Degenerate Solutions of Yamabe-Type Equations on Products of Spheres

TL;DR

This work analyzes Yamabe-type equations on the product manifold $S^n \times S^n$ endowed with a two-parameter family of metrics $G_{\delta}$ and demonstrates the existence of degenerate, diagonal $O(n+1)$-invariant solutions that genuinely couple both factors. By reducing the problem to a cohomogeneity-one setting via the isoparametric function $f(p,q)=\langle p,q\rangle$, the authors obtain an ODE in $t\in[-1,1]$ governed by Gegenbauer polynomials and apply the Crandall–Rabinowitz bifurcation theorem to identify bifurcation points $\lambda_k = \frac{k(k+n-1)}{q-2}(1+\frac{1}{\delta})$. For each even $k$, there exist bifurcating branches with precisely $k$ nodal domains, and a minimal $\lambda^*$ on the branch yields a degenerate, non-constant solution $u^*=w^*+1$ that depends nontrivially on both factors. The results extend prior studies by revealing a new class of two-factor dependent solutions, enriching the understanding of Yamabe-type problems on product manifolds.

Abstract

We study Yamabe-type equations on the product of two spheres $(S^n \times S^n, G_δ)$, where $G_δ$ is a family of Riemannian metrics parametrized by $δ> 0$. 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.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Lemma 2.4
• proof