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Curve shortening flow on Riemann surfaces with conical singularities

Nikolaos Roidos, Andreas Savas-Halilaj

Abstract

We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic equation. In this case, we establish short time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and obtain some collapsing and convergence results.

Curve shortening flow on Riemann surfaces with conical singularities

Abstract

We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic equation. In this case, we establish short time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and obtain some collapsing and convergence results.

Paper Structure

This paper contains 19 sections, 30 theorems, 389 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Curves on Riemann surfaces with singular metrics
  3. Geometry of singular surfaces
  4. Curves on singular surfaces
  5. Conical singularities
  6. A Gauss-Bonnet formula
  7. Functional analytic methods for parabolic problems
  8. Sectorial operators and functional calculus
  9. Maximal $L^q$-regularity for parabolic equations
  10. Cone differential operators
  11. Cone operators
  12. Short time existence and regularity of the flow
  13. Short time existence
  14. Evolution equations and some geometric consequences
  15. Evolution equations
  16. ...and 4 more sections

Key Result

Theorem A

Let $(\Sigma,\operatorname{g})$ be a Riemann surface where $\operatorname{g}$ is a metric with conformal conical $($not necessarily distinguished $)$ points $\{p_1,\dots,p_n\}$, $n\ge 2$, of orders $-1<\beta_1\le\cdots\le \beta_n<0$, respectively, and let $\gamma\colon[0,1]\to\Sigma$ be a continuous Then, there exist $T>0$ and a unique $\varGamma\colon[0,1]\times[0,T]\to\Sigma$ such that:

Figures (8)

  • Figure 1: Degenerate curve shortening flow.
  • Figure 2: The cone $V_{\theta}$.
  • Figure 7: Geodesic curves for $\beta=-1$.
  • Figure 8: Tear drop curve.
  • Figure 9: Curvilinear triangle.
  • ...and 3 more figures

Theorems & Definitions (69)

  • Theorem A
  • Remark 1.1
  • Theorem B
  • Remark 1.2
  • Theorem C
  • Theorem D
  • Theorem E
  • Example 2.1
  • Lemma 2.2
  • proof
  • ...and 59 more