Abstract
This paper is a survey on bubbling phenomena occurring in some geometric problems. We present here a few problems from conformal geometry, gauge theory and contact geometry and we give the main ideas of the proofs and important results. We focus in particular on the Yamabe type problems and the Weinstein conjecture, where A. Bahri made a huge contribution by introducing new methods in variational theory.
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Ahmedou, M.O.; El Mehdi, K.O.: On an elliptic problem with critical non-linearity in expanding annuli. J. Funct. Anal. (2) 163, 29–62 (1999)
Albers, P.; Frauenfelder, U.: Leaf-wise intersections and Rabinowitz Floer homology. J. Topol. Anal., 77-98 (2010)
Ambrosetti, A.; Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Atiyah, M.F.; Bott, R.: The YangMills equations over Riemann surfaces. Philos. Trans. Roy. Soc. Lond. Ser. A 308, 523–615 (1983)
Aubin, T.: Le probléme de Yamabe concernant la courbure scalaire (French). C. R. Acad. Sci. Paris Sr. A–B 280, Aii, A721–A724 (1975)
Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (1998)
Aubin, T.: Equations différentielles non linéaires et probléme de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)
Bahri, A.: Another proof of the Yamabe conjecture for locally conformally flat manifolds. Nonlinear Anal. Theory, Methods Appl. 20(10), 1261–1278 (1993)
Bahri, A.: Critical Points at Infinity in Some Variational Problems. Pitman Research Notes in Mathematics Series, 182. Longman, Harlow (1989)
Bahri, A.: Classical and Quantic Periodic Motions of Multiply Polarized Spin-particles, Pitman Research Notes in Mathematics Series, 378. Longman, Harlow (1998)
Bahri, A.: Flow Lines and Algebraic Invariants in Contact form Geometry. Progress in Nonlinear Differential Equations and their Applications, 53. Birkhäuser Boston, Inc., Boston, MA (2003)
Bahri, A.: Pseudo-Orbits of Contact Forms, Pitman Research Notes in Mathematics Series (173). Longman Scientic and Technical, London (1988)
Bahri, A.: On the contact homology of the first exotic contact form of J. Gonzalo and F. Varela. Arab. J. Math. 3(2), 211289 (2014)
Bahri, A.: Critical points at infinity in the Yamabe changing-sign equations (Preprint)
Bahri, A.: A Linking\(/S^{1}\) -equivariant variational argument in the space of dual Legendrian curves and the proof of the Weinstein conjecture on \(S^3\) in the large. Adv. Nonlinear Stud. 15, 497–526 (2015)
Bahri, A.; Brezis, H.: Topics in geometry: in memory of Joseph D’Atri. Gindikin, S. (ed.), 1–99.
Bahri, A.; Chanillo, S.: The difference of topology at infinity in changing-sign Yamabe problems on \(S^3\) (the case of two masses). Comm. Pure Appl. Math. 54(4), 450–478 (2001)
Bahri, A.; Coron, J.-M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Comm. Pure Appl. Math. 41(3), 253–294 (1988)
Bahri, A.; Xu, Y.: Recent Progress in Conformal Geometry, ICP Advanced Texts in Mathematics, 1. Imperial College Press, London (2007)
Ben Ayed, M.; Chtioui, H.; Hammami, M.: A Morse lemma at infinity for yamabe type problems on domains. Ann. I. H. Poincar AN 20(4), 543–577 (2003)
Cieliebak, K.; Frauenfelder, U.: A Floer homology for exact contact embeddings. Pac. J. Math. 239(2), 251–316 (2009)
Cieliebak, K.; Frauenfelder, U.: Morse homology on noncompact manifolds. J. Korean Math. Soc. 48(4), 749–774 (2011)
Cieliebak, K.; Frauenfelder, U.; Oancea, A.: Rabinowitz Floer homology and symplectic homology. Ann. Sci. Ec. Norm. Super. (4) 43(6), 957–1015 (2010)
del Pino, M.; Musso, M.; Pacard, F.; Pistoia, A.: Torus action on \(S^n\) and sign changing solutions for conformally invariant equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12(1), 209–237 (2013)
Ding, W.Y.: On a conformally invariant elliptic equation on \({\mathbb{R}}^n\). Comm. Math. Phys. 107(2), 331–335 (1986)
Ding, W.Y.: Positive solutions of \(\Delta u + u^{\frac{n+2}{n2}}=0\) on contractible domains. J. Partial Differ. Equ. 2(4), 83–88 (1989)
Donaldson, S.K.: An application of gauge theory to four-dimensional topology. J. Differ. Geom. 18, 279–315 (1983)
Donaldson, S.K.; Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford University Press, Oxford (1990)
Fadell, E.; Rabinowitz, P.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45(2), 139–174 (1978)
Friedrich, T.: Dirac Operators in Riemannian Geometry. Grad. Stud. Math., vol. 25, Amer. Math. Soc., Providence, RI (2000)
Gamara, N.: The CR Yamabe conjecture the case \(n=1\). J. Eur. Math. Soc. 3, 105–137 (2001)
Gamara, N.; Yacoub, R.: CR Yamabe conjecture—the conformally flat case. Pac. J. Math. 201(1), 121–175 (2001)
