Description:

Title: Non-persistence of Strongly Isolated Singularities and Hsiang's minimal hyperspheres

Abstract: In 1969, Shiing-Shen Chern proposed the spherical Bernstein problem, asking whether the equators in a round (n+1)-dimensional sphere are the only smooth, embedded minimal hyperspheres. In 1983, Hsiang provided a negative answer by constructing an infinite sequence of distinct embedded minimal hyperspheres in the round 4-dimensional sphere. This sequence arises from the desingularization of the Clifford football—the spherical suspension of a Clifford torus inside an equator—which has exactly two strongly isolated singular points.
About a decade ago, André Neves asked whether such a phenomenon persists under a small perturbation of the round metric. In this talk, I will discuss how to show the non-persistence of these strongly isolated singular points under a generic perturbation by analyzing the Fredholm index of the Jacobi operator for a certain class of varifolds. As a geometric application, we provide a negative answer to Neves’ question. This is based on joint work with Alessandro Carlotto and Zhihan Wang.