Pólya-Szegő Inequalities on Submanifolds with Small Total Mean Curvature

We establish Pólya-Szegő-type inequalities (PSIs) for Sobolev-functions defined on a regular <i>n</i>-dimensional submanifold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Σ</mml:mi></mml:math> (possibly with boundary) of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> -dimensional Euclidean space, under explicit upper bounds of the total mean curvature. The <i>p</i>-Sobolev and Gagliardo-Nirenberg inequalities, as well as the spectral gap in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msubsup><mml:mi>W</mml:mi> <mml:mn>0</mml:mn> <mml:mrow><mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi></mml:mrow> </mml:msubsup> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>Σ</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> are derived as corollaries. Using these PSIs, we prove a sharp <i>p</i>-Log-Sobolev inequality for minimal submanifolds in codimension one and two. The asymptotic sharpness of both the multiplicative constant appearing in PSIs and the assumption on the total mean curvature bound as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi></mml:mrow> </mml:math> is provided. A second equivalent version of our PSIs is presented in the appendix of this paper, introducing the notion of model space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mo>+</mml:mo></mml:msup> <mml:mo>,</mml:mo> <mml:msub><mml:mi>m</mml:mi> <mml:mrow><mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi></mml:mrow> </mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:math> of dimension <i>n</i> and total mean curvature bounded by <i>K</i>.