Optimal transport and geometric analysis

最佳传输和几何分析

• 批准号: EP/R004730/2
• 负责人: Andrea Mondino
• 金额: $ 2.13万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FR004730%2F2
• 关键词: Optimal; transport; geometric; analysis

The subject of study of differential geometry are smooth manifolds, which correspond to smooth curved objects of finite dimension.In modern differential geometry, it is becoming more and more common to consider sequences (or flows) of smooth manifolds. Typically the limits of such sequences (or flows) are non smooth anymore. It is then useful to isolate a natural class of non smooth objects which generalize the classical notion of smooth manifold, and which is closed under the process of taking limits.If the sequence of manifolds satisfy a lower bound on the sectional curvatures, a natural class of non-smooth objects which is closed under (Gromov-Hausdorff) convergence is given by special metric spaces known as Alexandrov spaces; if instead the sequence of manifolds satisfy a lower bound on the Ricci curvatures, a natural class of non-smooth objects, closed under (measured Gromov-Hausdorff) convergence, is given by special metric measure spaces (i.e. metric spaces endowed with a reference volume measure) known as RCD(K,N) spaces. These are a 'Riemannian' refinement of the so called CD(K,N) spaces of Lott-Sturm-Villani, which are metric measure spaces with Ricci curvature bounded below by K and dimension bounded above by N in a synthetic sense via optimal transport.In the proposed project we aim to understand in more detail the structure, the analytic and the geometric properties of RCD(K,N) spaces. The new results will have an impact also on the classical world of smooth manifolds satisfying curvature bounds.

差异几何形状的研究主题是平滑的歧管，它们对应于有限维度的平滑曲面对象。在现代微分几何形状中，考虑平滑歧管的序列（或流）越来越普遍。通常，此类序列（或流）的极限不再平滑。然后，隔离一类自然的非平滑物体，从而在限制限制的过程中封闭了经典的概念，如果歧管的顺序满足了截面曲率的下限，则在截面曲率上满足，这是一种自然的非平滑物体，它是通过（Gromov-hausdorff）封闭的（Gromov-hausdorff）被特殊的alexcand clemencand clemencand clemencand clemencand clemencand clesecand clesecand clemencand clemencand clemencand clemencand。相反，如果歧管的序列满足RICCI曲率上的下限，则一类天然的非平滑物体（测量的Gromov-Hausdorff）收敛下封闭的非平滑物体，由特殊的度量测量空间（即具有参考体积度量）的特殊度量度量空间（即称为RCD（K，N）空间）给出。 These are a 'Riemannian' refinement of the so called CD(K,N) spaces of Lott-Sturm-Villani, which are metric measure spaces with Ricci curvature bounded below by K and dimension bounded above by N in a synthetic sense via optimal transport.In the proposed project we aim to understand in more detail the structure, the analytic and the geometric properties of RCD(K,N) spaces.新结果也将对满足曲率界限的平滑流形的古典世界产生影响。

项目成果

Entropy-Transport distances between unbalanced metric measure spaces

• DOI: 10.1007/s00440-022-01159-4
• 发表时间: 2020-09
• 期刊: Probability Theory and Related Fields
• 影响因子: 2
• 作者: Nicolò De Ponti;Andrea Mondino

Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below

• DOI: 10.1016/j.matpur.2019.10.005
• 发表时间: 2018-07
• 期刊: Journal de Mathématiques Pures et Appliquées
• 作者: Andrea Mondino;Daniele Semola

Existence and Regularity of Spheres Minimising the Canham-Helfrich Energy

最小化 Canham-Helfrich 能量的球体的存在性和规律性

• DOI: 10.1007/s00205-020-01497-4
• 发表时间: 2020
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Mondino A

NEW FORMULAS FOR THE LAPLACIAN OF DISTANCE FUNCTIONS AND APPLICATIONS

• DOI: 10.2140/apde.2020.13.2091
• 发表时间: 2020-01-01
• 期刊: ANALYSIS & PDE
• 影响因子: 2.2
• 作者: Cavalletti, Fabio;Mondino, Andrea
• 通讯作者: Mondino, Andrea

Foliation by Area-constrained Willmore Spheres Near a Nondegenerate Critical Point of the Scalar Curvature

标量曲率非简并临界点附近区域约束威尔莫尔球体的叶理化

• DOI: 10.1093/imrn/rny203
• 发表时间: 2020
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Ikoma N