Index formulas for higher order Loewner vector fields

Abstract

Let ∂ z ¯ be the Cauchy–Riemann operator and f be a C n real-valued function in a neighborhood of 0 in R 2 in which ∂ z ¯ n f ≠ 0 for all z ≠ 0 . In such cases, ∂ z ¯ n f is known as a Loewner vector field due to its connection with Loewner's conjecture that the index of such a vector field is bounded above by n. The n = 2 case of Loewner's conjecture implies Carathéodory's conjecture that any C 2 -immersion of S 2 into R 3 must have at least two umbilics. Recent work of F. Xavier produced a formula for computing the index of Loewner vector fields when n = 2 using data about the Hessian of f. In this paper, we extend this result and establish an index formula for ∂ z ¯ n f for all n ⩾ 2 . Structurally, our index formula provides a defect term, which contains geometric data extracted from Hessian-like objects associated with higher order derivatives of f.