Registration

Pak Tung Ho 何柏通

Title: Yamabe flow and its soliton on manifolds with boundary

Abstract: I will first talk about the convergence rate of the Yamabe flow on manifolds with boundary. Then I will talk about the Yamabe soliton on manifolds with boundary. I will look at it from equation point of view and discuss some of its geometric properties. This is joint work with Jinwoo Shin (Korea Institute For Advanced Study).

 Lunch break

Chia-Chieh Jay Chu 朱家杰

Title: A numerical method for surfaces' PDEs and mean curvature flows

Abstract:  Partial differential equations on surfaces have wild applications in many areas, such as material science, surfactant problems, image processing and biology. These PDEs usually originate from minimizing energy functions defined on surfaces.  This work targets applications that use implicit or non-parametric representations of closed surfaces or curves and require numerical solution for minimization problems defined on the surfaces. The energy function defined on surfaces can be extended to the energy function defined on the nearby tubular neighborhood that gives the same energy when input the constant-along-normal extension. Furthermore, the extended energy function gives the same minimizer as which the original energy function gives in the sense of restriction on the surface. This new approach connects the original energy function to an extended energy function and provides a good framework to solve PEDs numerically on Cartesian grids.

Recently, we apply the methodology for mean curvature problems. Our method uses the sign distance function defined in a narrowband near the moving interface to represent the evolution of the curve. We derive the equivalent evolution equations of distance function in the narrowband. The novelty of the work is to determine the equivalent evolution equation on Cartesian girds without extra conditions or constraints. The proposed method extends the differential operators appropriately so that the solutions on the narrowband are the distance function of the solution to the original mean flow solution. Furthermore, the extended solution carries the correct geometric information, such as distance and curvature, on Cartesian grids. Some experiments confirm that the proposed method is convergent numerically.

This is a joint work with Richard Tsai and Ming-Chih Lai, Shih-Hsuan Hsu, Chun-Chieh Lin, Kim Kai Run.

 Tea break

Nicolau Sarquis Aiex 艾尼克

Title: Singularities of Minimal hypersurfaces with bounded Index

Abstract: We prove a quantitative estimate on the size of a tubular neighbourhood of the singular set of a minimal hypersurface. This is a generalization of the work of Naber-Valtorta in the case of area minimizing hypersurfaces.

These quantitative estimates control not only the size of the singular set but also the behaviour of the hypersurface near singularities. We will discuss motivation and the main idea of the proof.

Forum discussion

Symposium Banquet

Place: 清華大學風雲樓四樓芸彩薈