Lebedev–Milin-type inequality for the critical conformal curvature functional

Let $(B^{n+1},dx^2)$ be the unit ball in Euclidean $(n+1)$-space, and let $k=\frac{n+1}{2}$. Define

$$\mathcal{C}_{k-1}:=\left\{g=u^2dx^2\mid \sigma_j^g\geq 0,\ H_j^g>0,\ 1\leq j\leq k-1\right\}.$$

For $g=e^{2u}dx^2$, let $\mathcal{F}_k(g)$ denote the critical conformal primitive of $(\sigma_k;H_k)$, namely

$$\mathcal{F}_k(g):=\int_0^1\left\{\int_{B^{n+1}}u\sigma_k^{g_s}\,\operatorname{dvol}_{g_s}+\oint_{S^n}uH_k^{g_s}\,\operatorname{dvol}_{\iota^*g_s}\right\}\,ds,$$

where $g_s=e^{2su}dx^2$. Lebedev–Milin-type inequality. For every $g\in\mathcal{C}_{k-1}$,

$$\mathcal{F}_k(g)\geq \frac{2^{(n+1)/n}}{n(n+1)}\omega_n\log\frac{\operatorname{Vol}_{\iota^*g}(S^n)}{\omega_n},$$

with equality if and only if $g$ is flat.

The inequality is the critical-dimensional analogue of the preceding sharp trace inequality and is related to sharp Onofri-type inequalities on closed spheres. According to the supplied status evidence, it was proved by adapting Escobar's Obata-type argument proving the classical trace inequality.