Bifurcation for the Constant Scalar Curvature Equation and Harmonic Riemannian Submersions
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Bifurcation for the Constant Scalar Curvature Equation and Harmonic Riemannian Submersions Nobuhiko Otoba1
· Jimmy Petean2
Received: 18 June 2018 © Mathematica Josephina, Inc. 2019
Abstract We study bifurcation for the constant scalar curvature equation along a one-parameter family of Riemannian metrics on the total space of a harmonic Riemannian submersion. We provide an existence theorem for bifurcation points and a criterion to see that the conformal factors corresponding to the bifurcated metrics must be indeed constant along the fibers. In the case of the canonical variation of a Riemannian submersion with totally geodesic fibers, we characterize discreteness of the set of all degeneracy points along the family and give a sufficient condition to guarantee that bifurcation necessarily occurs at every point where the linearized equation has a nontrivial solution. In the model case of quaternionic Hopf fibrations, we show that SU(2)-symmetry-breaking bifurcation does not occur except at the round metric. Keywords Constant scalar curvature · Bifurcation for potential operators · Symmetry-breaking bifurcation · Horizontal Laplacian · Canonical variation · Riemannian submersion with totally geodesic fibers · Hopf fibrations Mathematics Subject Classification 53C20
1 Introduction It is well known that every conformal class on a closed manifold carries a Riemannian metric of constant scalar curvature (cf. [1,18,21,23]) while such metrics of unit volume within a conformal class are not necessarily unique (e.g. [12,19]). More recently, de
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Nobuhiko Otoba [email protected] Jimmy Petean [email protected]
1
Universität Regensburg, 93040 Regensburg, Deutschland
2
Centro de Investigación en Matemáticas, Jalisco S/N, Col. Valenciana, CP: 36023 Guanajuato, Gto, Mexico
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N. Otoba, J. Petean
Lima et al. [7] introduced a setup of bifurcation problem for the constant scalar curvature equation and studied direct product Riemannian manifolds from this perspective. Related work about local bifurcation on the total space of a Riemannian submersion with totally geodesic fibers include [4,5]. For global aspects of bifurcation in this context, see [8,9,16]. We recall that, for a smooth map ϕ : (M, g) → (N , h) between Riemannian manifolds, the following are equivalent (see [6,22]): (1) ϕ is a Riemannian submersion with minimal fibers. (2) ϕ is simultaneously a Riemannian submersion and a harmonic map. (3) ϕ is Laplacian-commuting, that is, ϕ ∗ ◦ h = g ◦ ϕ ∗ . Such a map ϕ is called a harmonic Riemannian submersion. In particular, a Riemannian submersion with totally geodesic fibers is harmonic. In this article, we study bifurcation phenomena for the constant scalar curvature equation in the presence of harmonic Riemannian submersions. More precisely, we consider a family of harmonic Riemannian submersions of constant scalar curvature and study bifurcation points for the constant scalar curvature equation. Bifurcation points are the elements of the family where new conformal metrics of constant scalar curvature appear;
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