Differential Geometry in the Large

Owen Dearricott is Honorary Research Fellow at La Trobe University, Australia. A Riemannian geometer best known for his work constructing metrics of positive sectional curvature in dimension seven, he was a co-author of a proceedings volume of the 2010 mini-meeting in Differential Geometry at CIMAT, Guanajuato. Wilderich Tuschmann holds the Differential Geometry Professorial Chair at Karlsruhe Institute of Technology, Germany. He is a geometer with research interests in global differential geometry and geometric topology. He co-authored a scientific biography of the Russian mathematician Sofya Kovalevskaya (1993) and Moduli Spaces of Riemannian Metrics (2015). Yuri Nikolayevsky is Associate Professor at La Trobe University, Victoria. He is a differential geometer best known for his work on Osserman manifolds and homogeneous geometry. He has organised numerous geometry workshops in Australia and was the director of the Australian Mathematical Sciences Institute Summer School at La Trobe University in 2020. Thomas Leistner is Associate Professor at the University of Adelaide. He is a differential geometer who works on Lorentzian and pseudo-Riemann geometry, conformal geometry and holonomy theory. Diarmuid Crowley is Associate Professor at the University of Melbourne. He is a differential topologist with special expertise in the classification of 7-manifolds via smooth invariants.

• Introduction Owen Dearricott, Wilderich Tuschmann, Yuri Nikolayevsky, Thomas Leistner and Diarmuid Crowley; Part I. Geometric Evolution Equations and Curvature Flow: 1. Real geometric invariant theory Christoph Böhm and Ramiro A. Lafuente; 2. Convex ancient solutions to mean curvature flow Theodora Bourni, Mat Langford and Giuseppe Tinaglia; 3. Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in Minkowski space and the cross curvature flow Paul Bryan, Mohammad N. Ivaki and Julian Scheuer; 4. A mean curvature flow for conformally compact manifolds A. Rod Gover and Valentina-Mira Wheeler; 5. A survey on the Ricci flow on singular spaces Klaus Kröncke and Boris Vertman; Part II. Structures on Manifolds and Mathematical Physics: 6. Some open problems in Sasaki geometry Charles P. Boyer, Hongnian Huang, Eveline Legendre and Christina W. Tønnesen-Friedman; 7. The prescribed Ricci curvature problem for homogeneous metrics Timothy Buttsworth and Artem Pulemotov; 8. Singular Yamabe and Obata problems A. Rod Gover and Andrew K. Waldron; 9. Einstein metrics, harmonic forms and conformally Kähler geometry Claude LeBrun; 10. Construction of the supersymmetric path integral: a survey Matthias Ludewig; 11. Tight models of de-Rham algebras of highly connected manifolds Lorenz Schwachhöfer; Part III. Recent Developments in Non-Negative Sectional Curvature: 12. Fake lens spaces and non-negative sectional curvature Sebastian Goette, Martin Kerin and Krishnan Shankar; 13. Collapsed three-dimensional Alexandrov spaces: a brief survey Fernando Galaz-García, Luis Guijarro and Jesús Núñez-Zimbrón; 14. Pseudo-angle systems and the simplicial Gauss–Bonnet–Chern theorem Stephan Klaus; 15. Aspects and examples on quantitative stratification with lower curvature bounds Nan Li; 16. Universal covers of Ricci limit and RCD spaces Jiayin Pan and Guofang Wei; 17. Local and global homogeneity for manifolds of positive curvature Joseph A. Wolf.

'The high-quality surveys and original work in this book give a convenient path to understand some recent exciting developments in global Differential Geometry and Geometric Analysis. This should be of great value to graduate students entering the field, as well as to more experienced researchers looking for an updated perspective on a wide range of topics, ranging from nonnegative curvature and Alexandrov spaces to geometric flows and equivariant geometry.' Renato G. Bettiol, Lehman College, The City University of New York