Hostname: page-component-7479d7b7d-m9pkr Total loading time: 0 Render date: 2024-07-09T05:05:40.698Z Has data issue: false hasContentIssue false

SPACE OF RICCI FLOWS (II)—PART A: MODULI OF SINGULAR CALABI–YAU SPACES

Published online by Cambridge University Press:  28 December 2017

XIUXIONG CHEN
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA School of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China; xiu@math.sunysb.edu
BING WANG
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA; bwang@math.wisc.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish the compactness of the moduli space of noncollapsed Calabi–Yau spaces with mild singularities. Based on this compactness result, we develop a new approach to study the weak compactness of Riemannian manifolds.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

References

Abresch, U. and Gromoll, D., ‘On complete manifolds with nonnegative Ricci curvature’, J. Amer. Math. Soc. 3(2) (1990), 355374.Google Scholar
Anderson, M., ‘Convergence and rigidity of manifolds under Ricci curvature bounds’, Invent. Math. 102 (1990), 429445.Google Scholar
Bakry, D., ‘On Sobolev and logarithmic inequalities for Markov semigroups’, inNew Trends in Stochastic Analysis (Charingworth, 1994) (World Scientific Publishing, River Edge, NJ, 1997), 4375.Google Scholar
Bakry, D. and Emery, M., ‘Diffusions hypercontractives’, Seminaire de probabilities XIX (1983/84), 177206.Google Scholar
Cheeger, J., ‘Differentiability of Lipschitz functions on metric measure spaces’, Geom. Funct. Anal. 9 (1999), 428517.CrossRefGoogle Scholar
Cheeger, J., Degeneration of Riemannian Metrics Under Ricci Curvature Bounds, Publications of the Scuola Normale Superiore, Edizioni della Normale, October 1, 2001.Google Scholar
Cheeger, J., ‘Integral Bounds on curvature, elliptic estimates and rectifiability of singular sets’, Geom. Funct. Anal. 13 (2003), 2072.Google Scholar
Cheeger, J. and Colding, T. H., ‘Lower bounds on Ricci curvature and the almost rigidity of warped products’, Ann. of Math. (2) 144(1) (1996), 189237.Google Scholar
Cheeger, J. and Colding, T. H., ‘On the structure of spaces with Ricci curvature bounded below. I’, J. Differential Geom. 45 (1997), 406480.Google Scholar
Cheeger, J., Colding, T. H. and Tian, G., ‘On the singularities of spaces with bounded Ricci curvature’, Geom. Funct. Anal. 12 (2002), 873914.CrossRefGoogle Scholar
Cheeger, J. and Naber, A., ‘Lower bounds on Ricci curvature and quantitative behavior of singular sets’, Invent. Math. 191(2) (2013), 321339.Google Scholar
Cheeger, J. and Simons, J., ‘Differential characters and geometric invariants’, inGeometry and Topology (College Park, MD, 1983/84), Lecture Notes in Mathematics, 1167 (Springer, 1985), 5080.Google Scholar
Colding, T. H., ‘Ricci curvature and volume convergence’, Ann. of Math. (2) 145 (1997), 477501.CrossRefGoogle Scholar
Colding, T. H. and Naber, A., ‘Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications’, Ann. of Math. (2) 176(2) (2012), 11731229.Google Scholar
Chen, X. X. and Wang, B., ‘Space of Ricci flows (I)’, Comm. Pure Appl. Math. 65(10) (2012), 13991457.CrossRefGoogle Scholar
Chen, X. X. and Wang, B., ‘Space of Ricci flows (II)’, Preprint, 2014, arXiv:1405.6797.Google Scholar
Chen, X. X. and Wang, B., ‘Space of Ricci flows (II)—Part B: weak compactness of the flows’, Preprint.Google Scholar
Chen, X. X. and Wang, B., ‘Further details of “Space of Ricci flows (II)”’, Preprint, available at www.math.wisc.edu/∼bwang/HTfurtherdetails.pdf.Google Scholar
