Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

The Guest Editors for this special issue are

Lan-Hsuan Huang (University of Connecticut, USA)
Blaine Lawson (Stony Brook University, USA)
Pengzi Miao (University of Miami, USA)
Paolo Piazza (Sapienza Università di Roma, Italy)
Richard Schoen (University of California at Irvine, USA)
Christina Sormani (City University of New York, USA)
Guofang Wei (University of California at Santa Barbara, USA)

This special issue of SIGMA is dedicated to Misha Gromov on his 75th Birthday. We invited key participants from his IAS Emerging Topics Working Group on Scalar Curvature and Convergence as well as other leading experts to submit survey articles concerning open problems on the following topics:

  • synthetic notions of Ricci and scalar curvature;
  • geometric properties of Riemannian manifolds with bounds on their Ricci and scalar curvature;
  • low regularity notions of Ricci and scalar curvature;
  • geometric Stability and Almost Rigidity of rigidity theorems involving Ricci and scalar curvature;
  • Gromov–Hausdorff and intrinsic flat convergence of sequences of Riemannian manifolds with bounds on their Ricci or scalar curvature and the properties of their limit spaces.

In this volume there are 22 articles exploring these topics and related problems. We are grateful to the authors who contributed to the volume and to the referees for their careful reviews.

Everyone was also invited to record lectures for 2020 Virtual Workshop on Ricci and Scalar Curvature. Twenty-four lectures were selected, shared, and discussed online in the Summer for 2020. Selected postdocs participating in the workshop were funded in part by NSF DMS 1612049. We published their talks as well. We thank all the speakers and participants for contributing to the success of the event.

We thank Misha Gromov for all his inspiring work and brilliant ideas. We chose SIGMA because its articles are free for all to read and there are no publishing fees for the authors. We hope that everyone will enjoy both these articles and the freely available videos of the 2020 Virtual Workshop on Ricci and Scalar Curvature.

The Guest Editors        

Papers in this Issue:

Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants
Boris Botvinnik, Paolo Piazza and Jonathan Rosenberg
SIGMA 17 (2021), 062, 39 pages   [ abs   pdf ]
On Scalar and Ricci Curvatures
Gerard Besson and Sylvestre Gallot
SIGMA 17 (2021), 046, 42 pages   [ abs   pdf ]
Homotopy Invariance of the Space of Metrics with Positive Scalar Curvature on Manifolds with Singularities
Boris Botvinnik and Mark G. Walsh
SIGMA 17 (2021), 034, 27 pages   [ abs   pdf ]
Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian
Masayuki Aino
SIGMA 17 (2021), 017, 29 pages   [ abs   pdf ]
Curvature-Dimension Condition Meets Gromov's $n$-Volumic Scalar Curvature
Jialong Deng
SIGMA 17 (2021), 013, 20 pages   [ abs   pdf ]
On the 2-Systole of Stretched Enough Positive Scalar Curvature Metrics on $\mathbb{S}^2\times\mathbb{S}^2$
Thomas Richard
SIGMA 16 (2020), 136,   7 pages   [ abs   pdf ]
Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary
Annegret Burtscher, Christian Ketterer, Robert J. McCann and Eric Woolgar
SIGMA 16 (2020), 131, 29 pages   [ abs   pdf ]
Positive Scalar Curvature due to the Cokernel of the Classifying Map
Thomas Schick and Vito Felice Zenobi
SIGMA 16 (2020), 129, 12 pages   [ abs   pdf ]
Defining Pointwise Lower Scalar Curvature Bounds for $C^0$ Metrics with Regularization by Ricci Flow
Paula Burkhardt-Guim
SIGMA 16 (2020), 128, 10 pages   [ abs   pdf ]
Width, Largeness and Index Theory
Rudolf Zeidler
SIGMA 16 (2020), 127, 15 pages   [ abs   pdf ]
Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing
Shaosai Huang, Xiaochun Rong and Bing Wang
SIGMA 16 (2020), 123, 25 pages   [ abs   pdf ]
The Measure Preserving Isometry Groups of Metric Measure Spaces
Yifan Guo
SIGMA 16 (2020), 114, 14 pages   [ abs   pdf ]
Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds
Aaron Naber
SIGMA 16 (2020), 104,   8 pages   [ abs   pdf ]
Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces
Chao Li
SIGMA 16 (2020), 099,   8 pages   [ abs   pdf ]
Covariant vs Contravariant Methods in Differential Geometry
Maung Min-Oo
SIGMA 16 (2020), 095, 10 pages   [ abs   pdf ]
About Bounds for Eigenvalues of the Laplacian with Density
Aïssatou Mossèle Ndiaye
SIGMA 16 (2020), 090,   8 pages   [ abs   pdf ]
The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature
Jiayin Pan
SIGMA 16 (2020), 078, 16 pages   [ abs   pdf ]
Gromov Rigidity of Bi-Invariant Metrics on Lie Groups and Homogeneous Spaces
Yukai Sun and Xianzhe Dai
SIGMA 16 (2020), 068,   6 pages   [ abs   pdf ]
The Bochner Technique and Weighted Curvatures
Peter Petersen and Matthias Wink
SIGMA 16 (2020), 064, 10 pages   [ abs   pdf ]
Nonnegative Scalar Curvature and Area Decreasing Maps
Weiping Zhang
SIGMA 16 (2020), 033,   7 pages   [ abs   pdf ]
NNSC-Cobordism of Bartnik Data in High Dimensions
Xue Hu and Yuguang Shi
SIGMA 16 (2020), 030,   5 pages   [ abs   pdf ]
Collapsed Ricci Limit Spaces as Non-Collapsed RCD Spaces
Shouhei Honda
SIGMA 16 (2020), 021, 10 pages   [ abs   pdf ]

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