Using Picard's theorem and the Leray-Schauder fixed point theorem to reinvestigate the area-preserving convex curve flow in the plane which is considered as a coupled system and thus different from the setting han...Using Picard's theorem and the Leray-Schauder fixed point theorem to reinvestigate the area-preserving convex curve flow in the plane which is considered as a coupled system and thus different from the setting handled by Gage.展开更多
We consider the area-preserving mean curvature flow with free Neumann boundaries. We show that a rotationally symmetric n-dimensional hypersurface in R^(n+1)between two parallel hyperplanes will converge to a cylinder...We consider the area-preserving mean curvature flow with free Neumann boundaries. We show that a rotationally symmetric n-dimensional hypersurface in R^(n+1)between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the maximal principle to obtain gradient and curvature estimates, leading to long-time existence of the flow and convergence to a constant mean curvature surface.展开更多
We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter.Without imposing on any non-degeneracy assumption,we prove a formal KAM theorem for the mappings,which implies many previou...We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter.Without imposing on any non-degeneracy assumption,we prove a formal KAM theorem for the mappings,which implies many previous KAM-type results under some non-degeneracy conditions.Moreover,by this formal KAM theorem,we can also obtain some new interesting results under some weaker non-degeneracy conditions.Thus,the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.展开更多
Area-preserving parameterization is now widely applied,such as for remeshing and medical image processing.We propose an efficient and stable approach to compute area-preserving parameterization on simply connected ope...Area-preserving parameterization is now widely applied,such as for remeshing and medical image processing.We propose an efficient and stable approach to compute area-preserving parameterization on simply connected open surfaces.From an initial parameterization,we construct an objective function of energy.This consists of an area distortion measure and a new regularization,termed as the Tutte regularization,combined into an optimization problem with sliding boundary constraints.The original area-preserving problem is decomposed into a series of subproblems to linearize the boundary constraints.We design an iteration framework based on the augmented Lagrange method to solve each linear constrained subproblem.Our method generates a high-quality parameterization with area-preserving on facets.The experimental results demonstrate the efficacy of the designed framework and the Tutte regularization for achieving a fine parameterization.展开更多
文摘Using Picard's theorem and the Leray-Schauder fixed point theorem to reinvestigate the area-preserving convex curve flow in the plane which is considered as a coupled system and thus different from the setting handled by Gage.
文摘We consider the area-preserving mean curvature flow with free Neumann boundaries. We show that a rotationally symmetric n-dimensional hypersurface in R^(n+1)between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the maximal principle to obtain gradient and curvature estimates, leading to long-time existence of the flow and convergence to a constant mean curvature surface.
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.11871146,11671077)the Innovation Project for college postgraduates in Jiangsu Province(No.KYZZ160113).
文摘We consider a class of analytic area-preserving mappings Cm-smoothly depending on a parameter.Without imposing on any non-degeneracy assumption,we prove a formal KAM theorem for the mappings,which implies many previous KAM-type results under some non-degeneracy conditions.Moreover,by this formal KAM theorem,we can also obtain some new interesting results under some weaker non-degeneracy conditions.Thus,the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.
基金supported by Anhui Center for Applied Mathematics,the NSF of China (No.11871447)the special project of strategic leading science and technology of CAS (No.XDC08010100)the National Key Research and Development Program of MOST of China (No.2018AAA0101001).
文摘Area-preserving parameterization is now widely applied,such as for remeshing and medical image processing.We propose an efficient and stable approach to compute area-preserving parameterization on simply connected open surfaces.From an initial parameterization,we construct an objective function of energy.This consists of an area distortion measure and a new regularization,termed as the Tutte regularization,combined into an optimization problem with sliding boundary constraints.The original area-preserving problem is decomposed into a series of subproblems to linearize the boundary constraints.We design an iteration framework based on the augmented Lagrange method to solve each linear constrained subproblem.Our method generates a high-quality parameterization with area-preserving on facets.The experimental results demonstrate the efficacy of the designed framework and the Tutte regularization for achieving a fine parameterization.
