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Lower order tensors in non-Kähler geometry and non-Kähler geometric flow. (English) Zbl 1354.53075

Summary: In recent years, Streets and Tian introduced a series of curvature flows to study non-Kähler geometry. In this paper, we study how to construct the second-order curvature flows in a uniform way, under some natural assumptions which hold in Streets and Tian’s works. As a result, by classifying the lower order tensors, we classify the second-order curvature flows in almost Hermitian, almost Kähler, and Hermitian geometries in certain sense. In particular, the Symplectic Curvature Flow is the unique way to generalize Ricci Flow on almost Kähler manifolds.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)

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References:

[1] Boling, J.: Homogeneous solutions of pluriclosed flow on closed complex surfaces. arXiv:1404.7106 (2014) · Zbl 1347.53036
[2] http://mathoverflow.net/questions/72906/how-to-deduce-this-equation-for-a-4-dim-almost-kahler-manifold · Zbl 1214.53055
[3] Cao, H.D.: Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359-372 (1985) · Zbl 0574.53042 · doi:10.1007/BF01389058
[4] Dai, S.: A curvature flow unifying symplectic curvature flow and pluriclosed flow. Pac. J. Math. 277(2), 287-311 (2015) · Zbl 1331.53047 · doi:10.2140/pjm.2015.277.287
[5] Enrietti, N.: Static SKT metrics on Lie groups. Manuscripta Math. 140(3-4), 557-571 (2013) · Zbl 1308.32026 · doi:10.1007/s00229-012-0552-3
[6] Enrietti, N., Fino, A., Vezzoni, L.: The pluriclosed flow on nilmanifolds and Tamed symplectic forms. J. Geom. Anal. 25(2), 883-909 (2015) · Zbl 1325.53084 · doi:10.1007/s12220-013-9449-y
[7] Fernández-Culma, E.: Soliton almost Kähler structures on 6-dimensional nilmanifolds for the symplectic curvature flow. J. Geom. Anal. 25(4), 2736-2758 (2015) · Zbl 1345.57029 · doi:10.1007/s12220-014-9534-x
[8] Fino, A., Vezzoni, L.: Special Hermitian metrics on compact solvmanifolds. J. Geom. Phys. 91, 40-53 (2015) · Zbl 1318.53049 · doi:10.1016/j.geomphys.2014.12.010
[9] Gauduchon, P.: Hermitian connections and Dirac operators. Bollettino della Unione Matematica Italiana-B 2, 257-288 (1997) · Zbl 0876.53015
[10] Gill, M.: Convergence of the parabolic complex Monge-Amp ére equation on compact Hermitian manifolds. Commun. Anal. Geom. 19(2), 277-303 (2011) · Zbl 1251.32035 · doi:10.4310/CAG.2011.v19.n2.a2
[11] Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255-306 (1982) · Zbl 0504.53034
[12] Lauret, J.: Curvature flows for almost-Hermitian Lie groups. Trans. Am. Math. Soc. 367, 7453-7480 (2015) · Zbl 1323.53078 · doi:10.1090/S0002-9947-2014-06476-3
[13] Lauret, J., Will, C.: On the symplectic curvature flow for locally homogeneous manifolds. arXiv:1405.6065 (2014) · Zbl 1366.53049
[14] Lê, H.V., Wang, G.F.: Anti-complexified Ricci flow on compact symplectic manifolds. J. Reign Angew. Math. 530, 17-31 (2011) · Zbl 0985.53037
[15] Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifolds. Ann. Math. 65(3), 391-404 (1957) · Zbl 0079.16102 · doi:10.2307/1970051
[16] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159 (2002) · Zbl 1130.53001
[17] Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math/0303109 (2003) · Zbl 1130.53002
[18] Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math/0307245 (2003) · Zbl 1130.53003
[19] Pook, J.: Homogeneous and locally homogeneous solutions to symplectic curvature flow. arXiv:1202.1427 (2012) · Zbl 1273.00015
[20] Smith, D.: Stability of the almost Hermitian curvature flow. Thesis (Ph.D.), Michigan State University, pp. 60 (2013) · Zbl 1308.32026
[21] Streets, J.: Pluriclosed flow on generalized Kähler manifolds with split tangent bundle. arXiv:1405.0727 (2014) · Zbl 1393.53066
[22] Streets, J.: Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized Kähler manifolds. Comm. Partial Differ. Equ. 41(2), 318-374 (2016) · Zbl 1347.53055 · doi:10.1080/03605302.2015.1116560
[23] Streets, J.: Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces. arXiv:1601.02981 (2016) · Zbl 1371.53069
[24] Streets, J., Tian, G.: A parabolic flow of pluriclosed metrics. Int. Math. Res. Not. 16, 3101-3133 (2010) · Zbl 1198.53077
[25] Streets, J., Tian, G.: Regularity results for pluriclosed flow. Geom. Topol. 17(4), 2389-2429 (2013) · Zbl 1272.32022 · doi:10.2140/gt.2013.17.2389
[26] Streets, J., Tian, G.: Hermitian curvature flow. J. Eur. Math. Soc. 13(3), 601-634 (2011) · Zbl 1214.53055 · doi:10.4171/JEMS/262
[27] Streets, J., Tian, G.: Symplectic curvature flow. J. Reine Angew. Math. 696, 143-185 (2014) · Zbl 1305.53083
[28] Streets, J., Tian, G.: Generalized K ähler geometry and the pluriclosed flow. Nucl. Phys. B 858(2), 366-376 (2012) · Zbl 1246.53091 · doi:10.1016/j.nuclphysb.2012.01.008
[29] Streets, J., Warren, M.: Evans-Krylov estimates for a nonconvex Monge-Ampère equation. arXiv:1410.2911 (2014) · Zbl 1342.35124
[30] Tossati, V., Weinkove, B.: On the evolution of a Hermitian metric by its Chern-Ricci form. J. Differ. Geom. 99(1), 125-163 (2015) · Zbl 1317.53092
[31] Vezzoni, L.: On Hermitian curvature flow on almost complex manifolds. Differ. Geom. Appl. 29(5), 709-722 (2011) · Zbl 1225.53030 · doi:10.1016/j.difgeo.2011.07.006
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