Geometry & Topology

A formal Riemannian structure on conformal classes and uniqueness for the $\sigma_2$–Yamabe problem

Matthew Gursky and Jeffrey Streets

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Abstract

We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the σ 2 –Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.

Article information

Source
Geom. Topol., Volume 22, Number 6 (2018), 3501-3573.

Dates
Received: 12 June 2017
Revised: 29 September 2017
Accepted: 10 November 2017
First available in Project Euclid: 29 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.gt/1538186742

Digital Object Identifier
doi:10.2140/gt.2018.22.3501

Mathematical Reviews number (MathSciNet)
MR3858769

Zentralblatt MATH identifier
06945131

Subjects
Primary: 58J05: Elliptic equations on manifolds, general theory [See also 35-XX]
Secondary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 58B20: Riemannian, Finsler and other geometric structures [See also 53C20, 53C60]

Keywords
fully nonlinear Yamabe problem uniqueness

Citation

Gursky, Matthew; Streets, Jeffrey. A formal Riemannian structure on conformal classes and uniqueness for the $\sigma_2$–Yamabe problem. Geom. Topol. 22 (2018), no. 6, 3501--3573. doi:10.2140/gt.2018.22.3501. https://projecteuclid.org/euclid.gt/1538186742


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References

  • Z Błocki, On geodesics in the space of Kähler metrics, from “Advances in geometric analysis” (S Janeczko, J Li, D H Phong, editors), Adv. Lect. Math. 21, International, Somerville, MA (2012) 3–19
  • S Brendle, Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc. 21 (2008) 951–979
  • S Brendle, F C Marques, Blow-up phenomena for the Yamabe equation, II, J. Differential Geom. 81 (2009) 225–250
  • S Brendle, J A Viaclovsky, A variational characterization for $\sigma_{n/2}$, Calc. Var. Partial Differential Equations 20 (2004) 399–402
  • E Calabi, X X Chen, The space of Kähler metrics, II, J. Differential Geom. 61 (2002) 173–193
  • S-Y A Chang, M J Gursky, P Yang, An a priori estimate for a fully nonlinear equation on four-manifolds, J. Anal. Math. 87 (2002) 151–186
  • S-Y A Chang, M J Gursky, P C Yang, An equation of Monge–Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. 155 (2002) 709–787
  • S-Y A Chang, P C Yang, The inequality of Moser and Trudinger and applications to conformal geometry, Comm. Pure Appl. Math. 56 (2003) 1135–1150
  • X Chen, The space of Kähler metrics, J. Differential Geom. 56 (2000) 189–234
  • X X Chen, G Tian, Geometry of Kähler metrics and foliations by holomorphic discs, Publ. Math. Inst. Hautes Études Sci. 107 (2008) 1–107
  • B Chow, P Lu, L Ni, Hamilton's Ricci flow, Graduate Studies in Mathematics 77, Science, Beijing (2006)
  • S K Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, from “Northern California Symplectic Geometry Seminar” (Y Eliashberg, D Fuchs, T Ratiu, A Weinstein, editors), Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc., Providence, RI (1999) 13–33
  • S K Donaldson, Conjectures in Kähler geometry, from “Strings and geometry” (M Douglas, J Gauntlett, M Gross, editors), Clay Math. Proc. 3, Amer. Math. Soc., Providence, RI (2004) 71–78
  • L C Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982) 333–363
  • L Gårding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959) 957–965
  • D Gilbarg, N S Trudinger, Elliptic partial differential equations of second order, Grundl. Math. Wissen. 224, Springer (1977)
  • B Guan, The Dirichlet problem for complex Monge–Ampère equations and regularity of the pluri-complex Green function, Comm. Anal. Geom. 6 (1998) 687–703
  • P Guan, J Viaclovsky, G Wang, Some properties of the Schouten tensor and applications to conformal geometry, Trans. Amer. Math. Soc. 355 (2003) 925–933
  • P Guan, G Wang, A fully nonlinear conformal flow on locally conformally flat manifolds, J. Reine Angew. Math. 557 (2003) 219–238
  • M J Gursky, J Streets, A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow, preprint (2015)
  • M J Gursky, J Streets, Variational structure of the $v_k$–Yamabe problem, preprint (2016)
  • W He, The Gursky–Streets equations, preprint (2017)
  • N V Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983) 75–108 In Russian; translated in Math. USSR-Izv. 22 (1984) 67–97
  • J M Lee, T H Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987) 37–91
  • J Lelong-Ferrand, Transformations conformes et quasiconformes des variétés riemanniennes; application à la démonstration d'une conjecture de A Lichnerowicz, C. R. Acad. Sci. Paris Sér. A-B 269 (1969) A583–A586
  • T Mabuchi, $K$–energy maps integrating Futaki invariants, Tohoku Math. J. 38 (1986) 575–593
  • T Mabuchi, Some symplectic geometry on compact Kähler manifolds, I, Osaka J. Math. 24 (1987) 227–252
  • M Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962) 333–340
  • D Pollack, Nonuniqueness and high energy solutions for a conformally invariant scalar equation, Comm. Anal. Geom. 1 (1993) 347–414
  • R C Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geometry 8 (1973) 465–477
  • R M Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, from “Topics in calculus of variations” (M Giaquinta, editor), Lecture Notes in Math. 1365, Springer (1989) 120–154
  • S Semmes, Complex Monge–Ampère and symplectic manifolds, Amer. J. Math. 114 (1992) 495–550
  • W Sheng, N S Trudinger, X-J Wang, The $k$–Yamabe problem, from “Surveys in differential geometry” (H-D Cao, S-T Yau, editors), volume 17, International, Boston, MA (2012) 427–457
  • J A Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J. 101 (2000) 283–316
  • J A Viaclovsky, Conformally invariant Monge–Ampère equations: global solutions, Trans. Amer. Math. Soc. 352 (2000) 4371–4379
  • J A Viaclovsky, Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds, Comm. Anal. Geom. 10 (2002) 815–846
  • J Viaclovsky, Conformal geometry and fully nonlinear equations, from “Inspired by S S Chern” (P A Griffiths, editor), Nankai Tracts Math. 11, World Sci., Hackensack, NJ (2006) 435–460