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 $v \in TM$.

For an open 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.

This invariant appears in Kei Irie's paper 'Displacement energy of unit cotangent bundles' in Section 2, http://arxiv.org/abs/1106.2199

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 $T^*M$.

My question is if this invariant appears elsewhere in the literature or is related to other known invariants?

  • $\begingroup$ If $M$ is compact, we can take $K=M$, and any smooth function on $M$ has a critical point. Hence, there does not exist a smooth function $f$ such that $|\nabla f(x)|_g\geq 1$, $\forall x\in M$. How do you define $S(g, M)$ in this case? $\endgroup$ Jan 13 '13 at 11:53
  • $\begingroup$ @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 $K \subset M$. $\endgroup$
    – Cadoi
    Jan 13 '13 at 16:04

Remark. Now that Cadoi has disclosed the origin of the invariant (which was absent from the original formulation of the problem) my musings in trying to guess the symplectic geometry behind it seem a bit silly. I'll leave the post anyway, since it maybe useful in understanding the question. However, maybe there is something to be said for including motivation and sources (right from the start) in MO questions.

Dear Cadoi,

This is just a comment on your question, but since it is a bit long and I can see what I'm TeXing I'll post it as an answer. I myself have not seen this invariant before, however it has a symplectic look about it and may turn out to be some capacity in disguise. Note that if you define a Riemannian invariant by taking a symplectic capacity and evaluating it on unit codisc bundles, then you get a size invariant. This is because isometries lift to symplectic transformations that preserve the unit codisc bundle and because capacities are monotone.

The reason I say your invariant looks symplectic is because, seen geometrically, the differential of $f$ defines a Lagrangian submanifold of the cotangent bundle and the condition that the norm of the gradient of $f$ be at least $1$ on $K$ means this Lagrangian submanifold does not intersect the unit co-disc bundle over $K$. The quantity $\|f\|$ is a relative symplectic invariant of the pair of Lagrangians given by the zero section and the differential of $f$. I don't exactly remember Viterbo's notation for this, but it is in his paper Symplectic topology as the geometry of generating functions.

Just for the fun of it, and without implying that this will give something non-trivial, I'll attempt a "symplectization" of your invariant, at least to the extent that it might give size invariant of Riemannian or Finsler metrics that gives the same value on two metrics that have symplectomorphic codisc bundles:

Let $D^*M$ be the unit codisc bundle of your metric and let $K \subset D^*M$ be a compact subset. Consider the set $P_K$ of pairs of Lagrangian submanifolds $(E,F)$ such that (1) $E$ is contained in $D^*M$, (2) $F$ does not intersect $K$, and (3) both $E$ and $F$ are Hamiltonian isotopic to the zero section.

Define $S(D^*M,K)$ as the infimum of the Viterbo capacities of all pairs of Lagrangians $(E,F) \in P_K$ and now define $S(D^*M)$ as the supremum of the quantities $S(D^*M,K)$ as $K$ ranges over all compact subsets of $D^*M$.

If you restrict $E$ to always be the zero section, $F$ to the the graph of the differential of a function, and $K$ to the the codisc bundle over some compact subset of $M$ this is exactly your invariant.

Caveat emptor, this definition may require some fidgeting (like restricting the class of compact sets $K$) to make it non-obviously trivial.

  • $\begingroup$ Dear Cadoi, have you checked, that your invariant is non-zero? $\endgroup$
    – ubunke
    Jan 13 '13 at 7:42
  • $\begingroup$ @Ubunke: if I understand Cadoi's construction correctly, it should give different values when we take the real line with its standard metric $ds$ and the metric $exp(-x^2)ds$. $\endgroup$ Jan 13 '13 at 9:40
  • $\begingroup$ Yes, but in the higher dimensional case I suspect that you can make the derivative large at every point without varying the function to much. $\endgroup$
    – ubunke
    Jan 13 '13 at 13:14
  • $\begingroup$ A good test case could be the two-dimensional case of the example I propose in my previous comment. I like the idea behind Cadoi's invariant, but I also wonder whether it will give something interesting in practice. $\endgroup$ Jan 13 '13 at 14:00
  • 1
    $\begingroup$ @Cadoi: never mind, I have no right to complain given that I had fun reverse-engineering the invariant. $\endgroup$ Jan 14 '13 at 18:30

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