Mean Curvature Flow and Nonlinear Heat Equations
平均曲率流和非线性热方程
基本信息
- 批准号:1707270
- 负责人:
- 金额:$ 30.02万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2017
- 资助国家:美国
- 起止时间:2017-08-01 至 2021-07-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Mean curvature flow is an equation that drives the evolution of a surface in the direction of steepest descent for the surface's area. As a closed surface evolves to decrease its area as rapidly as possible, convex points will move inwards, concave points move outwards, and the speed is slower where the surface is flatter. The area will decrease and eventually go to zero in finite time. In particular, any closed surface becomes extinct in finite time and, thus, singularities always occur. This flow originated in the materials science literature and has been intensely studied in both pure and applied mathematics. The key to understanding the mean curvature flow is to understand the singularities it goes through.These projects focus on a number of related aspects of the singularities: (1) Which singularities occur for a generic flow or a generic family of flows? (2) What does the flow look like near a singularity? When is the blow up unique? (3) Is there a canonical neighborhoods theorem? (4) What is the size and structure of the singular set? When is the singular set a nice submanifold? (5) Are singularities in Euclidean 3-space generically isolated? What about singular times?
平均曲率流是一个方程,它驱动曲面在曲面面积的最陡下降方向上演化。当一个封闭的曲面尽可能快地缩小它的面积时,凸点会向内移动,凹点会向外移动,并且在曲面较平坦的地方速度较慢。面积将减小,并最终在有限时间内变为零。特别是,任何闭曲面在有限时间内消失,因此,奇点总是出现。 这种流动起源于材料科学文献,并在纯数学和应用数学中得到了深入的研究。理解平均曲率流的关键是理解它所经过的奇点,这些项目集中在奇点的一些相关方面:(1)对于一般流或一般流族,哪些奇点会发生? (2)奇点附近的流动是什么样的?什么时候爆炸是独一无二的?(3)是否存在典范邻域定理? (4)奇异集的大小和结构是什么?什么时候奇异集是一个好的子流形?(5)欧氏三维空间中的奇点是一般孤立的吗?奇异时刻呢?
项目成果
期刊论文数量(6)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Arnold‐Thom Gradient Conjecture for the Arrival Time
Arnold-Thom 到达时间梯度猜想
- DOI:10.1002/cpa.21824
- 发表时间:2019
- 期刊:
- 影响因子:3
- 作者:Colding, Tobias Holck;Minicozzi, William P.
- 通讯作者:Minicozzi, William P.
Liouville properties
刘维尔房产
- DOI:10.4310/iccm.2019.v7.n1.a10
- 发表时间:2019
- 期刊:
- 影响因子:0
- 作者:Colding, Tobias H.;Minicozzi, William P.
- 通讯作者:Minicozzi, William P.
Dynamics of closed singularities
闭合奇点的动力学
- DOI:10.5802/aif.3343
- 发表时间:2019
- 期刊:
- 影响因子:0
- 作者:Colding, Tobias Holck;Minicozzi II, William P.
- 通讯作者:Minicozzi II, William P.
Sharp frequency bounds for eigenfunctions of the Ornstein–Uhlenbeck operator
Ornstein-Uhlenbeck 算子特征函数的尖锐频率界限
- DOI:10.1007/s00526-018-1405-z
- 发表时间:2018
- 期刊:
- 影响因子:2.1
- 作者:Colding, Tobias Holck;Minicozzi, William P.
- 通讯作者:Minicozzi, William P.
