A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem

A. Colesanti; M. Focardi; P. Guan; P. Salani

A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem

Analysis of PDEs 2026-02-26 v2 Metric Geometry

Authors: A. Colesanti , M. Focardi , P. Guan , P. Salani

Abstract

We study the mixed Christoffel problem for $C^{2,+}$ convex bodies providing sufficient conditions for its solution. Key to our approach is a constant rank theorem, following the approach developed in \cite{Guan-Ma-2003} to address the Christoffel problem, in order to ensure that the solution to a related second order linear PDE on the sphere is indeed geometric, that is, it is the support functions of a $C^{2,+}$ convex body.

Keywords

constant rank theorem linear elliptic equations sphere mixed Christoffel problem convex bodies support functions

Cite

@article{arxiv.2512.08670,
  title  = {A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem},
  author = {A. Colesanti and M. Focardi and P. Guan and P. Salani},
  journal= {arXiv preprint arXiv:2512.08670},
  year   = {2026}
}

Comments

Theorem 1.3 of the first version of the paper has been corrected and modified (see Corollaries 1.5 and 1.6 of the new version)

A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem

Abstract

Keywords

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