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

TL;DR

This paper proves an index formula for hemispheres of $C^2$-regular convex surfaces in Euclidean 3-space, leading to an affirmative solution of Carathéodory's conjecture in this regularity class.

Contribution

It introduces a new index formula for convex surfaces and applies it to confirm Carathéodory's conjecture for $C^2$-regular surfaces.

Findings

Established an index formula for hemispheres of convex surfaces

Confirmed Carathéodory's conjecture for $C^2$-regular convex surfaces

Used Lorentz--Minkowski space techniques in the proof

Abstract

Carath\'eodory'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.