Moser–Trudinger type inequalities for complex Monge–Ampère operators and Aubin’s “hypothèse fondamentale”

Robert J. Berman; Bo Berndtsson

doi:10.5802/afst.1704

Robert J. Berman ¹ ; Bo Berndtsson ¹

¹ Mathematical Sciences - Chalmers University of Technology and University of Gothenburg - SE-412 96 Gothenburg, Sweden

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 3, pp. 595-645.

Résumé

We prove a version of Aubin’s “Hypothese fondamentale” concerning the existence of Moser–Trudinger type inequalities on any integral compact Kähler manifold $X$ . In the case of the anti-canonical class on a Fano manifold the constants in the inequalities are shown to only depend on the dimension of $X$ (but there are counterexamples to the precise value proposed by Aubin). In the different setting of pseudoconvex domains in complex space we also obtain a quasi-sharp version of the inequalities and relate it to Brezis–Merle type inequalities for the complex Monge–Ampère operator, recently considered by Demailly and Åhag–Cegrell–Kołodziej–Phạm–Zeriahi. The inequalities are shown to be sharp for $S^{1}$ -invariant functions on the unit ball.

Publié le : 2022-07-01

DOI : 10.5802/afst.1704

Affiliations des auteurs :

Robert J. Berman ¹ ; Bo Berndtsson ¹

¹ Mathematical Sciences - Chalmers University of Technology and University of Gothenburg - SE-412 96 Gothenburg, Sweden

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Robert J. Berman; Bo Berndtsson. Moser–Trudinger type inequalities for complex Monge–Ampère operators and Aubin’s “hypothèse fondamentale”. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 3, pp. 595-645. doi : 10.5802/afst.1704. https://afst.centre-mersenne.org/articles/10.5802/afst.1704/

Bibliographie
Cité par

[1] Per Åhag; Urban Cegrell; Rafal Czyz On Dirichlet’s principle and problem, Math. Scand., Volume 110 (2012) no. 2, pp. 235-250 | DOI | MR | Zbl

[2] Per Åhag; Urban Cegrell; Sławomir Kołodziej; Hoàng Hiệp Phạm; Ahmed Zeriahi Partial pluricomplex energy and integrability exponents of plurisubharmonic functions, Adv. Math., Volume 222 (2009) no. 6, pp. 2036-2058 | DOI | MR | Zbl

[3] Thierry Aubin Réduction du cas positif de l´equation de Monge–Ampère sur les variétés kahleriennes compactes à la démonstration d’une inégalité, J. Funct. Anal., Volume 57 (1984), pp. 143-153 | DOI | MR | Zbl

[4] Thierry Aubin Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer, 1998 | DOI | Zbl

[5] William Beckner Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. Math., Volume 138 (1993) no. 1, pp. 213-242 | DOI | MR | Zbl

[6] Slimane Benelkourchi; Vincent Guedj; Ahmed Zeriahi Plurisubharmonic functions with weak singularities, Complex analysis and digital geometry (Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist.), Volume 86, Univ. Uppsala, 2009, pp. 57-74 | MR | Zbl

[7] Robert J. Berman Analytic torsion, vortices and positive Ricci curvature (2010) (https://arxiv.org/abs/1006.2988)

[8] Robert J. Berman A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kahler–Einstein metrics, Adv. Math., Volume 248 (2013), pp. 1254-1297 | DOI | Zbl

[9] Robert J. Berman; Bo Berndtsson Symmetrization of Plurisubharmonic and Convex Functions, Indiana Univ. Math. J., Volume 63 (2014) no. 2, pp. 345-365 | DOI | MR | Zbl

[10] Robert J. Berman; Bo Berndtsson The volume of Kähler–Einstein Fano varieties and convex bodies, J. Reine Angew. Math., Volume 723 (2017), pp. 127-152 | Zbl

[11] Robert J. Berman; Sébastien Boucksom; Vincent Guedj; Ahmed Zeriahi A variational approach to complex Monge–Ampère equations, Publ. Math., Inst. Hautes Étud. Sci., Volume 117 (2013) no. 1, pp. 179-245 | DOI | Numdam | Zbl

[12] Robert J. Berman; Sébastien Boucksom; Philippe Eyssidieux; Vincent Guedj; Ahmed Zeriahi Kähler–Einstein metrics and the Kähler-Ricci flow on log Fano varieties, J. Reine Angew. Math., Volume 751 (2019), pp. 27-89 | DOI | Zbl

[13] Robert J. Berman; Jean-Pierre Demailly Regularity of plurisubharmonic upper envelopes in big cohomology classes, Perspectives in analysis, geometry, and topology (Progress in Mathematics), Volume 296, Birkhäuser, 2012, pp. 39-66 | DOI | MR | Zbl

[14] Robert J. Berman; Gerard Freixas i Montplet An arithmetic Hilbert-Samuel theorem for singular hermitian line bundles and cusp forms, Compos. Math., Volume 150 (2014) no. 10, pp. 1703-1728 | DOI | MR | Zbl

