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For their detection, we first transform the Eulerian advection\u2013diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or heat equation. By this coordinate transformation, the reversible effects of advection are separated from the irreversible joint effects of advection and diffusion. In this framework, LCSs express themselves as (boundaries of) metastable sets under the Lagrangian diffusion process. In the case of spatially homogeneous isotropic diffusion, averaging the time-dependent family of Lagrangian diffusion operators yields Froyland\u2019s dynamic Laplacian. In the associated geometric heat equation, the distribution of heat is governed by the dynamically induced intrinsic geometry on the material manifold, to which we refer as the<jats:italic>geometry of mixing<\/jats:italic>. We study and visualize this geometry in detail, and discuss connections between geometric features and LCSs viewed as<jats:italic>diffusion barriers<\/jats:italic>in two numerical examples. Our approach facilitates the discovery of connections between some prominent methods for coherent structure detection: the dynamic isoperimetry methodology, the variational geometric approaches to elliptic LCSs, a class of graph Laplacian-based methods and the effective diffusivity framework used in physical oceanography.<\/jats:p>","DOI":"10.1007\/s00332-020-09626-9","type":"journal-article","created":{"date-parts":[[2020,5,19]],"date-time":"2020-05-19T14:19:16Z","timestamp":1589897956000},"page":"1849-1888","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["A Geometric Heat-Flow Theory of Lagrangian Coherent Structures"],"prefix":"10.1007","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9403-6511","authenticated-orcid":false,"given":"Daniel","family":"Karrasch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Johannes","family":"Keller","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,5,19]]},"reference":[{"key":"9626_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1029-0","volume-title":"Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences","author":"R Abraham","year":"1988","unstructured":"Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences, vol. 75, 2nd edn. 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