Covariant vs Contravariant Methods in Differential Geometry

Maung Min-Oo

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

This is a short essay about some fundamental results on scalar curvature and the two key methods that are used to establish them.

Détail

Keywords: scalar curvature, spinors, Dirac operator.

Bibliographie

[1] Bartnik R., “The mass of an asymptotically flat manifold”, Comm. Pure Appl. Math., 39 (1986), 661–693 | DOI | MR | Zbl

[2] Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften, 298, Springer-Verlag, Berlin, 1992 | DOI | MR | Zbl

[3] Besson G., Courtois G., Gallot S., “Entropies et rigidités des espaces localement symétriques de courbure strictement négative”, Geom. Funct. Anal., 5 (1995), 731–799 | DOI | MR | Zbl

[4] Brendle S., Rigidity phenomena involving scalar curvature, Int. Press, Boston, MA, 2012, 179–202, arXiv: 1008.3097 | DOI | MR | Zbl

[5] Brendle S., Marques F. C., Neves A., “Deformations of the hemisphere that increase scalar curvature”, Invent. Math., 185 (2011), 175–197, arXiv: 1004.3088 | DOI | MR | Zbl

[6] Gromov M., “Positive curvature, macroscopic dimension, spectral gaps and higher signatures”, Functional Analysis on the Eve of the 21st Century (New Brunswick, NJ, 1993), v. II, Progr. Math., 132, Birkhäuser Boston, Boston, MA, 1996, 1–213 | DOI | MR | Zbl

[7] Gromov M., “Dirac and Plateau billiards in domains with corners”, Cent. Eur. J. Math., 12 (2014), 1109–1156, arXiv: 1811.04318 | DOI | MR | Zbl

[8] Gromov M., “Metric inequalities with scalar curvature”, Geom. Funct. Anal., 28 (2018), 645–726, arXiv: 1710.04655 | DOI | MR | Zbl

[9] Gromov M., Scalar curvature of manifolds with boundaries: natural questions and artificial constructions, arXiv: 1811.04311

[10] Gromov M., Lawson Jr. H.B., “Spin and scalar curvature in the presence of a fundamental group. I”, Ann. of Math., 111 (1980), 209–230 | DOI | MR | Zbl

[11] Gromov M., Lawson Jr. H.B., “The classification of simply connected manifolds of positive scalar curvature”, Ann. of Math., 111 (1980), 423–434 | DOI | MR | Zbl

[12] Gromov M., Lawson Jr. H.B., “Positive scalar curvature and the Dirac operator on complete Riemannian manifolds”, Inst. Hautes Études Sci. Publ. Math., 58 (1983), 83–196 | DOI | MR | Zbl

[13] Hang F., Wang X., “Rigidity and non-rigidity results on the sphere”, Comm. Anal. Geom., 14 (2006), 91–106 | DOI | MR | Zbl

[14] Hang F., Wang X., “Rigidity theorems for compact manifolds with boundary and positive Ricci curvature”, J. Geom. Anal., 19 (2009), 628–642, arXiv: 0911.0380 | DOI | MR | Zbl

[15] Llarull M., “Sharp estimates and the Dirac operator”, Math. Ann., 310 (1998), 55–71 | DOI | MR | Zbl

[16] Lohkamp J., Minimal smoothings of area minimizing cones, arXiv: 1810.03157

[17] Min-Oo M., “Scalar curvature rigidity of asymptotically hyperbolic spin manifolds”, Math. Ann., 285 (1989), 527–539 | DOI | MR | Zbl

[18] Schoen R., Yau S.-T., “Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature”, Ann. of Math., 110 (1979), 127–142 | DOI | MR | Zbl

[19] Schoen R., Yau S.-T., “On the proof of the positive mass conjecture in general relativity”, Comm. Math. Phys., 65 (1979), 45–76 | DOI | MR | Zbl

[20] Schoen R., Yau S.-T., “Proof of the positive mass theorem. II”, Comm. Math. Phys., 79 (1981), 231–260 | DOI | MR | Zbl

[21] Schoen R., Yau S.-T., Positive scalar curvature and minimal hypersurface singularities, arXiv: 1704.05490

[22] Witten E., “A new proof of the positive energy theorem”, Comm. Math. Phys., 80 (1981), 381–402 | DOI | MR | Zbl