[15] Bo Berndtsson Curvature of vector bundles associated to holomorphic fibrations, Ann. Math., Volume 169 (2009) no. 2, pp. 531-560 | DOI | MR | Zbl

[16] Bo Berndtsson Positivity of direct image bundles and convexity on the space of Kähler metrics, J. Differ. Geom., Volume 81 (2009) no. 3, pp. 457-482 | Zbl

[17] Bo Berndtsson A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry, Invent. Math., Volume 200 (2015) no. 1, pp. 149-200 | DOI | Zbl

[18] Bo Berndtsson A comparison principle for Bergman kernels, Analysis meets geometry (Trends in Mathematics), Birkhäuser, 2017, pp. 121-126 | DOI | Zbl

[19] Bo Berndtsson; Mihai Păun Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J., Volume 145 (2008) no. 2, pp. 341-378 | DOI | MR | Zbl

[20] Zbigniew Błocki On the $L^{p}$ -stability for the complex Monge–Ampère operator, Mich. Math. J., Volume 42 (1995) no. 2, pp. 269-275 | Zbl

[21] Sébastien Boucksom; Philippe Eyssidieux; Vincent Guedj; Ahmed Zeriahi Monge–Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | DOI | Zbl

[22] Haïm Brézis; Frank Merle Uniform estimates and blow-up behavior for solutions of $- Δ u = V (x) e^{u}$ in two dimensions, Commun. Partial Differ. Equations, Volume 16 (1992) no. 8-9, pp. 1223-1253 | DOI | MR | Zbl

[23] Emanuele Caglioti; Pierre-Louis Lions; Carlo Marchioro; Mario Pulvirenti A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Commun. Math. Phys., Volume 143 (1992) no. 3, pp. 501-525 | DOI | MR | Zbl

[24] Frédéric Campana Connexite rationnelle des varietes de Fano, Ann. Sci. Éc. Norm. Supér., Volume 25 (1992) no. 5, pp. 539-545 | DOI | Numdam | MR | Zbl

[25] Urban Cegrell Pluricomplex energy, Acta Math., Volume 180 (1998) no. 2, pp. 187-217 | DOI | MR | Zbl

[26] Urban Cegrell Approximation of plurisubharmonic functions in hyperconvex domains, Complex analysis and digital geometry (Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist.), Volume 86, Univ. Uppsala, 2009, pp. 125-129 | MR | Zbl

[27] Urban Cegrell; Leif Persson The Dirichlet problem for the complex Monge–Ampère operator: stability in $L^{2}$ , Mich. Math. J., Volume 39 (1992) no. 1, pp. 145-151 | Zbl

[28] Urban Cegrell; Ahmed Zeriahi Subextension of plurisubharmonic functions with bounded Monge–Ampère mass, C. R. Acad. Sci. Paris, Volume 336 (2003) no. 4, pp. 305-308 | DOI | Zbl

[29] Ivan A. Cheltsov Log canonical thresholds of Fano threefold hypersurfaces (Russian), Izv. Math., Volume 73 (2009) no. 4, pp. 77-152

[30] Tamás Darvas; Eleonora Di Nezza; Chinh H. Lu On the singularity type of full mass currents in big cohomology classes, Compos. Math., Volume 154 (2018) no. 2, pp. 380-409 | DOI | MR | Zbl

[31] Philippe Delanoë Radially symmetric boundary value problems for real and complex elliptic Monge–Ampère equations, J. Differ. Equations, Volume 58 (1985), pp. 318-344 | DOI | Zbl

[32] Jean-Pierre Demailly Estimates on Monge–Ampère operators derived from a local algebra inequality, Complex analysis and digital geometry (Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist.), Volume 86, Univ. Uppsala, 2009, pp. 131-143 | Zbl

[33] Jean-Pierre Demailly; János Kollár Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér., Volume 34 (2001) no. 4, pp. 525-556 | DOI | Numdam | Zbl

[34] Jean-Pierre Demailly; Hoàng Hiệp Phạm A sharp lower bound for the log canonical threshold, Acta Math., Volume 212 (2014) no. 1, pp. 1-9 | DOI | MR | Zbl

[35] Eleonora Di Nezza; Vincent Guedj; Chinh H. Lu Finite entropy vs finite energy, Comment. Math. Helv., Volume 96 (2021) no. 2, pp. 389-419 | DOI | MR | Zbl

[36] Weiyue Ding Remarks on the existence problem of positive Kähler–Einstein metrics, Math. Ann., Volume 282 (1988) no. 3, pp. 463-472 | DOI | Zbl

[37] Weiyue Ding; Gang Tian The generalized Moser–Trudinger Inequality, Nonlinear analysis and microlocal analysis (Nankai Series in Pure, Applied Mathematics and Theoretical Physics), Volume 2, World Scientific, 1993, pp. 57-70 | Zbl