Cheng, S. Y. and Yau, S. T., ‘Differential equations on Riemannian manifolds and their geometric applications’, Comm. Pure Appl. Math. 28(3) (1975), 333354.Google Scholar
Coulhon, T. and Saloff-Coste, L., ‘Isopérimétrie pour les groupes et les variétés’, Rev. Mat. Iberoam. 9(2) (1993), 293314.CrossRefGoogle Scholar
Croke, C. B., ‘Some isoperimetric inequalities and eigenvalue estimates’, Ann. Sci. Éc. Norm. Supér. 13 (1980), 419435.Google Scholar
Evans, L. C., Partial Differential Equations, 2nd edn, Graduate Studies in Mathematics, 19 (American Mathematical Society, 2010).Google Scholar
Falconer, K., Fractal Geometry, Mathematical Foundations and Applications (John Wiley and Sons, 1990).Google Scholar
Fukushima, M., Oshima, Y. and Takeda, M., Dirichlet Forms and Symmetric Markov Processes (de Gruyter, Berlin, 1994).Google Scholar
Gigli, N., ‘The splitting theorem in non-smooth context’, Preprint, 2013, arXiv:1302.5555.Google Scholar
Greene, B., The Elegant Universe (W.W. Norton and Co., 2003), ISBN 0-393-05858-1.Google Scholar
Grigor’yan, A. A., ‘The heat equation on noncompact Riemannian manifolds (Russian)’, Mat. Sb. 182(1) (1991), 5587. English translation: Math. USSR, Sb. 72(1) (1992), 47–77.Google Scholar
Hamilton, R. S., ‘A matrix Harnack estimate for the heat equation’, Comm. Anal. Geom. 1(1) (1993), 113126.Google Scholar
Hamilton, R. S., ‘A compactness property for solutions of the Ricci flow’, Amer. J. Math. 117(3) (1995), 545572.Google Scholar
Kinnunen, J. and Shanmugalingam, N., ‘Regularity of quasi-minimizers on metric spaces’, Manuscripta Math. 105 (2001), 401423.Google Scholar
Koskela, P., Rajala, K. and Shanmugalingam, N., ‘Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces’, J. Funct. Anal. 202 (2003), 147173.CrossRefGoogle Scholar
Koskela, P. and Zhou, Y., ‘Geometry and analysis of Dirichlet forms’, Adv. Math. 231(5) (2012), 27552801.Google Scholar
Li, P. and Yau, S. T., ‘On the parabolic kernel of the Schrödinger operator’, Acta Math. 156 (1986), 153201.Google Scholar
Perelman, G., ‘The entropy formula for the Ricci flow and its geometric applications’, Preprint, arXiv:math.DG/0211159.Google Scholar
Shanmugalingam, N., ‘Newtonian spaces: an extension of Sobolev spaces to metric measure spaces’, Rev. Mat. Iberoam. 16(2) (2000), 243279.CrossRefGoogle Scholar
Shanmugalingam, N., ‘Harmonic functions on metric spaces’, Illinois J. Math. 45(3) (2001), 10211050.Google Scholar
Saloff-Coste, L., ‘Uniformly elliptic operators on Riemannian manifolds’, J. Differential Geom. 36 (1992), 417450.Google Scholar
Saloff-Coste, L., ‘A note on Poincaré, Sobolev, and Harnack inequalities’, Int. Math. Res. Not. IMRN 2 (1992), 2738.Google Scholar
Saloff-Coste, L., ‘Sobolev inequalities in familiar and unfamiliar settings’, inSobolev Spaces in Mathematics I, Int. Math. Ser., 8 (Springer, New York, 2009), 299343.Google Scholar
Sturm, K. T., ‘Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and L p -Liouville properties’, J. Reine Angew. Math. 456 (1994), 173196.Google Scholar
Sturm, K. T., ‘Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations’, Osaka J. Math. 32(2) (1995), 275312.Google Scholar
Sturm, K. T., ‘Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality’, J. Math. Pures Appl. 75(9) (1996), 273297.Google Scholar
Wei, G. F., ‘Manifolds with a lower Ricci curvature bound’, Preprint, arXiv:math/0612107.Google Scholar
Zhu, S. H., ‘The comparison geometry of Ricci curvature’, inComparison Geometry 30 (MSRI Publications, 1997), 221262.Google Scholar