Logarithmic Sobolev inequalities on submanifolds of Euclidean space

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Complete $$\lambda $$λ-hypersurfaces of weighted volume-preserving mean curvature flow (2014, methodology): “Using Logarithmic Sobolev inequality for hypersurfaces in Euclidean space due to Ecker [13] and conformal theory, we can show Lemma 10.”
The rigidity theorems of self-shrinkers (2011, unknown): “…Integrating by parts, then we have (3.8) (cid:9) By (2.1), we have Δ | X | 2 = 2 n − | X N | 2 (see [5] or [8]), then From (3.8), we get (see also [9]) Combining (2.1), Sobolev inequality (3.5) and (3.9), we have (3.10) Combining the Cauchy inequality, (3.7) and (3.10), for any δ > 0, we have…”
Volume growth, eigenvalue and compactness for self-shrinkers (2011, unknown): “By Logarithmic type Sobolev inequalities on selfshrinkers one has [7] (3.”
Eigenvalues of the drifted Laplacian on complete metric measure spaces (2013, methodology): “ively. By deﬁnition, L = ∆f. The operator L was introduced by Colding-Minicozzi [9] to study self-shrinker hypersurfaces. Now we prove Theorem 3. Proof of Theorem 3. For self-shrinkers in Rn+p, Ecker [11] proved a logarithmic Sobolev inequality with respect to the measure µ. Precisely, in [11], Exam”