Hyperbolic surface ($K_{\mathrm{Gaussian}}=-1$)

A surface with a complete Riemannian metric of constant Gaussian curvature $-1$. Its local model is the Poincaré half-plane model; the orientable case is a quotient by a free properly discontinuous group action of orientation-preserving isometries.

Suppose a finite simple polygon in the curvature-minus-one hyperbolic plane has hyperbolic geodesic sides paired to produce a closed topological surface. Its metric descends to a smooth hyperbolic metric exactly when paired polygon sides have equal lengths and the sum of internal angles in each vertex equivalence class is $2\pi$. Equal lengths allow isometric gluing at open edges; at vertices the sectors assemble into a hyperbolic disc precisely when their total angle is $2\pi$. Other positive total angles produce a cone singularity of a surface metric. The area of a hyperbolic geodesic polygon and the Gauss-Bonnet theorem then give area $-2\pi \chi$ for the closed surface.

For a marked closed hyperbolic surface, the representation of its fundamental group whose image acts as the deck group on the hyperbolic plane. It is discrete and faithful, determined up to conjugation by the marking and the metric, and gives $X={\mathbb{H}}^2/\rho ({\pi }_1{\Sigma }_g)$. For a closed orientable surface it admits determinant-one lifts. It is different from the tangent-space action of the Riemannian holonomy group.

The unmarked length spectrum of a compact hyperbolic surface is the multiset of lengths of its closed geodesics, with a consistent convention for primitive curves and orientation. The marked version labels lengths by free homotopy classes. The Selberg trace formula relates the unmarked multiset to the Laplace-Beltrami operator spectrum.

For a closed hyperbolic surface with a specified marking of its fundamental group, assign to each nontrivial conjugacy class the length of its unique closed geodesic. Retaining the group label is stronger than retaining only the unmarked length multiset: it also records which lengths belong to intersecting curves or powers of particular curves.

For closed orientable hyperbolic surfaces of curvature $-1$, the spectrum of the Laplace-Beltrami operator, including multiplicities, determines the unmarked length spectrum of closed geodesics, and conversely. Count either oriented geodesics throughout or unoriented ones throughout. Primitive lengths with multiplicities determine the lengths of all iterates, and these determine the primitive lengths recursively.^[ref]

The Selberg trace formula connects the two lists: the spectral side sums functions of eigenvalues, while the geometric side sums explicit contributions from closed-geodesic lengths. Equality of either complete list forces equality of the other. The theorem concerns these spectral data; distinct surfaces can still have equal spectra.

For nonorientable hyperbolic surfaces, the trace contributions distinguish orientation-preserving and orientation-reversing geodesics. Their separate counts, rather than an unspecified undifferentiated convention, belong in the extended statement.^[ref]

In a compact smooth family of marked closed hyperbolic surfaces, only finitely many free homotopy classes of closed geodesics have length below a fixed bound somewhere in the family. A uniform bilipschitz equivalence with a fixed metric reduces this to local finiteness of its length spectrum.

For fixed genus $g\ge 2$ and systole lower bound $\epsilon >0$, a uniform finite length cutoff $L$ determines the entire unmarked length spectrum of a closed hyperbolic surface in the thick part, including multiplicities. Equality up to $L$ forces equality everywhere. This is a finite-determination statement for spectra and does not claim that all isospectral surfaces are isometric.

Suppose a compact parameter set carries a locally finite countable multiset of positive quantities, and a fixed injective transform of each labelled quantity is a polynomial in finitely many ambient coordinates. Assume only finitely many labels can have bounded value anywhere in the compact set. Then a uniform finite initial multiset determines the entire multiset. For each source label there are finitely many possible matching labels. Equality for finitely many labels in both directions is a finite union of algebraic sets, obtained from injective matchings and polynomial equalities. These algebraic sets decrease as more labels are included, so they stabilize by the Hilbert basis theorem. A compatible infinite matching in each direction follows by the König infinity lemma, and local finiteness makes multiplicities equal. Squared traces of hyperbolic holonomy representations give the application to the Buser finite length spectrum theorem.

