Abstract
In the chapter we study the metrics g t satisfying the Extrinsic Geometric Flow equation (see Sect. 3.2 Sections 3.4 and 3.5 collect results about existence and uniqueness of solutions (Theorems 3.1 and 3.2) and their proofs. The key role in proofs play hyperbolic PDEs and the generalized companion matrix studied in Sect. 3.3. In Sect. 3.6, we estimate the maximal existence time. In Sect. 3.7 we use the first derivative of functionals (when they are monotonous) to show convergence of metrics in a weak sense (Theorem 3.3). In Sect. 3.8 we study soliton solutions of the geometric flow equation (Theorems 3.4 and 3.5), and characterize them in the cases of umbilical foliations and foliations on surfaces (Theorems 3.6–3.8). Section 3.9 is devoted to applications and examples, including the geometric flow produced by the extrinsic Ricci curvature tensor (Theorem 3.9).
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© 2011 Vladimir Rovenski and Pawet Walczak
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Rovenski, V., Walczak, P. (2011). Extrinsic Geometric Flows. In: Topics in Extrinsic Geometry of Codimension-One Foliations. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9908-5_3
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DOI: https://doi.org/10.1007/978-1-4419-9908-5_3
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Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-9907-8
Online ISBN: 978-1-4419-9908-5
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