Existence of smooth subcritical minimizers for the sharp trace conjecture

Establish the existence and regularity of smooth positive minimizers in the curvature class \(\mathcal C_1\) for a sequence of subcritical quotients \(\mathcal Q_{q_i}\) with \(q_i\uparrow p_*\), in order to remove the conditional existence assumption and prove Case–Wang’s sharp \(\sigma_2\) Sobolev trace conjecture.

Background

The paper classifies smooth relative local minimizers of the subcritical boundary quotients $\mathcal Q_q$ and shows that, if smooth minimizers exist for a sequence of exponents approaching the critical trace exponent $p_*=2(n-1)/(n-2)$, then passage to the limit yields Case–Wang’s conjectured sharp trace inequality. The missing ingredient is precisely the existence and smooth regularity of such minimizers within the curvature-constrained class $\mathcal C_1$.