Abstract
In this paper and its sequel (Isobe in Morse–Floer theory for superquadratic Dirac equations, II: construction and computation of Morse–Floer homology, 2016), we study Morse–Floer theory for superquadratic Dirac functionals associated with a class of nonlinear Dirac equations on compact spin manifolds. We are interested in two topics: (i) relative Morse indices and its relation to compactness issues of critical points; (ii) construction and computation of the Morse–Floer homology and its application to the existence problem for solutions to nonlinear Dirac equations. In this part I, we focus on the topic (i). One of our main results is a compactness of critical points under the boundedness assumption of their relative Morse indices which is an analogue of the results of Bahri–Lions (Commun Pure Appl Math 45:1205–1215, 1992) and Angenent–van der Vorst (Math Z 231: 203–248, 1999) for Dirac functionals. To prove this, we give an appropriate definition of relative Morse indices for bounded solutions to \(\mathsf {D}_{g_{\mathbb {R}^{m}}}\psi =|\psi |^{p-1}\psi \) on \(\mathbb {R}^{m}\). We show that for \(m\ge 3\) and \(1<p<\frac{m+1}{m-1}\), the relative Morse index of any non-trivial bounded solution to that equation is \(+\infty \). We also give some useful properties of the relative Morse indices of Dirac functionals which will be used in the study of the topic (ii) above.
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References
Abbondandolo, A.: A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert space. Top. Methods Nonlinear Anal. 9, 325–382 (1997)
Abbondandolo, A.: Morse Theory for Hamiltonian Systems, vol. 425. CRC Press, Boca Raton (2001)
Abbondandolo, A., Majer, P.: Morse homology on Hilbert spaces. Commun. Pure Appl. Math. 54, 689–760 (2001)
Abbondandolo, A., Majer, P.: Ordinary differential operators in Hilbert spaces and Fredholm pairs. Math. Z. 243, 525–562 (2003)
Abbondandolo, A., Majer, P.: A Morse complex for infinite dimensional manifolds—part I. Adv. Math. 197, 321–410 (2005)
Abbondandolo, A., Majer, P.: Lectures on the Morse Complex for Infinite-dimensional Manifolds. Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology. Springer, Netherlands (2006)
Adams, R.: Sobolev Spaces. Academic Press, New York (1975)
Ammann, B.: A Variational Problem in Conformal Spin Geometry. Universität Hamburg, Habilitationsschift (2003)
Ammann, B.: The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions. Commun. Anal. Geom. 17, 429–479 (2009)
Ammann, B., Ginoux, N.: Dirac-harmonic maps from index theory. Calc. Var. 47, 739–762 (2013)
Ammann, B., Ginoux, N.: Examples of Dirac-harmonic maps after Jost–Mo–Zhu. (2011). http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/ammann/preprints/diracharm/diracharm_JostMoZhu09.pdf (preprint)
Ammann, B., Moroianu, A., Moroianu, S.: The Cauchy problems for Einstein metrics and parallel spinors. Commun. Math. Phys. 320, 173–198 (2013)
Angenent, S., van der Vorst, R.: A superquadratic indefinite elliptic system and its Morse–Conley–Floer homology. Math. Z. 231, 203–248 (1999)
Angenent, S., van der Vorst, R.: A priori bounds and renormalized Morse indices of solutions of an elliptic system. In: Ann. Inst. H. Poincaré, Anal. Non Linéaire, vol. 17, pp. 277–306 (2000)
Audin, M., Damian, M.: Morse Theory and Floer Homology. Universitext, Springer, Berlin (2014)
Bahri, A., Lions, P.L.: Solutions of superlinear elliptic equations and their Morse indices. Commun. Pure Appl. Math. 45, 1205–1215 (1992)
Booss-Bavnbek, B., Marcolli, M.: Bai-Ling Wang: weak UCP and perturbed monopole equations. Int. J. Math. 13, 987–1008 (2002)
Chen, Q., Jost, J., Wang, G.: Liouville theorems for Dirac-harmonic maps. J. Math. Phys. 48, 113517 (2007)
Chen, Q., Jost, J., Li, J., Wang, G.: Dirac-harmonic maps. Math. Z. 254, 409–432 (2006)
Conley, C., Zehnder, E.: Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure Appl. Math. 37, 207–253 (1984)
Daniele, G.: On the spectral flow for paths of essentially hyperbolic bounded operators on Banach spaces. arXiv preprint. arXiv:0806.4094 (2008)
