An isolated umbilical point of a Willmore surface
This paper proves that the index of an isolated umbilic point on a Willmore surface does not exceed 1/2. The author first gives a simple explanation of why we know that there do indeed exist umbilic points on Willmore surfaces with index equal to 1/2 and hence the estimate is sharp. The situation can be contrasted to constant mean curvature (CMC) surfaces, where the index of an isolated umbilic point is always negative. (Although not mentioned in this paper it is well known that, except on a round sphere, all umbilics on a CMC surface are isolated, whereas there are many examples of Willmore surfaces with non-isolated umbilics.) After reviewing the necessary background the proof of the main result consists of two parts, both of which require the surface to be Willmore. A nice example shows why special consideration must be given to extremely ``round'' umbilics. A detailed analysis, using the methods developed in a series of previous papers, of approximating homogeneous polynomials satisfying any possible limiting behavior yields the estimate needed for the second part. Understanding more examples and applications of the methods developed by the author, such as those presented in this paper may shed light on the Willmore, Caratheodory and Loewner conjectures. These conjectures and their relationship to the current work are discussed in the author's introduction.
scientific article; zbMATH DE number 2147089
scientific article