stadler.tex

\section{Total curvature of knots}

\begin{thm}{Theorem}
Assume a simple closed curve $\gamma$ in $\RR^m$ has total curvature at most $4\cdot\pi$.
Then the area minimizing disc spanned by $\gamma$ is embedded.
\end{thm}

\parit{Proof.}
Assume $\gamma$ is a polygonal line.
Let us attach to $\gamma$ a collar so that obtained space $\spc{U}$ will be $\CAT0$,
the minimal disc spanned by $\gamma$ together with the color will form a area minimizing surface.


\begin{thm}{Monotonicity formula}
Assume $\Sigma$ is an area minimizing surface in a complete length $\CAT0$ space $\spc{U}$
and $p\in\Sigma$.
Consider the function
\[r\mapsto\tfrac1{r^2}\cdot\area(\Sigma\cap\cBall[p,r]).\]
is nondecreasing.
\end{thm}

\parit{Proof.}
Let us show that 
\[4\cdot\pi\cdot(\length\Sigma\cap\Sphere[p,r])^2\ge \area(\Sigma\cap\cBall[p,r])\]
for all $r>0$.


\qeds