[38] Tommaso de Fernex; Lawrence Ein; Mircea Mustaţă Multiplicities and log canonical threshold, J. Algebr. Geom., Volume 13 (2004) no. 3, pp. 603-615 | DOI | MR | Zbl

[39] Luigi Fontana Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv., Volume 68 (1993) no. 3, pp. 415-454 | DOI | MR | Zbl

[40] Kento Fujita Optimal bounds for the volumes of Kähler–Einstein Fano manifolds, Am. J. Math., Volume 140 (2018) no. 2, pp. 391-414 | DOI | Zbl

[41] Basilis Gidas; Wei-Ming Ni; Louis Nirenberg Symmetry and related properties via the maximum principle, Commun. Math. Phys., Volume 68 (1979), pp. 209-243 | DOI | MR | Zbl

[42] Vincent Guedj; Boris Kolev; Nader Yeganefar Kähler–Einstein fillings, J. Lond. Math. Soc., Volume 88 (2013) no. 3, pp. 737-760 | DOI | Zbl

[43] Vincent Guedj; Ahmed Zeriahi Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. (2005) no. 4, pp. 607-639 | DOI | Zbl

[44] Vincent Guedj; Ahmed Zeriahi The weighted Monge–Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal., Volume 250 (2007) no. 2, pp. 442-482 | DOI | Zbl

[45] Jun-Muk Hwang On the degrees of Fano four-folds of Picard number 1, J. Reine Angew. Math., Volume 556 (2003), pp. 225-235 | MR | Zbl

[46] Said Ilias Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes, Ann. Inst. Fourier, Volume 33 (1983) no. 2, pp. 151-165 | DOI | Numdam | Zbl

[47] Michael K.-H. Kiessling Statistical mechanics of classical particles with logarithmic interactions, Commun. Pure Appl. Math., Volume 46 (1993) no. 1, pp. 27-56 | DOI | MR | Zbl

[48] János Kollár; Yoichi Miyaoka; Shigefumi Mori Rational connectedness and boundedness of Fano manifolds, J. Differ. Geom., Volume 36 (1992) no. 3, pp. 765-779 | MR | Zbl

[49] Sławomir Kołodziej The complex Monge–Ampère equation, Acta Math., Volume 180 (1998) no. 1, pp. 69-117 | DOI | Zbl

[50] Chi Li A Pohožaev identity and critical exponents of some complex Hessian equations, J. Partial Differ. Equations, Volume 29 (2016) no. 3, pp. 175-194 | DOI | MR | Zbl

[51] Peter Li On the Sobolev constant and the p-spectrum of a compact Riemannian manifold, Ann. Sci. Éc. Norm. Supér., Volume 13 (1980), pp. 451-468 | Numdam | MR | Zbl

[52] Li Ma; Juncheng C. Wei Convergence for a Liouville equation, Comment. Math. Helv., Volume 76 (2001) no. 3, pp. 506-514 | MR | Zbl

[53] Jürgen Moser A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., Volume 20 (1971), pp. 1077-1092 | DOI | MR | Zbl

[54] Duong H. Phong; Jian Song; Jacob Sturm; Ben Weinkove The Moser–Trudinger inequality on Kähler–Einstein manifolds, Am. J. Math., Volume 130 (2008) no. 4, pp. 1067-1085 | DOI | Zbl

[55] Alexander Rashkovskii Extremal cases for the log canonical threshold, C. R. Math. Acad. Sci. Paris, Volume 353 (2015) no. 1, pp. 21-24 | DOI | MR | Zbl

[56] Gábor Székelyhidi The partial $C^{0}$ -estimate along the continuity method, J. Am. Math. Soc., Volume 29 (2016) no. 2, pp. 537-560 | DOI | MR | Zbl

[57] Gábor Székelyhidi; Valentino Tosatti Regularity of weak solutions of a complex Monge–Ampère equation, Anal. PDE, Volume 4 (2011) no. 3, pp. 369-378 | DOI | Zbl

[58] Gang Tian On Kähler–Einstein metrics on certain Kähler manifolds with $c_{1} (M) > 0$ ., Invent. Math., Volume 89 (1987), pp. 225-246 | DOI | Zbl

[59] Gang Tian On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math., Volume 101 (1990) no. 1, pp. 101-172 | DOI | MR | Zbl

[60] Gang Tian Canonical Metrics in Kähler Geometry, Lectures in Mathematics, Birkhäuser, 2000 | DOI

[61] Gang Tian; Shing Tung Yau Kahler-Einstein metrics on complex surfaces with $C_{1} > 0$ , Commun. Math. Phys., Volume 112 (1987) no. 1, pp. 175-203 | DOI | Zbl

[62] Neil S. Trudinger On imbeddings into Orlicz spaces and some applications, J. Math. Mech., Volume 17 (1967), pp. 473-483 | MR | Zbl

[63] Steve Zelditch Book review of: Xiaonan Ma and George Marinescu, Holomorphic Morse inequalities and Bergman kernels, Bull. Am. Math. Soc., Volume 46 (2009) no. 2, pp. 349-361 | DOI | Zbl

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