Curve shortening flow with boundary

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Let $(M^2,g)$ be a 2-dimensional complete Riemannian manifold (e.g. $(mathbbR^2,delta_ij)$) and $p,qin M$ two points with $pneq q$. Let $gamma:Ito M$ be a smooth embedded curve starting at $p$ and ending at $q$.

When does the curve shortening flow starting at $gamma$ converge to a geodesic segment joining $p$ and $q$, i.e. the "shortest" curve joining the two points?

I am aware of Grayson's result that an embedded closed curve in a 2-manifold either shrinks to a round point or converges to a geodesic.

What results are known for curve shortening flow for line segments? Is the flow even well defined?

differential-geometry pde mean-curvature-flows

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asked Jul 22 at 16:34

rpf

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Let $(M^2,g)$ be a 2-dimensional complete Riemannian manifold (e.g. $(mathbbR^2,delta_ij)$) and $p,qin M$ two points with $pneq q$. Let $gamma:Ito M$ be a smooth embedded curve starting at $p$ and ending at $q$.

When does the curve shortening flow starting at $gamma$ converge to a geodesic segment joining $p$ and $q$, i.e. the "shortest" curve joining the two points?

I am aware of Grayson's result that an embedded closed curve in a 2-manifold either shrinks to a round point or converges to a geodesic.

What results are known for curve shortening flow for line segments? Is the flow even well defined?

differential-geometry pde mean-curvature-flows

share|cite|improve this question

asked Jul 22 at 16:34

rpf

929512

add a comment | 

up vote
4
down vote

favorite

2

up vote
4
down vote

favorite

2

2

Let $(M^2,g)$ be a 2-dimensional complete Riemannian manifold (e.g. $(mathbbR^2,delta_ij)$) and $p,qin M$ two points with $pneq q$. Let $gamma:Ito M$ be a smooth embedded curve starting at $p$ and ending at $q$.

When does the curve shortening flow starting at $gamma$ converge to a geodesic segment joining $p$ and $q$, i.e. the "shortest" curve joining the two points?

I am aware of Grayson's result that an embedded closed curve in a 2-manifold either shrinks to a round point or converges to a geodesic.

What results are known for curve shortening flow for line segments? Is the flow even well defined?

differential-geometry pde mean-curvature-flows

share|cite|improve this question

asked Jul 22 at 16:34

rpf

929512

Let $(M^2,g)$ be a 2-dimensional complete Riemannian manifold (e.g. $(mathbbR^2,delta_ij)$) and $p,qin M$ two points with $pneq q$. Let $gamma:Ito M$ be a smooth embedded curve starting at $p$ and ending at $q$.

When does the curve shortening flow starting at $gamma$ converge to a geodesic segment joining $p$ and $q$, i.e. the "shortest" curve joining the two points?

I am aware of Grayson's result that an embedded closed curve in a 2-manifold either shrinks to a round point or converges to a geodesic.

What results are known for curve shortening flow for line segments? Is the flow even well defined?

differential-geometry pde mean-curvature-flows

share|cite|improve this question

asked Jul 22 at 16:34

rpf

929512

share|cite|improve this question

share|cite|improve this question

asked Jul 22 at 16:34

rpf

929512

asked Jul 22 at 16:34

rpf

929512

asked Jul 22 at 16:34

rpf

929512

929512

add a comment | 

add a comment | 

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Two references that might help:

(1) "Midpoint geodesic polygon / Birkhoff curve shortening": Bowditch: "The convergence of the Birkhoff process seems to be an open question for Riemanninan 2-manifolds."

---

         

---

(2) The paper below explores what they call the disk flow, which
replaces arcs of a curve with geodesic segments:

Hass, Joel, and Peter Scott. "Shortening curves on surfaces." Topology 33, no. 1 (1994): 25-43. (PDF download from Semantic Scholar.)

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answered Jul 22 at 18:57

Joseph O'Rourke

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1 Answer
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1 Answer
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up vote
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Two references that might help:

(1) "Midpoint geodesic polygon / Birkhoff curve shortening": Bowditch: "The convergence of the Birkhoff process seems to be an open question for Riemanninan 2-manifolds."

---

         

---

(2) The paper below explores what they call the disk flow, which
replaces arcs of a curve with geodesic segments:

Hass, Joel, and Peter Scott. "Shortening curves on surfaces." Topology 33, no. 1 (1994): 25-43. (PDF download from Semantic Scholar.)

share|cite|improve this answer

answered Jul 22 at 18:57

Joseph O'Rourke

17.1k248103

add a comment | 

up vote
0
down vote

Two references that might help:

(1) "Midpoint geodesic polygon / Birkhoff curve shortening": Bowditch: "The convergence of the Birkhoff process seems to be an open question for Riemanninan 2-manifolds."

---

         

---

(2) The paper below explores what they call the disk flow, which
replaces arcs of a curve with geodesic segments:

Hass, Joel, and Peter Scott. "Shortening curves on surfaces." Topology 33, no. 1 (1994): 25-43. (PDF download from Semantic Scholar.)

share|cite|improve this answer

answered Jul 22 at 18:57

Joseph O'Rourke

17.1k248103

add a comment | 

up vote
0
down vote

up vote
0
down vote

Two references that might help:

(1) "Midpoint geodesic polygon / Birkhoff curve shortening": Bowditch: "The convergence of the Birkhoff process seems to be an open question for Riemanninan 2-manifolds."

---

         

---

(2) The paper below explores what they call the disk flow, which
replaces arcs of a curve with geodesic segments:

Hass, Joel, and Peter Scott. "Shortening curves on surfaces." Topology 33, no. 1 (1994): 25-43. (PDF download from Semantic Scholar.)

share|cite|improve this answer

answered Jul 22 at 18:57

Joseph O'Rourke

17.1k248103

Two references that might help:

(1) "Midpoint geodesic polygon / Birkhoff curve shortening": Bowditch: "The convergence of the Birkhoff process seems to be an open question for Riemanninan 2-manifolds."

---

         

---

(2) The paper below explores what they call the disk flow, which
replaces arcs of a curve with geodesic segments:

Hass, Joel, and Peter Scott. "Shortening curves on surfaces." Topology 33, no. 1 (1994): 25-43. (PDF download from Semantic Scholar.)

share|cite|improve this answer

answered Jul 22 at 18:57

Joseph O'Rourke

17.1k248103

share|cite|improve this answer

share|cite|improve this answer

answered Jul 22 at 18:57

Joseph O'Rourke

17.1k248103

answered Jul 22 at 18:57

Joseph O'Rourke

17.1k248103

answered Jul 22 at 18:57

Joseph O'Rourke

17.1k248103

17.1k248103

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add a comment | 

 

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