Let $\Omega$ be an open subset of a Riemannian manifold $M$. Assume that $\Sigma:=\partial \Omega$ is $C^2$. Let $U$ be a neighborhood of $\Omega$ such that $\exp_p(t\nu(p))$ is diffieomorphism, where $p\in\Sigma$ and $\nu$ is the outward unit normal vector field along $\Sigma$. We now consider the distance function $d_\Sigma(x):=d(x,\Sigma)$, $x\in U\setminus\Omega$ and the squared distance function $\eta_\Sigma(x):=d^2_\Sigma(x)$. Let denote the Hessian of $d_\Sigma$ by $S=\nabla\nabla d_\Sigma$. We have the Riccati euquation $$\dot{S}=-R(J(t),\nu(\gamma(t)))\nu(\gamma_t)-S^2.$$ If $M$ is a space form (e.g. $\mathbb R^n$, $S^n$, $\mathbb H^n$) and we know the eigenvalues of $S$ at time 0 (i.e. the principal curvature of $\Sigma$), then using the Riccati equation we can compare the eigenvalues of $S(s)$ (i.e. the principal curvature of $\Sigma_s=\{\exp_p(s\nu(p)):p\in \Sigma\}$) with the eigenvalues of $S(0)$. For example when $M=\mathbb R^n$, if denote the eigenvalues of $S(s)$ by $\kappa_i(s)$, then $$\kappa_i(s)=\frac{\kappa_i(0)}{1+s\kappa_i(0)}$$

My question is, what if we only know a lower sectional curvature bound $\alpha$? For example, can we have any explicit comparison between $\kappa_i(s)$ and $\bar{\kappa}_i(s)$ where $\bar{\kappa}_i(s)$ is the eigenvalues of $\bar{S}(s)$? Here $\bar{S}(s)$ is the solution to the Riccati equation in the space form of curvature $\alpha$ with the initial value $\bar{S}(0)=S(0)$. I have the same question for eigenvalues of $\nabla^2\eta_\Sigma$.

  • $\begingroup$ Could you give another Web-accessible reference for the evolution equation for the eigenvalues? $\endgroup$ Dec 2 '16 at 19:07
  • $\begingroup$ @TomCopeland I think that the note of Hermann Karcher on Riemannian comparison constructions would be useful. Please click here. Also check the references given in the link below (in the answer). $\endgroup$
    – Math101
    Dec 2 '16 at 19:24

I have found an answer to my question on Mathoverflow.net here. @Raziel gave the reference

H. L. Royden, MR 948079 Comparison theorems for the matrix Riccati equation, Comm. Pure Appl. Math. 41 (1988), no. 5, 739--746.

where I can find a comparison theorem for eigenvalues of $S$ but not for eigenvalues of $\nabla\nabla \eta_\Sigma$.

I keep the question and the answer on Mathoverflow since it might be useful for someone else.


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