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arXiv · 2504.14182

Degenerate Solutions of Yamabe-Type Equations on Products of Spheres

Hector Barrantes G.·Jorge Dávila
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Abstract

We study Yamabe-type equations on the product of two spheres $(S^n \times S^n, G_\delta)$, where $G_\delta$ is a family of Riemannian metrics parametrized by $\delta > 0$. Using bifurcation theory and isoparametric functions, we establish the existence of degenerate solutions that are invariant under the diagonal action of $O(n+1)$ and depend non-trivially on both factors. Our analysis relies on the properties of Gegenbauer polynomials and a careful application of local bifurcation techniques for simple eigenvalues. These results extend previous studies by demonstrating the emergence of solutions that do not solely depend on a single factor, thereby providing new insights into the structure of solutions for Yamabe-type problems on product manifolds.

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Hector Barrantes G., Jorge Dávila. 2025-04-19. Degenerate Solutions of Yamabe-Type Equations on Products of Spheres. https://arxiv.org/abs/2504.14182

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