Transversality theorems for harmonic forms. (English) Zbl 1071.58002

For a closed oriented manifold, the author studies the regularity and transversality properties of the zero set of harmonic forms for generic Riemannian metrics. For instance, he shows the following results by using transversality theory. Let \(M\) be a closed, oriented \(4\)-manifold with \(b^{+}_2(M)>0\). Let \(\text{ Met}^l(M)\) be the space of \(C^l\)-Hölder metrics on \(M\) for a large enough non-integer \(l\). Let \(Q^+\subset H^2(M;\mathbb{R})\times\text{ Met}^l(M)\) be the Banach submanifold consisting of pairs \(([\omega],g)\) such that \(\omega\) is a self-dual \(g\)-harmonic \(2\)-form with \([\omega]\neq 0\). Then there is a dense open set \({\mathcal U}\subset Q^+\) such that, if \(([\omega],g)\in {\mathcal U}\) with \(\omega\) \(g\)-harmonic, then \(\omega\) has regular zero set (it consists of disjoint circles). Let \(M\) be a closed, oriented \(4\)-manifold with \(b^{\pm}_2(M)>0\), and let \(\text{ Met}^l(M)\) be as above. Then there exists a dense open set \({\mathcal U}\subset H^2(M;\mathbb{R})\times\text{ Met}^l(M)\) such that if \(([\omega],g)\in {\mathcal U}\) with \(\omega\) \(g\)-harmonic, then \(\omega\) is neither self-dual nor anti-self-dual, and \(\omega\) has no zeros, has full rank away from a submanifold of codimension \(1\), and is self-dual/anti-self-dual on a union of disjoint circles. The author also considers the Dirichlet problem on a bounded domain in \({\mathbb R}^n\) and proves that, for a fixed boundary condition and a generic choice of metric, the solution has regular zero set.


58A14 Hodge theory in global analysis
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J32 Boundary value problems on manifolds
58J37 Perturbations of PDEs on manifolds; asymptotics
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[1] N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of the second order , J. Math. Pures Appl. 36 (1957), 235-249. · Zbl 0084.30402
[2] T. Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations , Grundlehren Math. Wiss., vol. 252, Springer-Verlag, New York, 1982. · Zbl 0512.53044
[3] S.K. Donaldson, Connections, cohomology, and the intersection forms of \(4\)-manifolds , J. Differential Geom. 24 (1986), 275-341. · Zbl 0635.57007
[4] S.K. Donaldson and P.B. Kronheimer, The geometry of four-manifolds , Oxford Math. Monographs, Oxford Univ. Press, New York, 1990. · Zbl 0820.57002
[5] G. Folland, Introduction to partial differential equations , Princeton Univ. Press, Princeton, NJ, 1976. · Zbl 0325.35001
[6] D. Fujiwara, Elliptic partial differential operators on a manifold , Appendix to K. Kodaira, Complex manifolds and deformation of complex structures , Grundlehren Math. Wiss., vol. 283, Springer-Verlag, New York, 1986.
[7] K. Honda, Local properties of self-dual harmonic \(2\)-forms on a \(4\)-manifold , preprint 1997.
[8] ——–, An openness theorem for harmonic \(2\)-forms on \(4\)-manifolds , Illinois J. Math. 44 (2000), 479-495. · Zbl 0970.58001
[9] S. Lang, Real and functional analysis , 3rd ed., Graduate Texts in Math., vol. 142 -1993. · Zbl 0831.46001
[10] C. LeBrun, Yamabe constants and the perturbed Seiberg-Witten equations , Comm. Anal. Geom. 5 (1997), 535-553. · Zbl 0901.53028
[11] J. Martinet, Sur les singularités des formes différentielles , Ann. Inst. Fourier (Grenoble) 20 (1970), 95-178. · Zbl 0189.10001 · doi:10.5802/aif.340
[12] T. Parker and S. Rosenberg, Invariants of conformal Laplacians , J. Differential Geom. 25 (1987), 199-222. · Zbl 0644.53038
[13] V.K. Patodi, Curvature and the eigenforms of the Laplace operator , J. Differential Geom. 5 (1971), 233-249. · Zbl 0211.53901
[14] C.H. Taubes, Self-dual connections on 4-manifolds with indefinite intersection matrix , J. Differential Geom. 19 (1984), 517-560. · Zbl 0552.53011
[15] ——–, The Seiberg-Witten and Gromov invariants , Math. Res. Lett. 2 (1995), 221-238. · Zbl 0854.57020 · doi:10.4310/MRL.1995.v2.n2.a10
[16] ——–, The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on \(S^1\times B^3\) , Geom. Topol. 2 (1998), 221-332 (electronic). · Zbl 0908.53013 · doi:10.2140/gt.1998.2.221
[17] ——–, Seiberg-Witten invariants and pseudo-holomorphic subvarieties for self-dual, harmonic \(2\)-forms , Geom. Topol. 3 (1999), 167-210 (electronic). · Zbl 1027.53111 · doi:10.2140/gt.1999.3.167
[18] ——–, Moduli spaces and Fredholm theory for pseudoholomorphic subvarieties associated to self-dual, harmonic \(2\)-forms , in Sir Michael Atiyah: A great mathematician of the twentieth century , Asian J. Math. 3 (1999), 275-324. · Zbl 0972.53055
[19] K. Uhlenbeck, Generic properties of eigenfunctions , Amer. J. Math. 98 (1976), 1059-1078. JSTOR: · Zbl 0355.58017 · doi:10.2307/2374041
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