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Recently Brendle-Huisken introduced a fully nonlinear flow $G$. Their aim wasto extend the surgery algorithm of Huisken-Sinestrari, into the Riemanniansetting. The aim of this paper is to go through the details on how to performneck detection for a closed, embedded hypersurface $M_0$ in $\mathbb{R}^{n+1}$undergoing this $G$-flow. In order to do this we make some adjustments toBrendle and Huiskens gradient estimate, after we have done this we can go on toargue as in Huisken-Sinestrari's paper Mean curvature flow with surgeries oftwo-convex hyperusrfaces, in order to classify two-convex surfaces undergoing$G$-flow.

Neck Detection for Two-Convex Hypersurfaces Embedded in Euclidean Space undergoing Brendle-Huisken G-Flow