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We study the connection between the magnetization profiles of modelsdescribed by a scalar field with marginal interaction term in a bounded domainand the solutions of the so-called Yamabe problem in the same domain, whichamounts to finding a metric having constant curvature. Taking the slab as areference domain, we first study the magnetization profiles at the uppercritical dimensions $d=3$, $4$, $6$ for different (scale invariant) boundaryconditions. By studying the saddle-point equations for the magnetization, wefind general formulas in terms of Weierstrass elliptic functions, extendingexact results known in literature and finding new ones for the case ofpercolation. The zeros and poles of the Weierstrass elliptic solutions can beput in direct connection with the boundary conditions. We then show that, forany dimension $d$, the magnetization profiles are solution of the correspondinginteger Yamabe equation at the same $d$ and with the same boundary conditions.The magnetization profiles in the specific case of the $4$-dimensional Isingmodel with fixed boundary conditions are compared with Monte Carlo simulations,finding good agreement. These results explicitly confirm at the upper criticaldimension recent results presented in [1].

Magnetization profiles at the upper critical dimension as solutions of the integer Yamabe problem