Four-manifolds with positive isotropic curvature

@article{Hamilton1997FourmanifoldsWP,
  title={Four-manifolds with positive isotropic curvature},
  author={Richard S. Hamilton},
  journal={Communications in Analysis and Geometry},
  year={1997},
  volume={5},
  pages={1-92}
}
  • R. Hamilton
  • Published 1997
  • Mathematics
  • Communications in Analysis and Geometry
1. Positive Isotropic Curvature 2 (1) The Result 2 (2) The Algebra of Isotropic Curvature 4 2. Curvature Pinching 6 (1) Pinching Estimates which are Preserved 6 (2) Pinching Estimates which Improve 13 (3) Necklike Curvature Pinching 21 3. The Geometry of Necks 27 (1) Harmonic Parametrizations by Spheres 27 (2) Geometric Necks 30 (3) Curvature Necks 35 (4) The Fundamental Group 41 (5) Finding Necks 44 4. Surgery 47 (1) How to do Surgery 47 (2) Curvature Changes under Surgery .49 (3) Pinching… 
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Highly Influential Citations
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216 Citations

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References

SHOWING 1-4 OF 4 REFERENCES
Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes
There has been much interest among differential geometers in finding relationships between curvature and topology of Riemannian manifolds. For the most part, efforts have been directed towards…
Ricci deformation of the metric on a Riemannian manifold
The interaction between algebraic properties of the curvature tensor and the global topology and geometry of a Riemannian manifold has been studied extensively. Of particular interest is the question…
Three-manifolds with positive Ricci curvature
Matrix Harnack estimate for the heat equation