Verifying Shen–Ye generalized Bonnet–Myers hypotheses on the slice Σ_{m−1} in dimension n = 7 for m ∈ {2,3,4}

Ascertain whether the Shen–Ye generalized Bonnet–Myers hypothesis can be verified on the weighted slicing component Σ_{m−1} produced in the authors’ construction when n = 7 and m ∈ {2,3,4}; specifically, determine whether there exists a positive function f on Σ_{m−1} satisfying the Shen–Ye inequality Ric_{Σ_{m−1}}(v,v) − τ f^{−1}Δ_{Σ_{m−1}} f + [τ − ((k−1)/4 + ε)τ^2] |∇_{Σ_{m−1}} ln f|^2 ≥ κ > 0 for appropriate τ > 0, ε ≥ 0, and κ > 0.

Background

A key step in the paper’s method is to obtain diameter bounds for certain slices arising in a weighted slicing procedure, using the Shen–Ye generalized Bonnet–Myers theorem. This requires verifying the existence of a positive function satisfying a conformal-Ricci-type inequality on the slice.

The authors achieve this verification in dimensions up to 6 (with delicate arguments for n = 6, m = 3), but they state they could not verify the necessary hypothesis in dimension 7 for m ∈ {2,3,4}, leaving open whether this approach extends to n = 7.