Given a closed 2-surface $M$ together with a Riemannian metric $g$. We pick a free homotopy class $\gamma \in \pi_1(M)$ and consider the set $C(\gamma)$ of all closed geodesics homotopic to $\gamma$.
Of course, the set $C(\gamma)$ may be infinite. The length of a geodesic loop gives a function on $C(\gamma)$. My question is whether this function is bounded (in terms of $g$)?
No such bound $C(\gamma)$ exists, even when $S$ is the two-sphere and even assuming that all geodesics considered are simple. Here is the example (which generalises to surfaces with genus).
Suppose that $S$ is the two-sphere. Pick four open disks $(D_i)_{i = 1}^4$ whose closures are closed disks and which are pairwise disjoint. (For example, use small round disks with respect to the usual round metric on $S$.) Let $P = S - \bigcup_i D_i$; so $P$ is a "four-holed sphere". We equip $P$ with a hyperbolic metric $g_P$ where all boundary components are geodesic. We now claim the following:
We now choose any riemannian metric $g_S$ on $S$ that extends $g_P$. Note that all of the previous geodesics (except perhaps the components of $\partial P$) remain geodesic with respect to the metric $g_S$. Since all of those geodesics are null-homotopic in $S$, we are done.
Morally: The riemannian surface $(S, g_S)$ has four "mushrooms" (the disks $D_i$). The geodesics used above wander around the surface avoiding the tops of the mushrooms.
This seems relevant to your question.
An isosceles tetrahedron (all four faces congruent) contains arbitrarily long closed simple geodesics. This paper proves the reverse: having such geodesics implies the surface is an isosceles tetrahedron.
Akopyan, Arseniy, and Anton Petrunin. "Long geodesics on convex surfaces." arXiv:1702.05172 (2017).
The authors say that their theorem "implies that a smooth convex surface does not have arbitrarily long simple closed geodesics."
Figure from Akopyan-Petrunin.
It is more reasonable to consider the situation when the genus of $M$ is finite, because when genus of $M$ is not finite, we can use some rescaling to expand some part of $M$ and the problem will be less meaningful.
In the compact situation, given $(M,g)$, $M$ is torus or sphere, then $C(\gamma)$ is bounded. This is easy to prove, follow a topology argument and local regularity of geodesic flow, in fact this thing describle in the graph1 can not happen.
For a general compact manifold $M$, genus of $M$ is great than 1, first it is easy to prove there is a metric $g$ on $M$ with constant -1 sectional curvature. So by prime geodesic theorem, $\sup_{\gamma \in H_1(M)}C(\gamma)=\infty$, but there is no contradiction with for general $M,g$ and a fix $\gamma\in H_1(M)$, $C(\gamma)<\infty$, in fact this also follow from the phnomenon of graph1 can not happen(regularity of the ODE describle geodesic flow)
graph1
For noncompact case, I think following claim could be proved, but I do not guarantee that the following argument is completely correct
Claim Given a orientable closed 2 -surface together with a Riemannian metric $(M,g )$, and genus of $M$ is $n$. If we can give a condition on $g$ to make the following situation do not happen, We pick a cycle $\gamma \in H_{1}(M)$, and define, $$C(\gamma)=\sup_{[c] = \ [\gamma]\in H_1(M)\\ c\ is\ closed\ geodesic}l(c)$$ where $l$ is the length function, then $C(\gamma)$ is finite for all $\gamma\in H_1(M)$.
As a comment, $(M,g)$ with variable pointwise negative curvature can not avoid the situation.
situation
proof
a fix closed 2 -surface together with a Riemannian metric $(M,g
)$ with genus $n$. Such famalliy could be clssification into $n+1$ finer types, to define a type, we need first claim there is a cononical basis in $H_1(M)$ given by $\gamma_1,...,\gamma_n$, for each $\gamma_i$ it is a loop around a hole given in the second picture. And there are two type for "hole", one type is compact "hole", which is defined by the hole can not be eliminate by glue a $D$ to $M$, and the other type is noncompact "hole", which could be eliminate by glue a $D$ to $M$.
And a finer type of $(M,g)$ with genus $n$ is just counting the number of compact holes, there is at most $k$ and at least $0$ compact hole, so there is $k+1$ finer type. For each type there is $k$ compact genus and $n-k$ non-compact genus, $0\leq k\leq n$.
Now assume $(M,g)$ with genus $n$ also has type $k$, i.e. there is $k$ compact holes in $M$, and $n-k$ noncompact holes in $M$, assume the generators in $H_1(M)$ is $\{\gamma^c_1,...,\gamma^c_k,\gamma^{nc}_1,...,\gamma^{nc}_{n-k}\}$ for a general cycle $\gamma$ in $H_1(M)$ could be represent by $\gamma=\sum_{i=1}^ka_i\gamma^c_i+\sum_{i=1}^{n-k}b_i\gamma^{nc}_i$, it could be prove that, $$C(\gamma)\lesssim \sum_{i=1}^ka_iC(\gamma^c_i)+\sum_{i=1}^{n-k}b_iC(\gamma^{nc}_i)$$ the constant in the previous inequality only depend on $(M,g)$. So we only need to prove $$C(\gamma^c_i)<\infty, C(\gamma^{nc}_i)<\infty, C([0])<\infty.$$ And if the metric $g$ is choosed such that the situation describle in the picture do not happen, and for example if contract $C([0])=\infty$, then there is a sequence of closed geodesic $w_1,w_2,...,w_n,...$ homotopic to zero and $l(w_1)<l(w_2)<l(w_3)<...<l(w_n)<..., \lim_{k\to \infty}l(w_k)=\infty$, on the other hand fix a compact subset of $M, M_k\subset M$ we can consider a family of covering $A_i, \cup_{a \in A_i}B_{r_i}(a)=M_k, \lim_{i\to \infty}r_i=0$, then for every scale $A_i$, there is a $B_{r_i}(a_i)$ s.t. $\limsup_jl((w_j)\cap B_{r_i}(a_i))=\infty$, and because geodesic flow is a hamiltonian system in $M\times TM$， so if we give some gerularity on $g$ then locally we can gain a uniformly upper bound on $l((w_j)\cap B_{r_i}(a_i))$, this derive the major part of $l(w_k)$ for $k$ suffice large is come fram place out of any compact set of $M$. So if $C([\gamma])$ is not bounded it must follow there is a infinite sequence of closed geodesic around a fix noncompact hole, which fall into the situation.
