Vol. 288, No. 1, 2017

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ISSN: 0030-8730
$C^1$-umbilics with arbitrarily high indices

Naoya Ando, Toshifumi Fujiyama and Masaaki Umehara

Vol. 288 (2017), No. 1, 1–26
Abstract

We show that C1-umbilics with arbitrarily high indices exist. This implies that more than C1-regularity is required to prove Loewner’s conjecture.

Keywords
umbilic, curvature line, Carathéodory's conjecture, Loewner's conjecture
Mathematical Subject Classification 2010
Primary: 53A05, 53C45
Secondary: 57R42, 37C10, 53A30
Milestones
Received: 7 August 2015
Revised: 25 September 2016
Accepted: 26 September 2016
Published: 8 April 2017
Authors
Naoya Ando
Graduate School of Science and Technology
Kumamoto University
2-39-1 Kurokami
Kumamoto 860-8555
Japan
Toshifumi Fujiyama
Department of Mathematical Informatics
Graduate School of Information Science and Technology
University of Tokyo
Tokyo 113-8656
Japan
Masaaki Umehara
Department of Mathematical and Computing Sciences
Tokyo Institute of Technology
2-12-1-W8-34, O-okayama
Meguro-ku
Tokyo 152-8552
Japan