The paper discusses the behavior of embedded plane curves under the heat equation, also known as curve shortening or heat flow on isometric immersions. The main theorem states that a smooth embedded curve in the plane will shrink smoothly to a point, with the limiting shape being a round circle. The proof involves several key steps: 1. **Existence and Smoothness**: The existence of a solution to the evolution equation is established, and it is shown that the curve remains smooth and embedded as long as its curvature remains bounded. 2. **Nonexistence of Corners**: A theorem by Richard Hamilton is generalized to show that if the curvature blows up, it must do so along an arc with a total curvature of at least \(\pi\). 3. **δ-Whisker Lemma**: This lemma prevents the curve from getting too close to itself along subarcs that turn through at least \(\pi\), ensuring that the curve does not develop singularities. 4. **Main Theorem Proof**: The proof of the main theorem is divided into three cases: - **Case I**: Spirals do not collapse. The argument shows that if a spiral shrinks to a point, it must first become convex. - **Case II**: If the curve shrinks to a point, it becomes convex before degenerating. - **Case III**: If the curve shrinks to a point and has a total curvature of exactly \(\pi\), it must become convex. The paper also includes detailed proofs and lemmas to support these arguments, demonstrating that the curve's curvature remains bounded and that the curve's total curvature converges to \(2\pi\). The results complete the proof that curve shortening shrinks embedded plane curves smoothly to points with a round limiting shape.The paper discusses the behavior of embedded plane curves under the heat equation, also known as curve shortening or heat flow on isometric immersions. The main theorem states that a smooth embedded curve in the plane will shrink smoothly to a point, with the limiting shape being a round circle. The proof involves several key steps: 1. **Existence and Smoothness**: The existence of a solution to the evolution equation is established, and it is shown that the curve remains smooth and embedded as long as its curvature remains bounded. 2. **Nonexistence of Corners**: A theorem by Richard Hamilton is generalized to show that if the curvature blows up, it must do so along an arc with a total curvature of at least \(\pi\). 3. **δ-Whisker Lemma**: This lemma prevents the curve from getting too close to itself along subarcs that turn through at least \(\pi\), ensuring that the curve does not develop singularities. 4. **Main Theorem Proof**: The proof of the main theorem is divided into three cases: - **Case I**: Spirals do not collapse. The argument shows that if a spiral shrinks to a point, it must first become convex. - **Case II**: If the curve shrinks to a point, it becomes convex before degenerating. - **Case III**: If the curve shrinks to a point and has a total curvature of exactly \(\pi\), it must become convex. The paper also includes detailed proofs and lemmas to support these arguments, demonstrating that the curve's curvature remains bounded and that the curve's total curvature converges to \(2\pi\). The results complete the proof that curve shortening shrinks embedded plane curves smoothly to points with a round limiting shape.