User cadoi - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T09:16:24Z http://mathoverflow.net/feeds/user/30597 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118754/known-size-invariant-for-riemannian-manifolds Known size invariant for Riemannian manifolds? Cadoi 2013-01-12T19:39:21Z 2013-01-14T18:28:23Z <p>Larry Guth in his 2010 ICM address mentions the notion of a size invariant of Riemannian metrics on a smooth manifold $M$. These are functions $S: Metrics(M) \to \mathbb{R}$ that are invariant under isometry and satisfy $S(g) \leq S(g')$ if $g \leq g'$, where $g \leq g'$ means $g(v,v) \leq g'(v,v)$ for all $v \in TM$. </p> <p>For an <b>open</b> Riemannian manifold $(M, g)$ and for a compact set $K \subset M$ define $$S_M(g, K) = \inf \{||f|| : f \in C^\infty_c(M) \mbox{ such that } |\nabla f(x)|_g \geq 1 \mbox{ for all } x\in K\}$$ where $||f|| = \max_M f - \min_M f$, and then define $$S_M(g) = \sup_{K\subset M} S_M(g, K).$$ One can prove that this is an size invariant.</p> <p>This invariant appears in Kei Irie's paper 'Displacement energy of unit cotangent bundles' in Section 2, <a href="http://arxiv.org/abs/1106.2199" rel="nofollow">http://arxiv.org/abs/1106.2199</a></p> <p>Irie proves that $S_M(g) \leq c_n r(M, g)$ where $c_n$ is a dimensional constant and $r(M,g)$ is the inner radius. Irie also lower bounds $S_M(g)$ in terms of the displacement energy of the unit disk cotangent bundle $D^*_g M$ in the symplectic manifold $T^*M$.</p> <p>My question is if this invariant appears elsewhere in the literature or is related to other known invariants? </p> http://mathoverflow.net/questions/118754/known-size-invariant-for-riemannian-manifolds/118760#118760 Comment by Cadoi Cadoi 2013-01-14T13:04:06Z 2013-01-14T13:04:06Z @alvarezpaiva: I should have included the source of this invariant from the start to be fair to Irie as well as any readers. I was/am in particular looking for other Riemannian geometry connections to this invariant, as opposed to the symplectic connection, so I thought I could steer the question towards the Riemannian side by not mentioning the symplectic side. This was a mistake and I am sorry. http://mathoverflow.net/questions/118754/known-size-invariant-for-riemannian-manifolds Comment by Cadoi Cadoi 2013-01-13T16:04:09Z 2013-01-13T16:04:09Z @Liviu Nicolaescu: This is why one specifies that $S_M(g)$ should be considered for open manifolds M. For non-open manifolds the best one can do is look at $S_M(g,K)$ for proper compact subsets $K \subset M$. http://mathoverflow.net/questions/118754/known-size-invariant-for-riemannian-manifolds/118760#118760 Comment by Cadoi Cadoi 2013-01-13T15:56:18Z 2013-01-13T15:56:18Z @Ubunke: Irie proves a non-trivial lower bound for $S_M(g)$ in terms of the displacement energy of the unit disk cotangent bundle $D^*M$ in $T^*M$. This is due to the symplectic view of this invariant mentioned by alvarezpaiva.