Concentration Phenomena for Conformal Metrics with Constant Q-Curvature

2024/02/22 by Alarcón, Salomón, Masnú, Simón, Montero, Pedro +1
#35B33 #35J60 #53C21 #58J05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.14675

Abstract

Let (M,g) be an analytic Riemannian manifold of dimension n ≥ 5. In this paper, we consider the so-called constant Q-curvature equation ε⁴Δ_g² u -ε² b Δ_g u +a u = u^p , in M, ugt;0, u∈ H²_g(M) where a,b are positive constants such that b²-4 a>0, p is a sub-critical exponent 10 is small enough, then positive solutions to the above constant Q-curvature equation are generated by a maximum or minimum point of the function τ_g, given by τ_g(ξ):= ∑_{i, j=1}ⁿ \frac∂² g_ξ^{i i}∂ z_j²(0), where g_ξ^{i j} denotes the components of the inverse of the metric g in geodesic normal coordinates. This result shows that the geometry of M plays a crucial role in finding solutions to the equation above and provides a metric of constant Q-curvature on a product manifold of the form (M× X, g+ε² h) where (M,g) is flat and closed, and (X,h) any m-dimensional Einstein Riemannian manifold, m≥ 3.

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