This paper shows such a sharp Sobolev principle that if \((\Sigma ,g)\) is a compact n-dimensional graphic submanifold of \({\mathbb {R}}^{n+m}\) , \(G=|\text {det}g|\) is the absolute value of the determinant of g, \(|B^n|\) is the volume of the open unit ball \(B^n\) in \({\mathbb {R}}^n\) , and f is a positive smooth function on \(\Sigma \) , then \(\begin{aligned} \left( \frac{\int _\Sigma |\nabla f|\, \mathrm{{d}}V+ \int _{\partial \Sigma } f\, dA}{n|B^n|}\right) ^\frac{1}{n-1}\ge \left( \frac{\int _\Sigma f^{\frac{n}{n-1}}G^{-\frac{1}{2} }\, \mathrm{{d}}V}{|B^n|}\right) ^{\frac{1}{n}}, \end{aligned}\) holds with equality when and only when \(\begin{aligned} {\left\{ \begin{array}{ll} f=\text {a constant on}\ \Sigma ;\\ G=1\ \text {on}\ \partial \Sigma ;\\ \Sigma =\text {the image of a round ball in}\ {\mathbb {R}}^n\ \text {under a smooth map}. \end{array}\right. } \end{aligned}\) Intrinsically, the foregoing inequality is indeed geometric thanks to the fact that \(f=G^\frac{n-1}{2n}=1\) implies the following new sharp isoperimetric inequality for the graphic submanifolds of \({\mathbb {R}}^{n+m}\) with the unit metric determinant \(\begin{aligned} \left( \frac{|\partial \Sigma |}{n|B^n|}\right) ^\frac{1}{n-1}\ge \left( \frac{|\Sigma |}{|B^n|}\right) ^{\frac{1}{n}}. \end{aligned}\)