Graham, C.; Jenne, R.; Mason, L.; Sparling, G.: Conformally invariant powers of the Laplacian I: existence. J. Lond. Math. Soc. 46, 557–565 (1992)
Geiges, H.; Gonzalo, J.: Contact geometry and complex surfaces. Invent. Math. 121(1), 147–209 (1995)
Gonzalo, J.; Varela, F.: Modéles globaux des variétés de contact, Third Schnepfenried geometry conference, vol. 1 (Schnepfenried, 1982), Asterisque, no. 107–108, pp. 163168, Soc. Math.France, Paris (1983)
Hirano, N.: A nonlinear elliptic equation with critical exponents: effect of geometry and topology of the domain. J. Diff. Equ. 182, 78–107 (2002)
Hofer, H.: Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114, 515–563 (1993)
Isobe, T.: Existence results for solutions to nonlinear Dirac equations on compact spin manifolds. Manuscripta Math. 135(3–4), 329–360 (2011)
Isobe, T.: Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds. J. Funct. Anal. 260(1), 253–307 (2011)
Jerison, D.; Lee, J.M.: The Yamabe problem on CR manifolds. J. Differ. Geom. 25(2), 167–197 (1987)
Jerison, D.; Lee, J.M.: Intrinsic CR normal coordinates and the CR Yamabe problem. J. Diff. Geom. 29, 303–343 (1989)
Lee, J.; Parker, T.: The Yamabe problem. Bull. Amer. Math. Soc. 17, 37–91 (1987)
Maalaoui, A.: Rabinowitz–Floer homology for super-quadratic Dirac equations on spin manifolds. J. Fixed Point Theory Appl. 13(1), 175–199 (2013)
Maalaoui, A.: Infinitely many solutions for the spinorial Yamabe problem on the round sphere. Nonlinear Differ. Equ. Appl. 23, Art ID 25 (2016)
Maalaoui, A.: Dynamics of the third exotic contact form on the sphere along a vector field in its kernel. Adv. Nonlinear Stud. 16, 315–332 (2016)
Maalaoui, A.; Martino, V.: Multiplicity result for a nonhomogeneous Yamabe type equation involving the Kohn Laplacian. J. Math. Anal. Appl. 399(1), 333–339 (2013)
Maalaoui, A.; Martino, V.: Existence and concentration of positive solutions for a super-critical fourth order equation. Nonlinear Anal. 75, 5482–5498 (2012)
Maalaoui, A.; Martino, V.: Changing sign solutions for the CR-Yamabe equation, differential and integral equations, vol. 25. Numbers 7–8, 601–609 (2012)
Maalaoui, A.; Martino, V.: Existence and multiplicity results for a non-homogeneous fourth order equation. Topol. Methods Nonlinear Anal. 40(2), 273–300 (2012)
Maalaoui, A.; Martino, V.: The Rabinowitz–Floer homology for a class of semilinear problems and applications. J. Funct. Anal. 269(12), 4006–4037 (2015)
Maalaoui, A.; Martino, V.: Homological approach to problems with jumping non-linearity. Nonlinear Anal. 144, 165–181 (2016)
Maalaoui, A.; Martino, V.: The topology of a subspace of the Legendrian curves on a closed contact \(3\) -manifold. Adv. Nonlinear Stud. 14, 393–426 (2014)
Maalaoui, A.; Martino, V.: Homology computation for a class of contact structures on \(T^3\). Calc. Var. Partial Differ. Equ. 50, 599–614 (2014)
Maalaoui, A.; Martino, V.; Pistoia, A.: Concentrating solutions for a sub-critical sub-elliptic problem, differential and integral equations, vol. 26. Numbers 11–12, 1263–1274 (2013)
Maalaoui, A.; Martino, V.; Tralli, G.: Complex group actions on the sphere and changing sign solutions for the CR-Yamabe equation. J. Math. Anal. Appl. 431, 126–135 (2015)
Martino, V.: A Legendre transform on an exotic \(S^3\). Adv. Nonlinear Stud. 11, 145–156 (2011)
Martino, V.: Legendre duality on hypersurfaces in Khler manifolds. Adv. Geom. 14(2), 277–286 (2014)
Rabinowitz, P.: Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31(2), 157–184 (1978)
Thom, R.: Sous-variétés et classes dhomologie des variétés différentiables. II. Résultats et applications. C. R. Acad. Sci. Paris 236, 573–575 (1953)
Taubes, C.H.: Path-connected Yang–Mills moduli spaces. J. Differ. Geom. 19, 337–392 (1984)
Taubes, C.H.: A framework for Morse theory for the Yang–Mills functional. Invent. math. 94, 327–402 (1988)
Trudinger, N.: Remarks concerning the conformai deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa 22, 265–274 (1968)
Schoen, R.: Conformai deformation of a Riemannian metric to constant scalar curvature. J. Diff. Geom. 20, 479–495 (1984)
Schoen, R.; Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65, 45–76 (1979)
Uhlenbeck, K.K.: Connections with \(L^{p}\) -bounds on curvature. Comm. Math. Phys. 83, 31–42 (1982)
Wehrheim, K.: Uhlenbeck Compactness. EMS Series of Lectures in Mathematics 2004.
Xu, Y.: A pseudo-gradient flow arising in contact form geometry. Adv. Nonlinear Stud. 15(2), 447–496 (2015)
Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)
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In memory of Prof. Abbas Bahri, a great mathematician from whom I am still learning.
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