2011/08/25 by Vladimir Rovenski, Rovenski, Vladimir · 1 citation
Mathematics · #53C12 #53C44 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.DG #msc:53C12 #msc:53C44
paper · pdf · doi:10.48550/arxiv.1108.5071
20 pages
openalex publication_date 2011/08/25 · arxiv created 2011/09/27 · arxiv updated 2011/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the geometry of a codimension-one foliation with a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies along the leaves of the foliation. Then the Extrinsic Geometric Flow depending on the second fundamental form of the foliation is introduced. Under suitable assumptions, this evolution yields the second order parabolic PDEs, for which the existence/uniquenes and in some cases convergence of a solution are shown. Applications to the problem of prescribing mean curvature function of a codimension-one foliation, and examples with harmonic and umbilical foliations (e.g., foliated surfaces) and with twisted product metrics are given.