Sobolev trace inequality on manifolds with non-negative Ricci curvature and convex boundary

We study some Sobolev trace inequalities on Riemannian manifolds with boundary. Using an adequate Bochner-Reilly type formula, we obtain some uniqueness result for compact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional Riemannian manifolds of non-negative Ricci curvature and strictly convex boundary, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 less-than-or-equal-to n less-than-or-equal-to 9"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>n</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>9</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2\leq n\leq 9</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . As an application of this result, we derive Sobolev and logarithmic trace inequalities with explicit constants.