Skip to main content
MathOverflow

Return to Question

deleted 8 characters in body
Source Link
Andrey Gogolev
Andrey Gogolev
  • 4.2k
  • 1
  • 22
  • 26

Let $M$ be a compact simply connected R. manifold. Let $x$ be a base point and let $\gamma$ be a smooth loop in $M$ starting and ending at $x$.

Is there a smooth, base point preserving retraction of $\gamma$ to $x$ such that every point on $\gamma$ travels a distance at most, say, $2diam(M)$?

I think the answer is yes. The scheme I have in mind is the following

Consider the cut locus $CL(x)$ of $x$. By perturbing the metric and the curve we can assume that $CL(x)$ is triangulable and $\gamma$ intersects it at finitely many points. Then slide these points off the $CL(x)$. Then the new curve does not intersect $CL(x)$ and can be contracted to $x$ following the geodesics.

This is probably too complicated...

Is there a simple proof? A reference?

Remark: 1) I would be happy with any estimate (not necessarily $2diam(M)$) which is independent of $\gamma$. 2) If there is a good curve shortening procedure that is not base-point-preserving please share it as well.

Let $M$ be a compact simply connected R. manifold. Let $x$ be a base point and let $\gamma$ be a smooth loop in $M$ starting and ending at $x$.

Is there a smooth, base point preserving retraction of $\gamma$ to $x$ such that every point on $\gamma$ travels a distance at most, say, $2diam(M)$?

I think the answer is yes. The scheme I have in mind is the following

Consider the cut locus $CL(x)$ of $x$. By perturbing the metric and the curve we can assume that $CL(x)$ is triangulable and $\gamma$ intersects it at finitely many points. Then slide these points off the $CL(x)$. Then the new curve does not intersect $CL(x)$ and can be contracted to $x$ following the geodesics.

This is probably too complicated...

Is there a simple proof? A reference?

Remark: 1) I would be happy with any estimate (not necessarily $2diam(M)$) which is independent of $\gamma$. 2) If there is a good curve shortening procedure that is not base-point-preserving please share it as well.

Let $M$ be a compact simply connected R. manifold. Let $x$ be a base point and let $\gamma$ be a smooth loop in $M$ starting and ending at $x$.

Is there a base point preserving retraction of $\gamma$ to $x$ such that every point on $\gamma$ travels a distance at most, say, $2diam(M)$?

I think the answer is yes. The scheme I have in mind is the following

Consider the cut locus $CL(x)$ of $x$. By perturbing the metric and the curve we can assume that $CL(x)$ is triangulable and $\gamma$ intersects it at finitely many points. Then slide these points off the $CL(x)$. Then the new curve does not intersect $CL(x)$ and can be contracted to $x$ following the geodesics.

This is probably too complicated...

Is there a simple proof? A reference?

Remark: 1) I would be happy with any estimate (not necessarily $2diam(M)$) which is independent of $\gamma$. 2) If there is a good curve shortening procedure that is not base-point-preserving please share it as well.

Source Link
Andrey Gogolev
Andrey Gogolev
  • 4.2k
  • 1
  • 22
  • 26

Effective contraction of a loop. Reference or a simple proof?

Let $M$ be a compact simply connected R. manifold. Let $x$ be a base point and let $\gamma$ be a smooth loop in $M$ starting and ending at $x$.

Is there a smooth, base point preserving retraction of $\gamma$ to $x$ such that every point on $\gamma$ travels a distance at most, say, $2diam(M)$?

I think the answer is yes. The scheme I have in mind is the following

Consider the cut locus $CL(x)$ of $x$. By perturbing the metric and the curve we can assume that $CL(x)$ is triangulable and $\gamma$ intersects it at finitely many points. Then slide these points off the $CL(x)$. Then the new curve does not intersect $CL(x)$ and can be contracted to $x$ following the geodesics.

This is probably too complicated...

Is there a simple proof? A reference?

Remark: 1) I would be happy with any estimate (not necessarily $2diam(M)$) which is independent of $\gamma$. 2) If there is a good curve shortening procedure that is not base-point-preserving please share it as well.