Conformal Deformation on Manifolds With Boundary
- Szu-yu Sophie Chen
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We consider natural conformal invariants arising from the Gauss–Bonnet formulas on manifolds with boundary, and study conformal deformation problems associated to them.
- M.T. Anderson, L 2 curvature and volume renormalization of AHE metrics on 4-manifolds. Math. Res. Lett. 8:1-2 (2001), 171–188.
- T.P. Branson, P.B. Gilkey, The functional determinant of a four-dimensional boundary value problem, Trans. Amer. Math. Soc. 344:2 (1994), 479–531. CrossRef
- L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155:3-4 (1985), 261–301.
- Chang S.-Y.A., Gursky M.J., Yang P.: An a priori estimate for a fully nonlinear equation on four-manifolds. J. Anal. Math. 87, 151–186 (2002) CrossRef
- S.-Y.A. Chang, M.J. Gursky, P. Yang, An equation of Monge-Amp‘ere type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (2) 155:3 (2002), 709–787. CrossRef
- S.-Y.A. Chang, J. Qing, P. Yang, On the topology of conformally compact Einstein 4-manifolds, Contemp. Math. 350, Amer. Math. Soc. (2004), 49–61.
- Chen S.-Y.S.: Local estimates for some fully nonlinear elliptic equations. Int. Math. Res. Not. 55, 3403–3425 (2005) CrossRef
- S.-Y.S. Chen, Boundary value problems for some fully nonlinear elliptic equations, Calc. Var. Partial Differential Equations 30:1 (2007), 1–15. CrossRef
- J.F. Escobar, The Yamabe problem on manifolds with boundary, J. Differential Geom. 35:1 (1992), 21–84.
- L.C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35:3 (1982), 333–363. CrossRef
- Gȧrding L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)
- Y. Ge, G. Wang, On a fully nonlinear Yamabe problem, Ann. Sci. École Norm. Sup. (4) 39:4 (2006), 569–598.
- D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, second edition, 1983.
- B. Guan, Conformal metrics with prescribed curvature functions on manifolds with boundary, preprint.
- P. Guan, C.-S. Lin, G. Wang, Application of the method of moving planes to conformally invariant equations, Math. Z. 247:1 (2004), 1–19. CrossRef
- Guan P., Wang G.: A fully nonlinear conformal flow on locally conformally flat manifolds. J. Reine Angew. Math. 557, 219–238 (2003)
- Guan P., Wang G.: Local estimates for a class of fully nonlinear equations arising from conformal geometry. Int. Math. Res. Not. 26, 1413–1432 (2003) CrossRef
- M.J. Gursky, The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE, Comm. Math. Phys. 207:1 (1999), 131–143. CrossRef
- M.J. Gursky, J.A. Viaclovsky, Prescribing symmetric functions of eigenvalues of Schouten tensor, Ann. of Math., to appear.
- M.J. Gursky, J.A. Viaclovsky, A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom. 63:1 (2003), 131–154.
- M.J. Gursky, J.A. Viaclovsky, Fully nonlinear equations on Riemannian manifolds with negative curvature, Indiana Univ. Math. J. 52:2 (2003), 399–419. CrossRef
- M.J. Gursky, J.A. Viaclovsky, Volume comparison and the sk-Yamabe problem, Adv. Math. 187:2 (2004), 447–487. CrossRef
- Q. Jin, Local Hessian estimates for some conformally invariant fully nonlinear equations with boundary conditions, Differential Integral Equations 20:2 (2007), 121–132.
- Q. Jin, A. Li, Y.Y. Li, Estimates and existence results for a fully nonlinear yamabe problem on manifolds with boundary, Calc. Var. Partial Differential Equations 28:4 (2007), 509–543. CrossRef
- N.V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47:1 (1983), 75–108.
- A. Li, Y.Y. Li, On some conformally invariant fully nonlinear equations. II. Liouville, Harnack and Yamabe, Acta Math. 195 (2005), 117–154.
- A. Li, Y.Y. Li, A fully nonlinear version of the Yamabe problem on manifolds with boundary, J. Eur. Math. Soc. (JEMS) 8:2 (2006), 295–316. CrossRef
- A. Li, Y.Y. Li, On some conformally invariant fully nonlinear equations, Comm. Pure Appl. Math. 56:10 (2003), 1416–1464. CrossRef
- Y.Y. Li, Degree theory for second order nonlinear elliptic operators and its applications, Comm. Partial Differential Equations 14:11 (1989), 1541–1578. CrossRef
- G.M. Lieberman, N.S. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc. 295:2 (1986), 509–546. CrossRef
- P.-L. Lions, N.S. Trudinger, Linear oblique derivative problems for the uniformly elliptic Hamilton–Jacobi–Bellman equation, Math. Z. 191:1 (1986), 1–15. CrossRef
- D.S. Mitrinovi , Analytic Inequalities, Springer-Verlag, New York, 1970.
- M. Perdigão do Carmo, Riemannian Geometry, Mathematics: Theory & Applications. Birkhäuser Boston Inc., 1992.
- J. Qing, On the rigidity for conformally compact Einstein manifolds, Int. Math. Res. Not. 21 (2003), 1141–1153. CrossRef
- R.C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math. J. 20 (1973), 373–383.
- Schoen R., Yau S.-T.: Lectures on Differential Geometry. International Press, Cambridge, MA (1994)
- W.-M. Sheng, N.S. Trudinger, X.-J. Wang, The Yamabe problem for higher order curvatures, J. Differential Geom. 77:3 (2007), 515–553.
- N.S. Trudinger, X.-J. Wang, On Harnack inequalities and singularities of admissible metrics in the Yamabe problem, preprint.
- Urbas J.I.E.: An expansion of convex hypersurfaces, J. Differential Geom. 33(1), 91–125 (1991)
- Viaclovsky J.A.: Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J. 101(2), 283–316 (2000) CrossRef
- P. Yang, personal communication.
- Conformal Deformation on Manifolds With Boundary
Geometric and Functional Analysis
Volume 19, Issue 4 , pp 1029-1064
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- fully nonlinear
- manifolds with boundary
- Neumann condition
- 53C21 (35J60, 58J05)
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- Szu-yu Sophie Chen (1) (2)
- Author Affiliations
- 1. Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA
- 2. Institute for Advanced Study, Princeton, NJ, 08540, USA