Abstract
The theme of these notes is centered around the use of the Dirac operator in geometry and physics, with the main focus on scalar curvature, Gromov’s K-area and positive mass theorems in General Relativity.
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References
R. Bartnik: “The mass of an asymptotically flat manifold”Comm. Pure Appl.Math.39 (1986), 661–693.
H. Bray: “Proof of the Riemannian Penrose Conjecture using the Positive Mass Theorem” preprint, 1999.
N. Berline, E. Getzler and M. Vergne: “Heat kernels and Dirac operators”, Grundlehren der math. Wiss. 298 (1992), Springer Verlag.
G. Besson, G. Courtois and S. Gallot: “Entropies et rigidites des espaces localemant symetriques de courbure strictement negative”Geom. Funct. Anal.5 (1995), 731–799.
H. Boualem and M. Herzlich: “Rigidity for even-dimensional asymptotically complex hyperbolic spaces”, preprint (2001).
M. Cai and G. Galloway: “Boundaries of zero scalar curvature in the AdS/CFT correspondence”, preprint (2000), hep-th/0003046.
P.T. Chrusciel and M. Herzlich: “The mass of asymptotically hyperbolic manifolds”, preprint (2001), math.DG/ 0110035.
S. Goette and U. Semmelmann: “Spin` structures and scalar curvature estimates”, preprint, 2000.
M. Gromov: “Positive curvature, macroscopic dimension, spectral gaps and higher signatures”, Functional Analysis on the eve of the 21st century, vol.II, Progress in Math., vol.132Birkhauser, Boston, (1996).
M. Gromov and H.B. Lawson: “Spin and scalar curvature in the presence of a fundamental group I”;Ann. Math. 111(1980), 209–230.
M. Gromov and H.B. Lawson: “The classification of simply connected manifolds of positive scalar curvature”; Ann.Math.111 (1980), 423–486.
M. Gromov and H.B. Lawson: “Positive scalar curvature and the Dirac operator on complete Riemannian manifolds”Publ. Math. I.H.E.S.58 (1983), 295–408.
V. Guillemin, E. Lerman and S. Sternberg: “Symplectic fibrations and the multiplicity diagrams”Cambridge Univ. Press1996.
M. Herzlich: “Scalar curvature and rigidity of odd-dimensional complex hyperbolic spaces”Math. Ann.312 (1998), 641–657.
M. Herzlich: “A Penrose-like inequality for the mass of Riemannian asymptotically flat manifolds”Comm. Math. Phys.188 (1997), 121–133.
G. Huisken and T. Ilmanen: “The Riemannian Penrose inequality”Internat.Math. Res. Notices20 (1997), 1045–1058.
G.T. Horowitz and R.C. Myers: “The AdS/CFT correspondence and a new positive energy conjecture for general relativity”Phys. Rev. D59 (1999), hep-th 9808079.
G. Huisken and T. Ilmanen: “The Riemannian Penrose inequality”Internat.Math. Res. Notices20 (1997), 1045–1058.
H.B. Lawson, and M.-L. Michelsohn: “Spin Geometry”; Princeton Math. Series 38, Princeton, 1989.
J. Lohkamp: “Curvature h-principles”; Ann.of Math.142.3 (1995), 457–498.The Dirac Operator in Geometry and Physics 87
M. Llarull: “Sharp estimates and the Dirac Operator”;Math. Ann. 310(1998), 55–71.
M. Llarull: “Scalar curvature estimates for (ri + 4k)-dimensional manifolds”Dif- ferential Geom. Appl. 6(1996), no. 4, 321–326.
M. Min-Oo: “Scalar curvature rigidity of asymptotically hyperbolic spin manifolds”Math. Ann. 285(1989), 527–539.
M. Min-Oo: “Scalar curvature rigidity of certain symmetric spaces”;Geometry topology and dynamics (Montreal),p. 127–136, CRM Proc. Lecture Notes, 15, A. M. 5.,1998.
T. Parker and C. Taubes: “On Witten’s proof of the positive energy theorem”Comm. Math. Phys. 84(1982), 223–238.
L. Polterovich: “Gromov’s K-area and symplectic rigidity”Geom. Funct. Anal. 6.4(1996), 726–739.
L. Polterovich: “Symplectic aspects of the first eigenvalue”J. Reine Angew. Math. 502(1998), 1–17.
O. Reula and K.P. Tod: “Positivity of the Bondi energy”J. Math. Phys. 25(1984), 1004–1008.
R. Schoen and S.T. Yau: “Existence of minimal surfaces and the topology of 3-dimensional manifolds with non-negative scalar curvature”Ann. Math. 110(1979), 127–142.
R. Schoen and S.T. Yau: “On the proof of the positive mass conjecture in general relativity ”Comm. Math. Phys. 65(1979), 45–76.
R. Schoen and S.T. Yau: “The energy and linear-momentum of space-times in general relativity ”Comm. Math. Phys. 79(1981), 47–51.
R. Schoen and S.T. Yau: “Proof of the positive mass theoremII”, Comm. Math. Phys. 79(1981), 231–260.
C. Vafa and E. Witten: “Eigenvalue inequalities for fermions in gauge theories”Comm. Math. Phys. 95(1984), 257–276.
E. Witten: “A new proof of the positive energy theorem”Comm. Math. Phys. 80(1981), 381–402.
E. Witten: “Anti-deSitter space and holography”, Adv.Theor. Math. Phys. 2(1998), 253–290. hep-th/9802150
E. Witten and S.T. Yau: “Connectedness of the boundary in the AdS/CFT cor-respondence”Comm. Math. Phys. 80(1981), 381–402. hep-th/9910245.
E. Woolgar: “The positivity of energy for asymptotically anti-de Sitter space-times”Classical and Quantum Gravity 11.7(1994), 1881–1900.
X. Zhang: “Rigidity of strongly asymptotically hyperbolic spin manifolds”,preprint, 2001.
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Min-Oo, M. (2003). The Dirac Operator in Geometry and Physics. In: Global Riemannian Geometry: Curvature and Topology. Advanced Courses in Mathematics CRM Barcelona. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8055-8_2
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DOI: https://doi.org/10.1007/978-3-0348-8055-8_2
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