zbMATH — the first resource for mathematics

Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized Kähler manifolds. (English) Zbl 1347.53055
Let \((M^{2n},g,J)\) be a Hermitian manifold. The metric is pluriclosed if \(\partial \bar{\partial} \omega=0\). The pluriclosed flow is a natural geometric flow on complex manifolds which preserves pluriclosed metrics. The author proves long-time existence and convergence results for this flow, and geometric and topological rigidity results which follow as corollaries. A long-time existence and convergence result for the pluriclosed flow in the setting of commuting generalized Kähler geometry is also derived. A generalized Käher structure on a compact manifold \(M\) is defined as a triple \((g,J_A,J_B)\) made of a Riemannian metric and two integrable complex structures satisfying the equations: \(d_A^c\omega_A=-d_B^c\omega_B\), \(dd_A^c\omega_A=-dd_B^c\omega_B=0\). To achieve these results the reduction of the pluriclosed flow to a degenerate parabolic equation for a \((1,0)\)-form is used. The convergence of the flow is proved using a generalization of Yau’s oscillation estimate.
Moreover the author develops a number of new a priori estimates for the pluriclosed flow which lead to general long-time existence and convergence result, given some topological constraints. This implies a rigidity result showing that under certain topological conditions, generalized Kähler structures are automatically covered by products of Calabi-Yau manifolds. The first principal estimate is an a priori \(C^\alpha\) estimate for the metric in the presence of upper and lower bounds on the metric. Considering of the pluriclosed flow as a parabolic system of equations for the Hermitian metric \(g\), this estimate is analogous to the DeGiorgi-Nash-Moser/Krylov-Safonov estimate for uniformly parabolic equations. We point out that these are only analogies, and that moreover, the DeGiorgi-Nash-Moser/Krylov-Safonov results are known to be false in general for the systems of equations which are considered here. The relations among the methods exploited here and in other settings are also analyzed.
Anyhow, by a careful study of the 1-form potential \(\alpha\) along the pluriclosed flow, the author discovers a judicious combination of the first derivatives of \(\alpha\) into a Hermitian \(2n\times 2n\) matrix \(W\) such that \(W(v,v)\) is a subsolution to a uniformly parabolic equation for every \(v\), and such that \(\det W=1\). This matrix \(W\) admits an interpretation as a “Born-Infeld” metric on the generalized tangent bundle \(T\oplus T^\ast\). This motivates the author to discuss the connection between the pluriclosed flow and generalized geometry. The second principle estimate is a general upper bound for the metric in terms of a lower bound. The proof exploits the very favorable evolution equations arising from the 1-form reduction of pluriclosed flow to control some torsion terms arising in the evaluation of metric quantities.

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
35R01 PDEs on manifolds
35K96 Parabolic Monge-Ampère equations
Full Text: DOI
[1] Aeppli, A. (1965). On the cohomology structure of Stein manifolds. Proceedings of the Conference on Complex Analysis (Minneapolis, Minn. 1964). Berlin: Springer, pp. 58–70. · doi:10.1007/978-3-642-48016-4_7
[2] DOI: 10.1007/s00220-007-0196-4 · Zbl 1135.53018 · doi:10.1007/s00220-007-0196-4
[3] Beauville A., Complex Analysis and Algebraic Geometry pp 61– (2000)
[4] DOI: 10.1007/BF01443359 · Zbl 0666.58042 · doi:10.1007/BF01443359
[5] DOI: 10.5802/aif.1674 · Zbl 0926.32025 · doi:10.5802/aif.1674
[6] Cherrier P., Bull. Sci. Math 111 (2) pp 343– (1987)
[7] DeGiorgi E., Classe di Scienze Fisiche, Matematicahe e Naturali 3 (3) pp 25– (1957)
[8] DeGiorgi E., Boll. Un. Mat. Ital 1968 pp 135– (1968)
[9] DOI: 10.1007/s12220-013-9449-y · Zbl 1325.53084 · doi:10.1007/s12220-013-9449-y
[10] DOI: 10.1002/cpa.3160350303 · Zbl 0469.35022 · doi:10.1002/cpa.3160350303
[11] DOI: 10.1016/0550-3213(84)90592-3 · doi:10.1016/0550-3213(84)90592-3
[12] Gauduchon P., C. R. Acad. Sci. Paris 285 pp 387– (1977)
[13] DOI: 10.4310/CAG.2011.v19.n2.a2 · Zbl 1251.32035 · doi:10.4310/CAG.2011.v19.n2.a2
[14] DOI: 10.4007/annals.2011.174.1.3 · Zbl 1235.32020 · doi:10.4007/annals.2011.174.1.3
[15] DOI: 10.1007/s00220-014-1926-z · Zbl 1304.53080 · doi:10.1007/s00220-014-1926-z
[16] DOI: 10.1093/qmath/hag025 · doi:10.1093/qmath/hag025
[17] Kobayashi S., Nagoya Math. J 77 pp 5– (1980)
[18] Krylov N.V., Izv. Akad. Mauk SSR Ser. Math 46 pp 487– (1982) · Zbl 0939.53024 · doi:10.4310/CAG.1999.v7.n4.a2
[19] Krylov N.V., Doklady Akademii Nauk SSSR 245 pp 18– (1979)
[20] Krylov N.V., Izvestiya Akademii Nauk SSSR. Ser. Mat 44 pp 161– (1980)
[21] DOI: 10.1007/BF02399203 · Zbl 0611.58045 · doi:10.1007/BF02399203
[22] DOI: 10.1142/3302 · doi:10.1142/3302
[23] DOI: 10.1002/cpa.3160130308 · Zbl 0111.09301 · doi:10.1002/cpa.3160130308
[24] DOI: 10.1073/pnas.43.8.754 · Zbl 0078.08704 · doi:10.1073/pnas.43.8.754
[25] DOI: 10.1088/1126-6708/1999/09/032 · Zbl 0957.81085 · doi:10.1088/1126-6708/1999/09/032
[26] Streets J., Int. Math. Res. Notices 2010 pp 3101– (2010)
[27] DOI: 10.4171/JEMS/262 · Zbl 1214.53055 · doi:10.4171/JEMS/262
[28] DOI: 10.1016/j.nuclphysb.2012.01.008 · Zbl 1246.53091 · doi:10.1016/j.nuclphysb.2012.01.008
[29] DOI: 10.2140/gt.2013.17.2389 · Zbl 1272.32022 · doi:10.2140/gt.2013.17.2389
[30] DOI: 10.1007/BF01168452 · Zbl 0519.35007 · doi:10.1007/BF01168452
[31] DOI: 10.1090/S0894-0347-2010-00673-X · Zbl 1208.53075 · doi:10.1090/S0894-0347-2010-00673-X
[32] DOI: 10.4310/AJM.2010.v14.n1.a3 · Zbl 1208.32034 · doi:10.4310/AJM.2010.v14.n1.a3
[33] DOI: 10.2307/2373880 · Zbl 0424.53040 · doi:10.2307/2373880
[34] DOI: 10.1002/cpa.3160310304 · 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.