In a series of papers, Associate Professor Lei Zhang and his collaborators have established several vanishing theorems for singular Liouville equations, which are deeply connected to problems in geometry and physics, most notably the renowned Nirenberg Problem. These equations also arise in models across physics, chemistry and biology and are known for posing some of the most difficult analytical challenges in the field.
Among the most intriguing and complex phenomena in this context is the “non-simple” blow-up behavior that occurs when the singular source is quantized — that is, restricted to discrete strength values. In such cases, smooth solutions may diverge in a chaotic and unpredictable way near a blow-up point. Zhang’s team, in a series of papers published in Journal of the European Mathematical Society, American Journal of Mathematics, and Advances in Mathematics, demonstrated that these non-simple blow-up scenarios can only arise under very special conditions — namely, when the coefficient function in the equation is nearly constant.
This work not only advances understanding of singular Liouville equations but also contributes significantly to the broader field of conformal geometry on Riemann surfaces. By ruling out many potential non-simple blow-up cases, Professor Zhang’s results have removed a major obstacle in connecting local curvature to global topological features — a central goal in conformal geometry. His theorems are already inspiring new lines of inquiry and are being actively studied by research groups worldwide.
