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Let $M^3$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the normal variation $\phi: \Sigma \times \Bbb{R}\to M$ given by

$$\phi(p,t)=\exp_p(tN(p)),$$

when $N$ is a normal vector field along to $\Sigma.$

I want to show that:

Lemma: There is $\delta>0$ such that $\phi:\Sigma\times [0,\delta) \to M$ is an immersion and even an embedding on $\Sigma\times [0,\delta)$.

I tried a sketch with tubular neighborhood tecniques but I don't conclude anything in this direction. In fact, I know that there is the tubular neighborhood of $\Sigma$, but I don't know how immerse this in $M$.

Anyone has a little help?

Thanks so much.

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2 Answers 2

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The normal exponential map will be regular up to the focal distance which is the reciprocal of the maximal principal curvature, or equivalently (for minimal surfaces) the square root of the maximum of minus the Gaussian curvature. If you think in terms of specific estimates it may be easier to get a solution.

This applies for a surface in Euclidean space. In the general situation the technicalities are more involved but you should try to understand the basic case first.

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You may end up reading a lot more than just what you wanted to know originally, but I think that Gromov's article Sign and geometric meaning of curvature is an amazing place to look for this sort of information. It will certainly explain the right way to think about this geometrically.

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