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Assume M is an oriented manifold and $W \subset M$ is a smooth compact embedded hypersurface without boundary.

Is there an example that the outer equidistants $W_t$ is not smooth for all $t$ in some oriented $M$ with non-positive sectional curvature $K_M \leq 0$?

And if we consider the case $M = \mathbb{R}^n$, why the outer equidistants $W_t$ is qualitively close to a perfectly round standard sphere for $t \gg 0$?

Thank you very much

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1 Answer 1

The answer to your first question is yes. Here is a trivial example: put $M = \mathbb R^3$ and put $W = S^2 \subset \mathbb R^3$. Choose the orientation on $\mathbb R^3$ that makes the region $|x| < 1$ the outside of $S^2$. Then $W_t$ is defined and smooth for $t < 1$ but singular at $t = 1$, as $W_1$ is just the origin. So part of your problem is that just saying "outer equidistant" by itself isn't very meaningful as I can change the orientation of $M$ without changing the sign of the curvature. However you can fix this by requiring $W$ to be convex, i.e. requiring that $$Hess_r(X,X) > 0$$ where $r(x)$ measures the distance from $x$ to $W$. Under this condition R. Hermann (and later F. Warner) has shown that $W_t$ is a smooth manifold for all time $t$. Now my previous counter example doesn't work, because if you change the orientation you reverse the sign of the hessian. Warner's proof is based on extending the Rauch comparison theorem to submanifolds; his paper is called "Extension of the Rauch Comparison Theorem to Submanifolds."

Regarding your second question, as I mentioned $W_t$ will, generally speaking, not be a manifold for large $t$ and therefore I think you need to clarify what you me an by "qualitatively close." Is there a specific situation you have in mind?

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