Summary of the Tianyuan Mathematical Research Center Workshop on “Geometric Variational Problems”
The Tianyuan Mathematical Research Center Workshop on “Geometric Variational Problems” was successfully held from January 18th-24th, 2026. This one-week workshop was convened by Professors Jiayu Li (李嘉禹，University of Science and Technology of China), Ruijun Wu (吴瑞军，Beijing Institute of Technology) and Jie Zhou (周杰，Capital Normal Univeristy). It brought together 27 participants from leading universities and research institutions working in differential geometry, geometric analysis and nonlinear analysis. In a vibrant and collaborative atmosphere, we have talked about the major advances in several closely related topics including harmonic maps, prescribing curvature equations and other conformal equations, mean curvature flows, Ricci flows, Yang-Mills(-Higgs) equations, Dirac type systems, and Mong-Ampere equations.
The scientific program is comprised of 20 invited talks and two half-day seminars by graduate students. We focus on the methods and tools which are transferable in different problems in geometry. We collect some of them in the following:
1, the classical variational methods (including Morse theory, degree theory, Jacobi theory) to obtain existence and (in)stabilities, to distinguish minimizers, stable solutions and saddles;
2. the perturbation methods, including Sacks-Uhlenbeck type perturbations, positive polynomial perturbations;
3. the blowup analysis for compactness of solution spaces, for PS sequences, for α-approximate solutions of harmonic map equations or Yang-Mills equations, or bubble solutions and singular solutions, for analysis of singularities, and to obtain energy identities and no-neck properties;
4. the parabolic method to obtain existence, to analyze and classify singular models or to deal with weak (non-smooth) metrics, including Ricci flow and Kahler-Ricci flow for metrics with nice curvature properties, De Turck-Ricci flow for positive mass in hyperbolic setting, symplectic and Lagrangian mean curvature flow for special submanifolds; modified negative gradient flow for prescribing scalar curvature; heat flow methods for mean field equations and Toda systems; surprisingly the discrete Ricci flow finds an effective application in machine learning;
5. refined estimates and asymptotic expansions to obtain existence or to exclude bubbling, finite dimensional Lyapunov-Schmidt reduction for the analysis of singular solutions;
6. techniques from metric geometry, geometric measure theory and conformal geometry, to obtain existence, regularity, fundamental solutions, classification of solition and tangent maps;
7. use of geometric properties of the manifold such as special symmetry, Killing field, for the reduction of order of equation and nonlinearities, and to construct solutions (for Yang-Mills equatioins or Monge-Ampere equations).
8.harmonic analysis and geometric measure theory tools (tangent measure, rectifiability, maximal function, covering argument, hardy space) to analysis higher dimensional singularities, especially to establish energy identities for density of blowup sets of higher dimensional harmonic maps.
Representations are around these topics from different perspective, shedding new light on the problems and methodology, and stimulating enthusiastic discussion. Note that these methods are classical, but in each problem there is new input which improves the situation and deepens our understanding.
The event successfully created an open platform for researchers at all levels to exchange the latest achievements and address pressing challenges. It significantly promoted scholarly communication and collaboration across disciplines, thereby actively contributing to the field’s academic progress. Furthermore, the workshop provided valuable opportunities for the professional growth of young scholars, who were able to engage closely with leading experts within the stimulating environment of the Tianyuan Center.
Participants expressed their sincere appreciation for the world-class platform and exceptional support provided by the Tianyuan Mathematics Research Center. The comprehensive insights and collaborations fostered throughout the event are expected to have a lasting impact. We extend our heartfelt gratitude to all organizers, participants, and staff members of TMRC who contributed to the success of this workshop.