Regularity results for pluriclosed flow

Streets, Jeffrey¹ ; Tian, Gang²

¹ Department of Mathematics, University of California, Irvine, Rowland Hall, Irvine, CA 92617, USA
² Department of Mathematics, Princeton University, Fine Hall, Princeton, NJ 08544, USA

Geometry & topology, Tome 17 (2013) no. 4, pp. 2389-2429.

Résumé

In [Int. Math. Res. Not. 16 (2010) 3101–3133] the authors introduced a parabolic flow of pluriclosed metrics. Here we give improved regularity results for solutions to this equation. Furthermore, we exhibit this equation as the gradient flow of the lowest eigenvalue of a certain Schrödinger operator, and show the existence of an expanding entropy functional for this flow. Finally, we motivate a conjectural picture of the optimal regularity results for this flow, and discuss some of the consequences.

Détail

DOI : 10.2140/gt.2013.17.2389

Classification : 32Q55, 53C44, 53C55
Keywords: geometric flow, Hermitian geometry

Bibliographie

[1] B Alexandrov, S Ivanov, Vanishing theorems on Hermitian manifolds, Differential Geom. Appl. 14 (2001) 251

[2] W P Barth, K Hulek, C A M Peters, A Van De Ven, Compact complex surfaces, Ergeb. Math. Grenzgeb. 4, Springer (2004)

[3] S Bochner, Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946) 776

[4] N Buchdahl, On compact Kähler surfaces, Ann. Inst. Fourier (Grenoble) 49 (1999) 287

[5] N Buchdahl, A Nakai–Moishezon criterion for non-Kähler surfaces, Ann. Inst. Fourier (Grenoble) 50 (2000) 1533

[6] E Calabi, On Kähler manifolds with vanishing canonical class, from: "Algebraic geometry and topology. A symposium in honor of S Lefschetz", Princeton Univ. Press (1957) 78

[7] J Cheeger, K Fukaya, M Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992) 327

[8] J Cheeger, M Gromov, M Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982) 15

[9] G Dloussky, K Oeljeklaus, M Toma, Class $\mathrm VII_0$ surfaces with $b_2$ curves, Tohoku Math. J. 55 (2003) 283

[10] I Enoki, On surfaces of class $\mathrm{VII}_{0}$ with curves, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980) 275

[11] M Feldman, T Ilmanen, L Ni, Entropy and reduced distance for Ricci expanders, J. Geom. Anal. 15 (2005) 49

[12] P Gauduchon, Le théorème de l'excentricité nulle, C. R. Acad. Sci. Paris Sér. A-B 285 (1977)

[13] B Guan, Q Li, Complex Monge–Ampere equations on Hermitian manifolds

[14] R S Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255

[15] R S Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. 7 (1999) 695

[16] S Ivanov, G Papadopoulos, Vanishing theorems and string backgrounds, Classical Quantum Gravity 18 (2001) 1089

[17] S Kobayashi, H H Wu, On holomorphic sections of certain Hermitian vector bundles, Math. Ann. 189 (1970) 1

[18] A Lamari, Le cône kählérien d'une surface, J. Math. Pures Appl. 78 (1999) 249

[19] G M Lieberman, Second order parabolic differential equations, World Scientific Publishing Co. (1996)

[20] I Nakamura, On surfaces of class $\mathrm{VII}_0$ with curves, Invent. Math. 78 (1984) 393

[21] T Oliynyk, V Suneeta, E Woolgar, A gradient flow for worldsheet nonlinear sigma models, Nuclear Phys. B 739 (2006) 441

[22] G Perelman, On the entropy formula for Ricci flow and its geometric applications

[23] D H Phong, N Sesum, J Sturm, Multiplier ideal sheaves and the Kähler–Ricci flow, Comm. Anal. Geom. 15 (2007) 613

[24] J Polchinski, String theory, I: an introduction to the bosonic string, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press (1998)

[25] N Šešum, Curvature tensor under the Ricci flow, Amer. J. Math. 127 (2005) 1315

[26] J Streets, Regularity and expanding entropy for connection Ricci flow, J. Geom. Phys. 58 (2008) 900

[27] J Streets, G Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. 2010 (2010) 3101

[28] J Streets, G Tian, Hermitian curvature flow, J. Eur. Math. Soc. 13 (2011) 601

[29] A Teleman, Donaldson theory on non-Kählerian surfaces and class VII surfaces with $b_2=1$, Invent. Math. 162 (2005) 493

[30] A Teleman, The pseudo-effective cone of a non-Kählerian surface and applications, Math. Ann. 335 (2006) 965

[31] G Tian, New results and problems on Kähler–Ricci flow, from: "Géométrie différentielle, physique mathématique, mathématiques et société, II", Astérisque 322, Soc. Math. France (2008) 71

[32] G Tian, Z Zhang, On the Kähler–Ricci flow on projective manifolds of general type, Chinese Ann. Math. Ser. B 27 (2006) 179

[33] B Weinkove, The Calabi–Yau equation on almost-Kähler four-manifolds, J. Differential Geom. 76 (2007) 317