Loewner local index conjecture for umbilic points

Prove the Loewner local index conjecture that 2 is the highest complex index arising from an umbilic point on a surface in R^3, thereby addressing the corresponding local index bound relevant to the Carathéodory conjecture.

Background

The paper relates complex points of real surfaces in the neutral Kähler manifold TS2 to umbilic points of surfaces in R3. In this correspondence, the complex index determines local behavior of the principal foliation, and index-2 complex points arise from umbilic points on rotationally symmetric surfaces.

The Loewner conjecture asserts that the complex index of an umbilic point on a surface in R3 cannot exceed 2. The paper states that this conjecture would imply the global Carathéodory conjecture, while only a weaker local index bound of 4 is known.

References

A long-standing local index Conjecture of Loewner, which implies a global Conjecture of Carathéodory, states that 2 is the highest complex index than can arise in this manner from an umbilic point on a surface in R3 [6][7][8][11]. The Conjecture of Loewner remains open, although the weaker local index bound of 4 has been proven in [10].

— The Bishop family of holomorphic discs: regularity and higher index  (2608.21068 - Guilfoyle et al., 21 Aug 2026) in Section 1, p. 3, paragraph beginning “In this setting, index 2 complex points”