A note on Huisken’s paper 1984

\mathbf{Thm(Myers):} If a Riemannian manifold has Ric\geq (n-1)K, then diam(M)\leq \frac{\pi}{K}.

Suppose we know a function h:M\to \mathbb{R}^+, M is compact and the gradient estimate

|\nabla h|\leq \eta^2 h^2_{\max} on M      (1)

and Ric_M\geq (n-1)c^2h^2 . Consider the relation \displaystyle h_{\min}\geq (1-\eta)h_{\max}.

\mathbf{Proof:}

Let x is a point on M, h achieves the maximum. Since (1), then

h(x)>(1-\eta)h_{\max} in B=B(x,\eta^{-1}h^{-1}_{\max}),

By the Myers thm, we get

\displaystyle diam(B)\leq \frac{\pi}{(1-\eta)h_{\max}}

But we know if \eta is small enough, then \displaystyle \eta^{-1}h^{-1}_{\max}>\frac{\pi}{(1-\eta)h_{\max}}. This forces diam(M)<\eta^{-1}h^{-1}_{\max}, then

\displaystyle h_{\min}\geq (1-\eta)h_{\max}.

\text{Q.E.D}\hfill \square

\mathbf{Remark:} Gerhard Huisken. Flow by mean curvature of convex surfaces into spheres.1984. p258 lemma 8.6.

The parameter \eta has relation with M and h_{\max}. We can not apply Myers thm directly. Guo Bin told me this method.

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