Index formulas for higher order Loewner vector fields

Let Ì¢è '__z be the Cauchy-Riemann operator and f be a real-valued Cⁿ function in a neighborhood of the origin in the complex plane for which the n^th order Cauchy-Riemann derivative of f is non-zero for all z other than 0 in that neighborhood. CarathÌÄå©odory conjectured that a sphere immersed in R³ must have at least two umbilics. This later motivated Loewner to conjecture an upper bound of n for the index of the n^th order Cauchy-Riemann derivative of f as a vector fields. Both of these conjectures remain open. Recent work of F. Xavier produced a formula for computing the index of such vector fields in the case n = 2 using data about the Hessian of f. In this paper, we extend this result and establish an index formula for the n^th order Cauchy-Riemann derivative that is valid for all n Ì¢'¡å´ 2. Structurally, our index formula provides a defect term for Loewner's conjecture containing geometric data extracted from Hessian-like objects associated with higher order derivatives of f. In particular, our index formula computes the Fredholm index of the Toeplitz operator on the unit circle whose symbol is the n^th order Cauchy-Riemann derivative of f.