Ding, Y.: Variational methods for strongly indefinite problems. In: Interdisciplinary Mathematical Sciences, vol. 7. World Scientific, Hackensack (2007)
Duistermaat, J.J.: On the Morse index in the calculus of variations. Adv. Math. 21, 173–195 (1976)
Esteban, M.J., Sèrè, E.: Stationary states of the nonlinear Dirac equation: a variational approach. Commun. Math. Phys. 171, 323–350 (1995)
Floer, A.: The unregularized gradient flow of the symplectic action. Commun. Pure Appl. Math. 41, 775–813 (1988)
Floer, A.: A relative Morse index for the symplectic action. Commun. Pure Appl. Math. 41, 393–407 (1988)
Floer, A.: Symplectic fixed points and holomorphic spheres. Commun. Math. Phys. 120, 575–611 (1989)
Floer, A.: An instanton-invariant for 3-manifolds. Commun. Math. Phys. 118, 215–240 (1988)
Friedrich, T.: Dirac operators in Riemannian geometry. In: Graduate Studies in Mathematics, vol. 25 (2000)
Ginoux, N.: The Dirac spectrum. In: Lecture Notes in Math, vol. 1976. Springer, Berlin (2009)
Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)
Isobe, T.: Existence results for solutions to nonlinear Dirac equations on compact spin manifolds. Manuscr. Math. 135, 329–360 (2011)
Isobe, T.: Nonlinear Dirac equations with critical nonlinearities on compact spin manifolds. J. Funct. Anal. 260, 253–307 (2011)
Isobe, T.: On the existence of nonlinear Dirac-geodesics on compact manifolds. Calc. Var. 43, 83–121 (2012)
Isobe, T.: A perturbation method for spinorial Yamabe type equations on \(S^m\) and its application. Math. Ann. 355, 1255–1299 (2013)
Isobe, T.: Spinorial Yamabe type equations on \(S^{3}\) via Conley index. Adv. Nonlinear Stud. 15, 39–60 (2015)
Isobe, T.: Morse–Floer Theory for Superquadratic Dirac Equations, II: Construction and Computation of Morse–Floer Homology. J. Fixed Point Theory Appl. (2016)
Jost, J.: Geometry and Physics. Springer, Berlin (2009)
Jost, J., Mo, X., Zhu, M.: Some explicit constructions of Dirac-harmonic maps. J. Geom. Phys. 59, 1512–1527 (2009)
Kryszewski, W., Szulkin, A.: An infinite dimensional Morse theory with applications. Trans. Am. Math. Soc. 349, 3181–3234 (1997)
Lawson, H.B., Michelson, M.L.: Spin Geometry. Princeton University Press, Princeton (1989)
Lesch, M.: The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators. In: Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds. Contemp. Math. vol. 366, pp. 193–224 (2005)
Maalaoui, A.: Rabinowitz–Floer homology for superquadratic Dirac equations on compact spin manifolds. J. Fixed Point Theory Appl. 13, 175–199 (2013)
Maalaoui, A., Martino, V.: The Rabinowitz–Floer homology for a class of semilinear problems and applications. J. Funct. Anal. 269, 4006–4037 (2015)
Mawhin, J., Willem, M.: Critical point theory and Hamiltonian systems. In: Applied Mathematical Sciences, vol. 74. Springer, New York (1989)
Raulot, S.: A Sobolev-like inequality for the Dirac operator. J. Funct. Anal. 26, 1588–1617 (2009)
Robbin, J., Salamon, D.: The spectral flow and the Maslov index. Bull. Lond. Math. Soc. 27, 1–33 (1995)
Salamon, D.: Lectures on Floer Homology. Symplectic Geometry and Topology (Park City, UT, 1997), vol. 7, pp. 143–229. IAS/Park City Math, Ser. (1999)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Szulkin, A.: Cohomology and Morse theory for strongly indefinite functionals. Math. Z. 209, 375–418 (1992)
Taylor, M.E.: Partial Differential Equations, Part I-Part III. Springer, New York (2011)
Thaller, B.: The Dirac Equations. Springer, Berlin (2010)
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This work was supported by JSPS KAKENHI Grant Numbers 22540222, 15K04947.
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Isobe, T. Morse–Floer theory for superquadratic Dirac equations, I: relative Morse indices and compactness. J. Fixed Point Theory Appl. 19, 1315–1363 (2017). https://doi.org/10.1007/s11784-016-0391-z
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DOI: https://doi.org/10.1007/s11784-016-0391-z