Abstract: In this talk, we discuss the volume-preserving mean curvature flows (VPMCF), where a hypersurface evolves with a velocity determined by its mean curvature, along with an additional constraint that ensures constant enclosed volume. Such flow and its variations have been extensively studied over the past half-century, with significant contributions from M. Gage, G. Huisken, B. Andrews, P. Guan, and many others. VPMCF is closely related to the isoperimetric problem and various optimal geometric inequalities. For instance, it provides an effective "path" for finding the minimizing set of the perimeter functional under a volume constraint. The constrained term in velocity presents distinct challenges depending on their nature and formulation of singularities. For example, the avoidance principle fails when the velocity involves a non-local term of VPMCF. Beyond closed hypersurfaces, we will focus on compact hypersurfaces with free or capillary boundaries, which naturally arise in variational problems and fluid mechanics.

Biography: Liangjun Weng is a postdoc at Uni. Pisa. His research interests lie in Fully nonlinear PDEs and Differential Geometry. He received Ph.D. from University of Science and Technology of China and University of Freiburg under the direction of Professors Xi-Nan Ma and Guofang Wang.

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