×
Sesum, Natasa

Linear and dynamical stability of Ricci-flat metrics. (English) Zbl 1103.53040

Duke Math. J. 133, No. 1, 1-26 (2006).
The author studies in this paper the problem of stability of solutions of the Ricci-flow equation \({{d}\over{dt}}g_{ij}=-2R_{ij}\) on a closed manifold \(M\) endowed with an initial smooth metric \(g_0\). The main result is that if \(g_0\) is dynamically stable it is linearly stable too. Furthermore, if \(g_0\) is linearly stable and integrable, then it is weakly dynamically stable. Dynamic stability means that there exist a \(C^k\)-neighborhood \({\mathcal U}\) of the metric \(g_0\) such that the Ricci-flow \(\widetilde{g}(t)\) of every metric \(\widetilde{g}\in{\mathcal U}\) exists for all times \(t\in[0,\infty)\) and converges to \(g_0\). Weak dynamic stability means that the Ricci-flow \(\widetilde{g}(t)\) of every metric \(\widetilde{g}\in{\mathcal U}\) exists to all times \(t\in[0,\infty)\) and converges. Linear stability means that \(\int_M(L(h),h)\,dV_{g_0}\leq 0\), where \(L\) is a suitable linear differential operator and \(h\) represents the linearized metric. Applications to \(K3\)-surfaces are also obtained.
Reviewer: Agostino Prástaro (Roma)

Cited in 2 Reviews
Cited in 31 Documents

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K55 Nonlinear parabolic equations

Keywords:

stability in Ricci-flows; K3-surfaces
PDFBibTeX XMLCite
\textit{N. Sesum}, Duke Math. J. 133, No. 1, 1--26 (2006; Zbl 1103.53040)
Full Text: DOI arXiv

References:

[1] A. L. Besse, Einstein Manifolds , Ergeb. Math. Grenzgeb. (3) 10 , Springer, Berlin, 1987. · Zbl 0613.53001
[2] F. A. Bogomolov, Hamiltonian Kählerian manifolds (in Russian), Dokl. Akad. Nauk SSSR 243 (1978), no. 5, 1101–1104.; English translation in Soviet Math. Dokl. 19 (1978), no. 6, 1462–1465.
[3] H. D. Cao, Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds , Invent. Math. 81 (1985), 359–372. · Zbl 0574.53042 · doi:10.1007/BF01389058
[4] H. D. Cao, R. Hamilton, and T. Ilmanen, Gaussian densities and stability for some Ricci solitons ,
[5] J. Cheeger and G. Tian, On the cone structure at infinity of Ricci-flat manifolds with Euclidean volume growth and quadratic curvature decay , Invent. Math. 118 (1994), 493–571. · Zbl 0814.53034 · doi:10.1007/BF01231543
[6] B. Chow and D. Knopf, The Ricci Flow: An Introduction , Math. Surveys Monogr. 110 , Amer. Math. Soc., Providence, 2004. · Zbl 1086.53085
[7] X. Dai, X. Wang, and G. Wei, On the stability of Riemannian manifold with parallel spinors , Invent. Math. 161 (2005), 151–176. · Zbl 1075.53042 · doi:10.1007/s00222-004-0424-x
[8] D. M. Deturck, Deforming metrics in the direction of their Ricci tensors , J. Differential Geom. 18 (1983), 157–162. · Zbl 0517.53044
[9] J. Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds , Amer. J. Math. 86 (1964), 109–160. JSTOR: · Zbl 0122.40102 · doi:10.2307/2373037
[10] P. Grisvard and G. Da Prato, Equations d’évolution abstraites non linéaires de type parabolique , Ann. Mat. Pura Appl. (4) 120 (1979), 329–396. · Zbl 0471.35036 · doi:10.1007/BF02411952
[11] C. Guenther, J. Isenberg, and D. Knopf, Stability of the Ricci flow at Ricci-flat metrics , Comm. Anal. Geom. 10 (2002), 741–777. · Zbl 1028.53043
[12] R. S. Hamilton, Three-manifolds with positive Ricci curvature , J. Differential Geom. 17 (1982), 255–306. · Zbl 0504.53034
[13] -, A compactness property for solutions of the Ricci flow , Amer. J. Math. 117 (1995), 545–572. JSTOR: · Zbl 0840.53029 · doi:10.2307/2375080
[14] G. Perelman, The entropy formula for the Ricci flow and its geometric applications , · Zbl 1130.53001
[15] N. Sesum, The limiting behaviour of the Ricci flow , Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Mass., 2004. · Zbl 1051.53036
[16] W.-X. Shi, Deforming the metric on complete Riemannian manifolds , J. Differential Geom. 30 (1989), 223–301. · Zbl 0676.53044
[17] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems , Ann. of Math. (2) 118 (1983), 525–571. JSTOR: · Zbl 0549.35071 · doi:10.2307/2006981
[18] G. Simonett, Center manifolds for quasilinear reaction-diffusion systems , Differential Integral Equations 8 (1995), 753–796. · Zbl 0815.35054
[19] Y. T. Siu, Every \(K3\) surface is Kähler , Invent. Math. 73 (1983), 139–150. · Zbl 0557.32004 · doi:10.1007/BF01393829
[20] G. Tian, “Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric” in Mathematical Aspects of String Theory (San Diego, 1986) , Adv. Ser. Math. Phys. 1 , World Scientific, Singapore, 1987, 629–646.
[21] A. N. Todorov, Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of \(K3\) surfaces , Invent. Math. 61 (1980), 251–265. · Zbl 0472.14006 · doi:10.1007/BF01390067
[22] -, “How many Kähler metrics has a \(K3\) surface?” in Arithmetic and Geometry, Vol. II: Geometry , Progr. Math. 36 , Birkhäuser, Boston, 1983, 451–463. · Zbl 0553.14017
[23] -, The Weil-Petersson geometry of the moduli space of, SU \((n\geq 3)\) (Calabi-Yau) manifolds, I, Comm. Math. Phys. 126 (1989), 325–346. · Zbl 0688.53030 · doi:10.1007/BF02125128
[24] S. T. Yau, Calabi’s conjecture and some new results in algebraic geometry , Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1798–1799. · Zbl 0355.32028 · doi:10.1073/pnas.74.5.1798
[25] -, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I , Comm. Pure Appl. Math. 31 (1978), 339–411. · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.