Volume gap between the minimal submanifold and the unit sphere

In 1984, Cheng et al. [Heat equations on minimal submanifolds and their applications, Amer. J. Math. 106(5) (1984) 1033–1065] proved that for a compact minimally immersed submanifold [Formula: see text] of [Formula: see text] there exists a positive constant [Formula: see text] depending only on [Formula: see text] such that [Formula: see text] In this paper, following the method of Cheng–Li–Yau, we first modify the coefficients in the constant [Formula: see text] to improve the volume gap. Further, we also enlarge our gap by applying an estimate of Cheng and Yang [Estimates on eigenvalues of Laplacian, Math. Ann. 331 (2005) 445–460] for eigenvalues of Laplacian.