Open problems and conjectures on submanifolds of finite type revisited

TL;DR

This paper revisits open problems and conjectures related to submanifolds of finite type, providing updated insights on longstanding questions in differential geometry.

Contribution

It offers an updated review and analysis of three conjectures concerning submanifolds of finite type, building on the author's earlier foundational work.

Findings

Updated status of three conjectures

New insights into submanifolds of finite type

Clarification of open problems in the field

Abstract

Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject had been collected in author's book [Total mean curvature and sub manifolds of finite type, World Scientific, NJ, 1984]. A list of ten open problems and three conjectures on submanifolds of finite type was published in 1981. The main purpose of this article is to provide some updated information on the three conjectures.