Rigidity of Immersed Gauss Curvature Shrinkers in Euclidean Space

For every $m\ge 2$, we prove that any connected closed oriented smooth immersion $X:M^m\to {\mathbb{R}}^{m+1}$ satisfying $$K=⟨X,\nu ⟩$$ is an embedding onto the unit sphere. No convexity is assumed. At a point maximizing $\mathrm{\mid }X\mathrm{\mid }$, we find a strictly convex component of $\{K>0\}$. If this component does not coincide with $M$, its polar immersion bounds an unbounded convex set. A finite area lemma then extends the local surface area formula for the polar body to a global measure identity. Using the vanishing first moment of surface area measure, we show that the origin lies in the interior of the polar body, contradicting the existence of recession directions. Thus the hypersurface is strictly convex, and the classification of strictly convex shrinkers completes the proof.