Perspectives in Scalar Curvature

Volume I contains a long article by Misha Gromov based on his many years of involvement in this subject. It came from lectures delivered in Spring 2019 at IHES. There is some background given. Many topics in the field are presented, and many open problems are discussed. One intriguing point here is the crucial role played by two seemingly unrelated analytic means: index theory of Dirac operators and geometric measure theory.

Very recently there have been some real breakthroughs in the field. Volume I has several survey articles written by people who were responsible for these results.

For Volume II, many people in areas of mathematics and physics, whose work is somehow related to scalar curvature, were asked to write about this in any way they pleased. This gives rise to a wonderful collection of articles, some with very broad and historical views, others which discussed specific fascinating subjects.

These two books give a rich and powerful view of one of geometry's very appealing sides.

• Volume I:
  • Four Lectures on Scalar Curvature (Misha Gromov)
  • Scalar Curvature and Generalized Callias Operators (Simone Cecchini and Rudolf Zeidler)
  • Convergence and Regularity of Manifolds with Scalar Curvature and Entropy Lower Bounds (Man-Chun Lee, Aaron Naber, and Robin Neumayer)
  • Level Set Methods in the Study of Scalar Curvature (Daniel Stern)
  • The Secret Hyperbolic Life of Positive Scalar Curvature (Joachim Lohkamp)
  • The Scalar Curvature of 4-Manifolds (Claude LeBrun)
• Volume II:
  • Classical Relations to Topology and the Dirac Operator:
    • Some Topological Implications of Positive Scalar Curvature and Generalizations:
      • Positive Scalar Curvature — Constructions and Obstructions (Stephan Stolz)
      • Positive Scalar Curvature on Pin ±- and Spin^c-Manifolds and Manifolds with Singularities (Boris Botvinnik and Jonathan Rosenberg)
      • Positive Scalar Curvature and Homotopy Theory (Boris Botvinnik and Johannes Ebert)
    • Complete Manifolds with Positive Scalar Curvature:
      • The Lichnerowicz Formula and Lower Bounds for the Scalar Curvature (Maung Min-Oo)
      • Deformed Dirac Operators and Scalar Curvature (Weiping Zhang)
      • Recent Results Concerning Topological Obstructions to Positive Scalar Curvature (Otis Chodosh and Chao Li)
      • Positive Scalar Curvature, Macroscopic Dimension, and Inessential Manifolds (Alexander Dranishnikov)
      • Scalar Curvature, Mass, and Other Asymptotic Invariants (Marc Herzlich)
      • Topological Characterization of Contractible 3-Manifolds with Positive Scalar Curvature (Jian Wang)
    • Manifolds with Boundary and Spaces of Metrics with Positive Scalar Curvature and Mean Curvature:
      • Boundary Conditions for Scalar Curvature (Christian Bär and Bernhard Hanke)
    • Minimal Varieties:
      • Small Two Spheres in Positive Scalar Curvature, Using Minimal Hypersurfaces (Thomas Richard and Jintian Zhu)
      • Minimal Surface Entropy of Negatively Curved Manifolds (Danny Calegari, Fernando C Marques and André Neves)
      • Marginally Outer Trapped Surfaces and Scalar Curvature Rigidity (Gregory J Galloway)
  • Positive Mass and Positive Energy:
    • Scalar Curvature, Spinors, Eigenvalues, and Mass (Oussama Hijazi, Sebastián Montiel and Simon Raulot)
    • Conserved Quantities in General Relativity: The Case of Initial Datasets with a Non-compact Boundary (Levi Lopes de Lima)
    • Dominant Energy Condition and Spinors on Lorentzian Manifolds (Bernd Ammann and Jonathan Glöckle)
    • Spacetime Harmonic Functions and Applications to Mass (Hubert Bray, Sven Hirsch, Demetre Kazaras, Marcus Khuri and Yiyue Zhang)
  • Positive Scalar Curvature on Generalized Spaces:
    • Polyhedra and Positive Scalar Curvature on Metric Spaces:
      • Conjectures on Convergence and Scalar Curvature (Christina Sormani)
      • Geometric Aspects of Quasi-Local Mass and Gromov's Fill-in Problem (Xue Hu and Yuguang Shi)
      • Interpreting Mass Via Riemannian Polyhedra (Pengzi Miao)
    • Distance Estimates:
      • Quantitative K-Theory, Positive Scalar Curvature, and Bandwidth (Hao Guo, Zhizhang Xie, and Guoliang Yu)
      • Waist Inequality for 3-Manifolds with Positive Scalar Curvature (Yevgeny Liokumovich and Davi Maximo)
    • Families and Foliations:
      • The Gromov–Lawson Index and the Baum–Connes Assembly Map (Moulay-Tahar Benameur)