change notation and integrate by parts

a bln

Huisken and Sinestrari 1999, pt. 4

leave a comment »

So I think I figured out basically what’s going on with those cones \Gamma_k. We needed them to be convex, so we could apply Hamilton’s maximum principle for systems.

Recall that the elementary symmetric polynomials on \mathbb{R}^n are defined by

S_k(\lambda)=\sum_{i_1<\cdots<i_k}\lambda_{i_1}\cdots\lambda_{i_k}

If we let S_{k,i} denote the polynomial formed by omitting from S_k without the terms involving \lambda_i. It’s immediate that

S_k(\lambda)=\lambda_iS_{k-1,i}+S_{k,i}

Recall also the cones \Gamma_k=\{\lambda\in\mathbb{R}^n|S_1(\lambda)> 0,\cdots,S_k(\lambda)>0\}. We have \Gamma_k\supset \Gamma_{k+1}.

We’re going to prove

Theorem. \Gamma_k are convex.

Proof. We’ll proceed by induction. The base case, \Gamma_1, is clear, since \Gamma_1 is a half-space.

Now suppose that \Gamma_k is convex. Consider the functions on \Gamma_{l-1}, Q_l(\lambda)=\frac{S_l}{S_{l-1}}, Q_{l,i}=\frac{S_{l,i}}{S_{l-1,i}}. In fact, \Gamma_l=\{\lambda| Q_l(\lambda)>0\}, i.e. \Gamma_{l} is the epigraph of Q_{l}.
Since we’ve assumed \Gamma_k is convex, this means that convexity of \Gamma_{k+1} is equivalent to \frac{\partial^2}{\partial \xi^2}Q_{k+1}(\lambda)\leq 0, where \xi\in\mathbb{R}^n,\lambda\in\Gamma_k.

The relation S_{k+1}=\lambda_iS_{k,i}+S_{k+1,i} implies

\lambda_i+Q_{k,i}=\frac{S_k}{S_{k-1,i}}

and

\sum_i\lambda_i^2 S_{k-1,i}=HS_k-(k+1)S_{k+1}

Dividing this last identity through by S_k, we have

(k+1)Q_{k+1}=\sum_i\lambda_i-\lambda_i^2\frac{S_{k-1,i}}{S_k}
=\sum_i\lambda_i-\lambda_i^2\frac{S_{k-1,i}}{S_{k,i}+\lambda_iS_{k-1,i}}
= \sum_i\lambda_i-\frac{\lambda_i^2}{\lambda_i+Q_{k,i}}

So that the function Q_{k+1}, whose concavity we’d like to establish, can be written in terms of the functions Q_{k,i}.

Now \frac{\partial^2}{\partial \xi^2}(Q_{k,i})(\lambda)=\frac{\partial^2}{\partial \overline{\xi}^2}Q_{k}(\overline{\lambda}), where the bar means setting the ith component to zero. It’s easy to check that \overline\lambda\in\Gamma_{k-1}, and by our inductive hypothesis, every second derivative of Q_k is nonpositive on this set. So we have \frac{\partial^2}{\partial \xi^2}(Q_{k,i})(\lambda)\leq 0, i.e. Q_{k,i} is concave on \Gamma_k.

The claim is now that the concavity of Q_{k,i} implies the concavity of \lambda_i-\frac{\lambda_i^2}{\lambda_i+Q_{k,i}}. This follows from a computation, but you can kind of see what’s going on by noting that Q_{k,i} and \lambda_i+Q_{k,i} are degree-1 rational functions with nonpositive second derivatives.

Consider the 1-dimensional case. A degree-1 rational function \phi(x) is one which is asymptotic to a line; the nonpositive second derivative means the graph of \phi lies above the graph of the line. For example, let’s let

\phi(x)=\frac{x^2+5x}{x+3}

Then \phi is asymptotic to the line y=x, but since \phi(x)=x\frac{x+5}{x+3}>x, we have that the second derivative is nonpositive everywhere. Here’s the graph of \phi:

Now consider what happens when we take the function \frac{x^2}{\phi{x}}, analogous to \frac{\lambda_i^2}{\lambda_i+Q_{k,i}}. In our example, we have \frac{x^2}{\phi{x}}=x\frac{x+3}{x+5}. This function, you may notice, is also asymptotic to y=x, but it’s asymptotic to it from below, and its second derivative is nonnegative.

But the function we’re interested in is \lambda_i-\frac{\lambda_i^2}{\lambda_i+Q_{k,i}}. Actually we’re interested in its second derivative, which means the linear term \lambda_i doesn’t matter, so we can discard it, and consider -\frac{\lambda_i^2}{\lambda_i+Q_{k,i}}, which is analogous to -\frac{x^2}{\phi(x)}, whose graph looks like

The relevant fact about this graph is that it has the same concavity as the original \phi!

So concavity of Q_{k,i}, which we have by the inductive hypothesis that \Gamma_k is convex, is enough to show that each \lambda_i-\frac{\lambda_i^2}{\lambda_i+Q_{k,i}} is concave, hence Q_{k+1} is the sum of concave functions, hence itself concave. But this implies \Gamma_{k+1} is convex, which completes the induction.

Written by thecooper

28 June 2008 at 12:54 am

Posted in math posts, mean curvature flow, moderate, prerequisites required

Tagged with , ,

«
»

Leave a Reply