Metric Diffusion Along Foliations

1. Front Matter

   Pages i-xi

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2. Wasserstein Distance

   • Szymon M. Walczak

   Pages 1-10

3. Foliations and Heat Diffusion

   • Szymon M. Walczak

   Pages 11-20

4. Compact Foliations

   • Szymon M. Walczak

   Pages 21-29

5. Metric Diffusion

   • Szymon M. Walczak

   Pages 31-48

6. Metric Diffusion for Non-compact Foliations: Remarks

   • Szymon M. Walczak

   Pages 49-52

7. Back Matter

   Pages 53-55

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Up-to-date research in metric diffusion along compact foliations is presented in this book. Beginning with fundamentals from the optimal transportation theory and the theory of foliations; this book moves on to cover Wasserstein distance, Kantorovich Duality Theorem, and the metrization of the weak topology by the Wasserstein distance. Metric diffusion is defined, the topology of the metric space is studied and the limits of diffused metrics along compact foliations are discussed. Essentials on foliations, holonomy, heat diffusion, and compact foliations are detailed and vital technical lemmas are proved to aide understanding.

Graduate students and researchers in geometry, topology and dynamics of foliations and laminations will find this supplement useful as it presents facts about the metric diffusion along non-compact foliation and provides a full description of the limit for metrics diffused along foliation with at least one compact leaf on the two dimensions.

• Wasserstein distance
• Metric diffusion
• Compact foliations
• heat diffusion
• non-compact foliations
• Holonomy
• Optimal Transportation Problem
• Harmonic measures

“The present book is a beautiful sample of results that can be proved on foliated manifolds by the theory of metric diffusion. … Graduate students and researchers in geometry, topology and dynamics of foliations will find this book especially valuable as it presents facts about the metric diffusion at the cutting edge of research.” (Glen E. Wheeler, Mathematical Reviews, May, 2018)

• National Science Center, Kraków, Poland

  Szymon M. Walczak

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