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I am currently trying to understand Wilking's paper "A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities" (DOI: 10.1515/crelle.2012.018, arXiv:1011.3561). If I may, I wish to clarify a few points where I am getting stuck:

  1. I am not sure how to prove Lemma 2.3. Wilking says that one can use estimates similar to Shi for linearized equations built from the PDE satisfied by $S_k - Hk$, where $Hk$ is a Harnack operator, and $S_k$ is an approximation as $k \to \infty$. Is this straightforward? Also, it is not at all clear to me why the sequence $S_k$ needs to be introduced at all for proving Theorem 2.1.

  2. On page 234, Wilking takes a closed convex set $C$ of maximal dimension inside the vector space $S^2_B(\mathfrak{g})$, and defines the cone at infinity as $$\partial_\infty C := \lim_{\lambda \to \infty}\frac{C}{\lambda}.$$ I am having trouble understanding exactly what this notation means.

Any help will be highly appreaciated. Thanks!

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To answer your second question, this is just the Gromov-Hausdorff limit of $C$ with the metric rescaled by $1/\lambda,$ where $\lambda \to \infty.$ In a metric sense, $C$ is "asymptotic" to $\partial_{\infty}C.$

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