Let $(M,g)$ be a Riemannian manifold, and let $N\subset M$ be an embedded sphere that is everywhere smooth except for a single point at which the embedding will only be $C^0$.

How much regularity can I obtain for the square of the distance function $$ F(x) := \inf\{d^2(x,y)| y\in N\}? $$

I would be surprised if this extra information had some effect on the answer, but you never know: The embedding is in high codimension; $M$ is dimension $2n+1$ and $N$ is dimension $n-1$.

Thank you very much for every comment, help or reference.

Best wishes Klaus


welcome to MO! It seems to me that you cannot expect anything more than the obvious, which is local Lipschitz regularity.

First, observe that even with a smooth embedding, there are problems at the some points (where the level hypersurface of the square distance fonction has a "double point"). In the $C^1$ case, you can avoid this in a neighborhood of the submanifold, but I think that if you lack regularity even at one point, then it can happen that these problematic points accumulate to the submanifold.

Take the following examples for a curve in $\mathbb{R}^2$ (or, taking a product, $\mathbb{R}^n$ for any $n$): $$\gamma(t)=\sqrt{|t|}\sin(1/t)$$ It defines a topological embedding of a line (which of course can be made into an embedding of a circle, or a higher-dimensional sphere easily) which only fails to be smooth at one point. It needs a bit of computation, but I think that there is no open set of values for which the square distance function is differentiable.

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