A Note on Common Zeroes of Laplace–Beltrami Eigenfunctions

In Memory of Professor S. KobayashiIn the process of finding Einstein metrics in dimension n ≥ 3, we can search critical metrics for the scalar curvature functional in the space of the fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer-Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer-Marsden conjecture said that if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by O. Kobayashi and J. Lafontaine. However, almost all the counter-examples are homogeneous. Multiple solutions to this system yield Killing vector fields. We show that the dimension of the solution space W can be at most n + 1, with equality implying that (M, g) is a sphere with constant sectional curvatures. Moreover, we show that the identity component of the isometry group has a factor SO(W ). We also show that geometries admitting FischerMarsden solutions are closed under products with Einstein manifolds after a rescaling. Therefore, we obtain a lot of non-homogeneous counter-examples to the Fischer-Marsden conjecture. We then prove that all the homogeneous manifold M with a solution are in this case. Furthermore, we also proved that a related Besse conjecture is true for the compact homogeneous manifolds.

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“…Besse [1, 4. (4). Indeed, since f is an eigenfunction of the Laplacian, we can write u := 1 + f and rewrite (4) as the critical metric equation…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

“…Actually, what [4] asserts is the following. Let u and v be eigenfunctions corresponding to the same eigenvalue.…”

Section: -5mentioning

confidence: 96%

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Killing fields generated by multiple solutions to the Fischer–Marsden equation

Cernéa

Guan

2015

Int. J. Math.

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“…We say that u is regular if zero is not a critical value for u. Proposition 2 (see [11]). Suppose n > 0, u ∈ H n .…”

Section: Because P(x)φ(x Y) Dσ(x) = P(y) For All Y ∈ Smentioning

confidence: 99%

“…Then each component of N u is a Jordan contour. According to [11], a pair of the nodal sets N u , N v , where u, v ∈ H R n and n > 0, have a non-void intersection; moreover, if u is regular, then each component of N u contains at least two points of N v . The set N u ∩ N v may be infinite but the family of such pairs (u, v) is closed and nowhere dense in 2 .…”

Section: Introductionmentioning

confidence: 99%

Some remarks on spherical harmonics

Gichev

2009

St. Petersburg Math. J.

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Abstract. Several observations on spherical harmonics and their nodal sets are presented: a construction for harmonics with prescribed zeros; a natural representation for harmonics on S 2 ; upper and lower bounds for the nodal length and the inner radius (the upper bounds are sharp); the sharp upper bound for the number of common zeros of two spherical harmonics on S 2 ; the mean Hausdorff measure of the intersection of k nodal sets for harmonics of different degrees on S m , where k ≤ m (in particular, the mean number of common zeros of m harmonics).

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Schrödinger Operators and the Zeros of Their Eigenfunctions

Schwartzman

2011

Commun. Math. Phys.

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A Note on Common Zeroes of Laplace–Beltrami Eigenfunctions

Cited by 5 publications

References 2 publications

Killing fields generated by multiple solutions to the Fischer–Marsden equation

Killing fields generated by multiple solutions to the Fischer–Marsden equation

Some remarks on spherical harmonics

Schrödinger Operators and the Zeros of Their Eigenfunctions

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