Sobolev and mean‐value inequalities on generalized submanifolds of Rⁿ

Finiteness of Index and Total Scalar Curvature for Minimal Hypersurfaces

“…This proposition follows immediately from Theorem 2 of P. Li and S.-T. Yau in [3]. We only need to replace the Sobolev inequality for Euclidean space, that they use, by the Sobolev inequality for minimal submanifolds, see [4]. In fact, in their paper P. Li and S-T. Yau point out that their argument is also valid on manifolds on which a Sobolev inequality holds.…”

Rigidity of Stable Marginally Outer Trapped Surfaces in Initial Data Sets

“…The argument to prove Theorem A.1 when Ω ⊂ R d a bounded regular domain, is based on the Sobolev inequality (A.2) Ω |u| dp r (x) then inequalities like (A.3) (and hence (A.2)) still hold true, while if Σ 0 is only known to be, say, a smooth submanifold (A.3) with locally bounded mean curvature then they should be replaced by the Michael-Simon inequality (see [MS73] and [HS74] for this extended version):…”

Upper Bounds for the First Eigenvalue of the Laplacian of Hypersurfaces in terms of Anisotropic Mean Curvatures

“…The proof of the lemma 7 uses a Nirenberg-Moser type of proof (see [2,3]) based on a Sobolev inequality due to Michael-Simon and Hoffman-Spruck (see [7], [8] and [11]). …”

“…An easy computation shows that |dφ 2α | 2αφ 2α−1 . Hence, using the Sobolev inequality (see [7], [8] and [11])…”

“…We will prove Theorem 3. For this, we will use a part of Theorem 2, namely the fact the X(M ) is close to the sphere for the Hausdorff distance and in particular (11). Before that, we prove the following lemma.…”