For a closed minimal submanifold in the unit sphere \((n<N)\) , we prove \(\begin{aligned} \textrm{Vol}(M^n) \ge \frac{n+1}{n+2} \int _{M}\left( 1+\varphi _{p}^2\right) \ge m\textrm{Vol}(\mathbb {S}^{n}), \end{aligned}\) where \(\varphi _{p}(x):=\langle f(x),p\rangle \) is the height function in direction \(p\in f(M)\) , m denotes the multiplicity of \(p\in f(M)\) and \(\textrm{Vol}\) denotes the Riemannian volume functional, and each equality holds if and only if M is totally geodesic. As an application, if the volume of \(M^n\) is less than or equal to the volume of any n-dimensional minimal Clifford torus, then \(M^n\) must be embedded, verifying the non-embedded case of Yau’s conjecture. In addition, we also get volume gaps for minimal hypersurfaces with constant scalar curvature, improving Cheng–Li–Yau’s classical volume gap in this case. Some other volume gaps and related pinching rigidities are also obtained.
