Title: Some spectrum estimates and their applications

 

Abstract: This talk explores spectral estimates for second-order elliptic operators arising from geometric variational problems and their consequences for stability and rigidity. I will focus on Yang–Mills connections and harmonic maps, viewed as critical points of natural energy functionals. Classical results of Simons and Xin show that nontrivial Yang–Mills fields on high-dimensional spheres and stable harmonic maps from spheres exhibit strong instability or triviality. I will describe refinements of these theorems via quantitative estimates on the spectrum of the associated Jacobi operators, including sharp upper bounds for the first eigenvalue and lower bounds on the dimension of negative eigenspaces. These results extend to minimal submanifolds in spheres and provide a unified perspective on Morse index phenomena in these settings. In the second part of the talk, I will discuss a result relating lower bounds of the Laplace spectrum to quasi-conformal diffeomorphisms. This connection leads to a proof of a theorem of Hamilton in full generality concerning convex hypersurfaces and illustrates how spectral information can control global geometric behavior. The overarching theme is that eigenvalue estimates serve as an effective bridge between analysis, topology, and geometric rigidity. 

 

Speaker Bio: Lei Ni is Distinguished Professor and Vice President in Academics at Zhejiang Normal University, and Distinguished/Above Scale Professor Emeritus at the University of California, San Diego, where he served on the faculty from 2002 to 2025 and chaired the Department of Mathematics from 2016 to 2020. Ni received his B.S. and M.S. from Fudan University and his Ph.D. from University of California, Irvine in 1998.

 

Professor Ni’s research lies in geometric analysis, differential geometry, complex geometry, and nonlinear partial differential equations, with fundamental contributions to Ricci flow, Kähler geometry, and Li–Yau–Hamilton inequalities, etc. He is the author of more than 70 research articles and co-author of the four-volume "The Ricci Flow: Techniques and Applications" (AMS Mathematical Surveys and Monographs) and "Hamilton’s Ricci Flow" (AMS Graduate Studies in Mathematics).

 

Ni's honors include election as a Fellow of the American Mathematical Society (2018), an Alfred P. Sloan Foundation Fellowship (2004–2006), and plenary invited addresses at major international conferences, including the 2024 International Congress of Chinese Mathematicians. He has mentored 20 doctoral students and postdoctoral scholars and has served many editorial boards, including Transactions of the AMS and International Journal of Mathematics.