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On a Calabi-type estimate for pluriclosed flow. (English) Zbl 1436.53074
Summary: The regularity theory for pluriclosed flow hinges on obtaining \(C^\alpha\) regularity for the metric assuming uniform equivalence to a background metric. This estimate was established in [14] by an adaptation of ideas from Evans-Krylov, the key input being a sharp differential inequality satisfied by the associated ‘generalized metric’ defined on \(T \oplus T^\ast\). In this work we give a sharpened form of this estimate with a simplified proof. To begin we show that the generalized metric itself evolves by a natural curvature quantity, which leads quickly to an estimate on the associated Chern connections analogous to, and generalizing, Calabi-Yau’s \(C^3\) estimate for the complex Monge-Ampère equation.
MSC:
53E30 Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows)
35K55 Nonlinear parabolic equations
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[1] V. Apostolov, J. Streets, The nondegenerate generalized Kähler Calabi-Yau problem, ArXiv e-prints, Mar. 2017.
[2] Calabi, E., Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Mich. Math. J., 5, 105-126 (1958) · Zbl 0113.30104
[3] De Giorgi, E., Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. (3), 3, 25-43 (1957) · Zbl 0084.31901
[4] De Giorgi, E., Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Unione Mat. Ital. (4), 1, 135-137 (1968) · Zbl 0155.17603
[5] Evans, L. C., Classical solutions of fully nonlinear, convex, second-order elliptic equations, Commun. Pure Appl. Math., 35, 3, 333-363 (1982) · Zbl 0469.35022
[6] Ivanov, S.; Papadopoulos, G., Vanishing theorems and string backgrounds, Class. Quantum Gravity, 18, 6, 1089-1110 (2001) · Zbl 0990.53078
[7] Krylov, N. V., Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR, Ser. Mat., 46, 3, 487-523 (1982), 670 · Zbl 0511.35002
[8] Krylov, N. V.; Safonov, M. V., An estimate for the probability of a diffusion process hitting a set of positive measure, Dokl. Akad. Nauk SSSR, 245, 1, 18-20 (1979)
[9] Krylov, N. V.; Safonov, M. V., A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR, Ser. Mat., 44, 1, 161-175 (1980), 239
[10] M.-C. Lee, J. Streets, Complex manifolds with negative curvature operator, ArXiv e-prints, Mar. 2019.
[11] Moser, J., A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Commun. Pure Appl. Math., 13, 457-468 (1960) · Zbl 0111.09301
[12] Nash, J., Continuity of solutions of parabolic and elliptic equations, Am. J. Math., 80, 931-954 (1958) · Zbl 0096.06902
[13] Phong, D. H.; Sesum, N.; Sturm, J., Multiplier ideal sheaves and the Kähler-Ricci flow, Commun. Anal. Geom., 15, 3, 613-632 (2007) · Zbl 1143.53064
[14] Streets, J., Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized Kähler manifolds, Commun. Partial Differ. Equ., 41, 2, 318-374 (2016) · Zbl 1347.53055
[15] Streets, J., Pluriclosed flow on manifolds with globally generated bundles, Complex Manifolds, 3, 1, 222-230 (2016) · Zbl 1351.32034
[16] Streets, J., Pluriclosed flow on generalized Kähler manifolds with split tangent bundle, J. Reine Angew. Math., 739, 241-276 (2018) · Zbl 1393.53066
[17] Streets, J.; Tian, G., A parabolic flow of pluriclosed metrics, Int. Math. Res. Not., 16, 3101-3133 (2010) · Zbl 1198.53077
[18] Yau, S. T., On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Commun. Pure Appl. Math., 31, 3, 339-411 (1978) · Zbl 0369.53059
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