Fixed points of endomorphisms and relations between metrics in preGarside monoids

Rodaro and Silva proved that the fixed points submonoid and the periodic points submonoid of a trace monoid endomorphism are always finitely generated. We show that for finitely generated left preGarside monoids, that includs finitely generated preGarside monoids, Garside monoids and Artin monoids, the fixed and periodic points submonoids of any endomorphism are also finitely generated left preGarside monoids under some condition, and in the case of Artin monoids, these submonoids are always Artin monoids too. We also prove algebraically some inequalities, equivalences and non-equivalences between three metrics in finitely generated preGarside monoids, and especially in trace monoids and Garside monoids

Branson's Q-curvature in Riemannian and Spin Geometry

On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's Q-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson's Q-curvature. On a closed n-dimensional manifold, $n\ge 5$, we compare the three basic conformally covariant operators : the Branson-Paneitz, the Yamabe and the Dirac operator (if the manifold is spin) through their first eigenvalues. Equality cases are also characterized.Comment: 14 pages, Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branso

A holographic principle for the existence of parallel spinor fields and an inequality of Shi-Tam type

Suppose that $\Sigma=\partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $( M,\langle\;,\;\rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $\Sigma$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric $\langle\;,\;\rangle_H=H^2\langle\;,\;\rangle$ is at least $n/2$ and equality holds if and only if there exists a parallel spinor field on $M$. As a consequence, if $\Sigma$ admits an isometric and isospin immersion $\phi$ with mean curvature $H_0$ as a hypersurface into another spin Riemannian manifold $M_0$ admitting a parallel spinor field, then \begin{equation} \label{HoloIneq} \int_\Sigma H\,d\Sigma\le \int_\Sigma \frac{H^2_0}{H}\, d\Sigma \end{equation} and equality holds if and only if both immersions have the same shape operator. In this case, $\Sigma$ has to be also connected. In the special case where $M_0=\R^{n+1}$, equality in (\ref{HoloIneq}) implies that $M$ is an Euclidean domain and $\phi$ is congruent to the embedding of $\Sigma$ in $M$ as its boundary. We also prove that Inequality (\ref{HoloIneq}) implies the Positive Mass Theorem (PMT). Note that, using the PMT and the additional assumption that $\phi$ is a strictly convex embedding into the Euclidean space, Shi and Tam \cite{ST1} proved the integral inequality \begin{equation}\label{shi-tam-Ineq} \int_\Sigma H\,d\Sigma\le \int_\Sigma H_0\, d\Sigma, \end{equation} which is stronger than (\ref{HoloIneq}) .Comment: arXiv admin note: text overlap with arXiv:1502.0408