An index formula for hemispheres of a $C^2$-regular convex closed surface in Euclidean $3$-space

Carathéodory's conjecture has long been regarded as one of the central problems in the classical theory of convex surfaces. In this paper, we establish an index formula for hemispheres of convex closed surfaces under $C^2$-regularity. The proof is based on studying a vertical section of the null hypersurfaces in Lorentz--Minkowski $4$-space associated with the originally given convex surface. As a consequence, the conjecture is affirmatively solved in the $C^2$-case.

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