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Let $M^n$ be a closed immersed hypersurface lying in a contractible ball$B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinchingHeintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$,1st eigenvalue and mean curvature of $M$, not only $M$ is Hausdorff close to ageodesic sphere $S(p_0,R_0)$ in $N$, but also the ``enclosed'' ball$B(p_0,R_0)$ is close to be of constant curvature, provided with a uniformcontrol on the volume and mean curvature of $M$. We raise a conjecture for $M$to be a diffeomorphic sphere, and give some positive partial answer.

Recognizing shape via 1st eigenvalue, mean curvature and upper curvature bound