  For a closed immersed minimal submanifold $M^n$ in the unit sphere
$\mathbb{S}^{N}$ $(n<N)$, we prove $$ {\rm Vol}(M^n) \geq
  \frac{m}{2}{\rm Vol}(\mathbb{S}^{n})+m\frac{\sqrt{n+1}}{n} {\rm
Vol}(\mathbb{S}^{n-1}),$$ where $m$ denotes the maximal multiplicity of
intersection points of $M^n$ in $ \mathbb{S}^{N}$ and ${\rm Vol}$ denotes the
Riemannian volume functional. 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 hypersurfaces under some conditions.
