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arXiv · 2010.13210

Variational Properties Of The Second Eigenvalue Of The Conformal Laplacian

Matthew J. Gursky·Samuel Pérez-Ayala
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Abstract

Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. Assume $[g]$ is a conformal class for which the Conformal Laplacian $L_g$ has at least two negative eigenvalues. We show the existence of a (generalized) metric that maximizes the second eigenvalue of $L_g$ over all conformal metrics (the first eigenvalue is maximized by the Yamabe metric). We also show that a maximal metric defines either a nodal solution of the Yamabe equation, or a harmonic map to a sphere. Moreover, we construct examples of each possibility.

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BibTeXRIS

Matthew J. Gursky, Samuel Pérez-Ayala. 2020-10-25. Variational Properties Of The Second Eigenvalue Of The Conformal Laplacian. https://doi.org/10.1016/j.jfa.2021.109371

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