Abstract: A Riemannian manifold (𝑀, 𝑔) is Ricci–pinched if Ric ≥ 0 and there exists a constant 𝜀 > 0 such that Ric ≥ 𝜀R𝑔. Hamilton’s pinching conjecture states that a complete, connected, Ricci–pinched Riemannian 3−manifold must be compact or flat. This conjecture has been addressed by Chen and Zhu, Lott, and Deruelle-Schulze-Simon, with a complete proof ultimately provided by Lee and Topping, relying on Ricci flow. Huisken and Körber have achieved an alternative proof using extrinsic geometric flows with an additional volume growth condition. Their approach is based on the monotonicity of the Willmore functional along the inverse mean curvature flow in Ricci–pinched manifolds. Rather than using the inverse mean curvature flow, one could consider 𝑝−harmonic potentials with 𝑝 ∈ (1, 2] and replace the Willmore functional with a suitable proxy, still achieving the conjecture under the extra growth condition.

In the talk, I will highlight the similarities and differences between these two methods, describe a version of the result for manifold with boundary and provide a unified perspective on a broad family of monotonicity formulas in (non)linear potential theory and along the inverse mean curvature flow.

For further information please contact elisur.magrini@unibocconi.it