Hypersurfaces of constant Gauss-Kronecker curvature with Li-normalization in affine space

Xin Nie (1) , Andrea Seppi (2)

1 Southeast University
2 IF - Institut Fourier

For convex hypersurfaces in the affine space $\mathbb{A}^{n+1}$ ($n\geq2$), A.-M.\ Li introduced the notion of $\alpha$-normal field as a generalization of the affine normal field. By studying a Monge-Amp\`ere equation with gradient blowup boundary condition, we show that regular domains in $\mathbb{A}^{n+1}$, defined with respect to a proper convex cone and satisfying some regularity assumption if $n\geq3$, are foliated by complete convex hypersurfaces with constant Gauss-Kronecker curvature relative to the Li-normalization. When $n=2$, a key feature is that no regularity assumption is required, and the result extends our recent work about the $\alpha=1$ case.

Domaines: Géométrie différentielle [math.DG]

Xin Nie, Andrea Seppi. Hypersurfaces of constant Gauss-Kronecker curvature with Li-normalization in affine space. Calculus of Variations and Partial Differential Equations, 2022, 62 (4), ⟨10.1007/s00526-022-02329-x⟩. ⟨hal-03421127⟩