In Search of Stable Geometric Structures
- DOI:10.1090/noti1993
- 发表时间:2019-07
- 期刊:
- 影响因子:0
- 作者:T. Colding;W. Minicozzi
- 通讯作者:T. Colding;W. Minicozzi
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William Minicozzi其他文献
William Minicozzi的其他文献
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{{ truncateString('William Minicozzi', 18)}}的其他基金
Singularities and rigidity in geometric evolution equations
几何演化方程中的奇异性和刚性
- 批准号:
2304684 - 财政年份:2023
- 资助金额:
$ 30.02万 - 项目类别:
Standard Grant
Dynamics and Singularities of Geometric Flows
几何流的动力学和奇点
- 批准号:
2005345 - 财政年份:2020
- 资助金额:
$ 30.02万 - 项目类别:
Continuing Grant
Mean curvature flow and geometric analysis
平均曲率流和几何分析
- 批准号:
1408398 - 财政年份:2013
- 资助金额:
$ 30.02万 - 项目类别:
Continuing Grant
Mean curvature flow and geometric analysis
平均曲率流和几何分析
- 批准号:
1206827 - 财政年份:2012
- 资助金额:
$ 30.02万 - 项目类别:
Continuing Grant
Minimal surfaces and geometric flows
最小表面和几何流
- 批准号:
0906233 - 财政年份:2009
- 资助金额:
$ 30.02万 - 项目类别:
Continuing Grant
FRG: Collaborative Research: Mean curvature flow as a tool in low dimensional topology
FRG:协作研究:平均曲率流作为低维拓扑的工具
- 批准号:
0853501 - 财政年份:2009
- 资助金额:
$ 30.02万 - 项目类别:
Standard Grant
Geometric Analysis and Nonlinear Elliptic PDE's
几何分析和非线性椭圆偏微分方程
- 批准号:
0623843 - 财政年份:2006
- 资助金额:
$ 30.02万 - 项目类别:
Standard Grant
Minimal surfaces and geometric analysis
最小曲面和几何分析
- 批准号:
0405695 - 财政年份:2004
- 资助金额:
$ 30.02万 - 项目类别:
Continuing Grant
Embedded Minimal Surfaces in Three Manifolds
三个流形中的嵌入式最小曲面
- 批准号:
0104187 - 财政年份:2001
- 资助金额:
$ 30.02万 - 项目类别:
Standard Grant
相似海外基金
Canonical mean curvature flow and its application to evolution problems
正则平均曲率流及其在演化问题中的应用
- 批准号:
23H00085 - 财政年份:2023
- 资助金额:
$ 30.02万 - 项目类别:
Grant-in-Aid for Scientific Research (A)
Toward applications of the crystalline mean curvature flow
晶体平均曲率流的应用
- 批准号:
23K03212 - 财政年份:2023
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Grant-in-Aid for Scientific Research (C)
Geometric analysis of mean curvature flow with dynamic contact angle structure
动态接触角结构平均曲率流动的几何分析
- 批准号:
23K12992 - 财政年份:2023
- 资助金额:
$ 30.02万 - 项目类别:
Grant-in-Aid for Early-Career Scientists
Singularities of Minimal Hypersurfaces and Lagrangian Mean Curvature Flow
最小超曲面的奇异性和拉格朗日平均曲率流
- 批准号:
2306233 - 财政年份:2023
- 资助金额:
$ 30.02万 - 项目类别:
Continuing Grant
Mean curvature flow of small sections of the tangent bundle
切束小截面的平均曲率流
- 批准号:
572922-2022 - 财政年份:2022
- 资助金额:
$ 30.02万 - 项目类别:
University Undergraduate Student Research Awards
Research of submanifolds by using the mean curvature flow and Lie group actions, and its application to theoretical physics
利用平均曲率流和李群作用研究子流形及其在理论物理中的应用
- 批准号:
22K03300 - 财政年份:2022
- 资助金额:
$ 30.02万 - 项目类别:
Grant-in-Aid for Scientific Research (C)
Mean Curvature Flow and Singular Minimal Surfaces
平均曲率流和奇异极小曲面
- 批准号:
2203132 - 财政年份:2022
- 资助金额:
$ 30.02万 - 项目类别:
Standard Grant
Singularities of Minimal Hypersurfaces and Lagrangian Mean Curvature Flow
最小超曲面的奇异性和拉格朗日平均曲率流
- 批准号:
2203218 - 财政年份:2022
- 资助金额:
$ 30.02万 - 项目类别:
Continuing Grant
Entropy in Mean Curvature Flow and Minimal Hypersurfaces
平均曲率流和最小超曲面中的熵
- 批准号:
2105576 - 财政年份:2021
- 资助金额:
$ 30.02万 - 项目类别:
Continuing Grant
Entropy in Mean Curvature Flow and Minimal Hypersurfaces
平均曲率流和最小超曲面中的熵
- 批准号:
2146997 - 财政年份:2021
- 资助金额:
$ 30.02万 - 项目类别:
Continuing Grant