【几何分析讨论班】Non-persistence of strongly isolated singularities and Hsiang's minimal hyperspheres

Title：Non-persistence of strongly isolated singularities and Hsiang's minimal hyperspheres

Speaker：李阳垟（美国圣母大学）

Time：2025年12月24日（星期三）, 上午10:30-11:30

Place：海纳苑2幢102

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.