C^1-umbilics with arbitrarily high indices

TL;DR

This paper demonstrates that C^1-umbilics can have arbitrarily high indices, indicating that higher regularity than C^1 is necessary to prove Loewner's conjecture.

Contribution

It establishes the existence of high-index C^1-umbilics, highlighting the need for more than C^1-regularity in related conjecture proofs.

Findings

Existence of C^1-umbilics with arbitrarily high indices

Implication that C^1-regularity is insufficient for Loewner's conjecture

Necessity of higher regularity in geometric analysis

Abstract

In this paper, the existence of C^1-umbilics with arbitrarily high indices is shown. This implies that more than C^1-regularity is required to prove Loewner's conjecture.