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Let $M$ be a complete non-compact manifold (possibly with boundary). Let $E$ be an open proper connected non-precompact subset of $M$ with smooth topological boundary, so that $\overline{E}$ is a non-compact complete manifold with boundary. Suppose that $E=M\setminus K$, where $K$ is a compact set that is the closure of a non-empty open set. If we now assume that $\overline{E}$ is a manifold with boundary of bounded geometry as described here, is it then also true that $M$ is of bounded geometry? I believe that it is true but does someone know a proof or a reference for that statement?

Added question: Is it essential that we assume that $\overline{E}$ is of bounded geometry and not $E$? What happends if we just assume that $E$ is of bounded geometry? Thanks in advance!

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Unless I am terribly mistaken, the answer is yes and the proof strategy is rather simple. The crucial observation is that the definition of bounded geometry depends on quantities that are continuous.

Consider first the injectivity radius function of the boundary, $r_{b}\colon\delta X\to \mathbb{R}$, $$r_b(x)=\sup\{\ t>0\mid \exp\colon B_{\delta X}(0_x,t)\to \delta X \quad \text{is a diffeomorphism}\ \}.$$ This is a continuous and positive function on $\delta X$ and it is bounded away from zero on $\delta X\setminus K$, so it must be bounded away from zero over all $\delta X$ because $K$ is compact. Denote the positive lower bound by $r_b(X)$; this is the injectivity radius of $X$.

Now define the ''normal collar injectivity radius'' $r_C\colon \delta X\to\mathbb{R}$ as $$r_c(x)=\sup\{\ t>0\mid \kappa\colon B_{\delta X}(x,r_b(X))\times[0,t)\to X\quad \text{is well-defined and a diffeomorphism} \ \},$$ where $\kappa(x,t)=\exp(t\nu_x)$ and $\nu_x$ is the unit inward normal vector. Again, this is continuous and bounded away from zero outside a compact set, hence bounded away from zero; we have obtained that $X$ satisfies the normal collar condition in Schick's definition.

A similar argument works for the remaining two conditions.

Answer to the added question: If $E$ is a manifold with boundary of bounded geometry with the metric that is the restriction of that of $\overline{E}$, then $E=\overline{E}$.

Assume for the sake of contradiction that $E$ is of bounded geometry and there is a point $x\in \overline{E}\setminus E$. Consider first the case where $x$ is in the closure of $\delta E$, so $x\in \delta \overline{E}$ and there is a sequence $(y_n)$ in $\delta E$ with $y_n\to x$. The injectivity radius $r_b$ of $\delta E$ is positive, so $B_{\delta \overline{E}}(y_n,r_b)\subset \delta E$ and $d_{\delta \overline{E}}(x,y_n)>r_b$ for all $n$, a contradiction.

Similar arguments apply when $x$ is a limit of points in the normal collar or a limit of points not in the normal collar.

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  • $\begingroup$ Thank you for your answer! Is it correct that $X=M$ in your text? $\endgroup$
    – Shaq155
    Aug 29 '21 at 21:35
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    $\begingroup$ Oh, dear! Yes, it should be $M$ instead of $X$. $\endgroup$
    – user44172
    Aug 30 '21 at 6:42
  • $\begingroup$ You are using here that the injectivity radius on a compact manifold is bounded below by a positive constant, right? $\endgroup$
    – Shaq155
    Aug 30 '21 at 8:49
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    $\begingroup$ Yes. Compact manifolds are automatically of bounded geometry, and therefore all the relevant quantities are bounded away from zero (in the case of the norm of the covariant derivative of the curvature tensor, bounded from above). $\endgroup$
    – user44172
    Aug 30 '21 at 8:52
  • $\begingroup$ Thank you again for your answer! Could you explain how the positive injectivity radius implies $d_{δ\overline{E}}(x,y_{n})>r_{b}$ for all $n$? $\endgroup$
    – Shaq155
    Sep 1 '21 at 10:47

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