the inequality ∫ J≥sys(α)² (via the auxiliary function ge(k)=J(k)+c|k| and Corollary 3.7) yields sys(α)²≤Vol(α)/|e| for e∈{1,2} and sys(α)²≤Vol(α)/2 for e≥2; equality holds iff α is Zoll, and when e>2 equality cannot occur; nevertheless the sharp constant 1/2 is attained as a supremum by a sequence (Proposition 4.1) . By contrast, the candidate solution misrepresents several core steps: (i) it claims periodic orbits arise at critical points of the potential with period 2h(s), while the paper shows closed orbits correspond to rational values of J′ and have period q(J−kJ′) (Corollary 3.7) ; (ii) it states a volume identity Vol(α)=C_e∫h with C_e depending on e, whereas the paper uses a lower bound Vol(α)≥2∫J and the constant 2 is geometric, not e-dependent (equation (3)) ; (iii) it invokes a sharp one-dimensional inequality in terms of min h that does not appear in the paper’s proof, which instead proceeds via the function ge and an integral estimate; and (iv) it calls Σ a global surface of section, whereas the paper constructs a union of invariant cylinders and works on the component intersecting the Legendrian set (Proposition 3.1) . The model’s final inequalities and equality cases match the paper’s statements (Theorem 1.2) but the derivational claims are incorrect or unsupported by the cited text .