Let $X=\Gamma \backslash {\mathbb{H}}^2$ be a closed orientable hyperbolic surface, where $\Gamma \le PS{L}_2(\mathbb{R})$ is a torsion-free discrete subgroup. Write ${\lambda }_j$ for the nonnegative eigenvalues of its Laplace-Beltrami operator, with multiplicities. A heat version of the Selberg trace formula is, for $t>0$,The outer geometric sum is over primitive nonidentity conjugacy classes in $\Gamma$, equivalently oriented primitive closed geodesics; a class and its inverse are both counted. The inner sum records their iterates. The first term is the identity contribution, proportional to hyperbolic area. This formula follows by taking the Gaussian test function $h(r)=e^{-t(r^2+1/4)}$ in the usual spectral-parameter formula, whose Fourier transform is the displayed Gaussian factor.^[ref]

The left side is the heat trace. Thus the same quantity can be computed from vibration frequencies or from closed-geodesic lengths and hyperbolic area. The Huber spectral equivalence theorem extracts the mutual determination of the two spectra. Quotients with cusps or cone points require additional terms, so the displayed version uses the closed smooth orientable setting.

A collar around a simple closed geodesic of length $\ell$ is an embedded annulus with metric $d{r}^2+{\ell }^2{\cosh }^{2}rd{t}^2$, where $t\in \mathbb{R}/\mathbb{Z}$. The collar lemma gives half-width $w=\mathrm{arcsinh}(1/\sinh (\ell /2))$. As $\ell$ tends to zero the collar gets wide, forcing crossing geodesics to become long.

In curvature $-1$, its length is $2\mathrm{arcosh}3$. The level-two principal congruence subgroup has traces congruent to two modulo four, so the smallest hyperbolic absolute trace is six, achieved by $\left(\begin{array}{cc}5& 2\\ 2& 1\end{array}\right)$. The hyperbolic translation length formula gives the result; the shortest closed geodesic is nonsimple.

A collection of disjoint essential simple closed curves that cuts a closed genus-$g$ surface into $2g-2$ surfaces of type pair of pants. There are $3g-3$ curves. Geodesic representatives are used for Fenchel–Nielsen coordinates.

An elementary move replaces one cuff inside the four-holed sphere or one-holed torus obtained by uniting its adjacent pants. The replacement is an essential curve meeting the old cuff minimally. It changes the pants decomposition graph without changing the underlying surface; the identity surface map identifies the two descriptions.

The graph of a pants decomposition has one vertex per pair of pants and one edge per glued cuff. It is a connected trivalent multigraph, possibly with loops. A closed surface of genus $g$ gives $2g-2$ vertices and $3g-3$ edges, so its first Betti number is $g$. The graph records a decomposition rather than the homeomorphism class alone, and omits cuff lengths and twists.

For fixed $g\ge 2$, a constant $B_g$ bounds all cuff lengths in some pants decomposition of every closed genus-$g$ hyperbolic surface. Together with Fenchel–Nielsen coordinates this implies Mumford's compactness theorem.

For a fixed pants decomposition, the positive cuff lengths and real twist parameters give coordinates on Teichmüller space. With twists measured in length units, a full Dehn twist changes ${\tau }_i$ by ${\ell }_i$; with angle units it changes the twist by $2\pi$.

After cutting a hyperbolic surface along a simple closed geodesic, reglue its two boundary copies after an arclength displacement $\tau$. The boundary length remains fixed, while the Riemannian metric varies. In marked Fenchel–Nielsen coordinates, the twist is real-valued; changing it by one boundary length corresponds to a Dehn twist in the marking. This construction also applies to a hyperbolic pair of pants with a fixed cone angle and two geodesic boundaries.

For a fixed pants decomposition of a closed genus-$g$ hyperbolic surface, its $3g-3$ cuff lengths determine all its Y-pieces. Two suitably chosen transverse simple closed curves at each cuff, one obtained from the other by a Dehn twist, determine the corresponding twist: hyperbolic seam identities express their lengths through shifted hyperbolic cosines of that twist, removing the sign ambiguity of just one transversal. Thus finitely many marked lengths determine the Fenchel–Nielsen coordinates. The unmarked length spectrum lacks these labels, which is why this fact alone does not imply global spectral uniqueness.

A simple closed curve that is a geodesic of length $\ell$ on a hyperbolic surface has an embedded collar of half-width $w$ with $\sinh w\sinh (\ell /2)=1$. In circumference-one coordinates its metric is $d{r}^2+{\ell }^2{\cosh }^{2}rd{t}^2$, and its conformal modulus of an annulus is $(2/\ell )\arctan (\mathrm{csch}(\ell /2))\sim \pi /\ell$.

The infimum of nonconstant closed geodesic lengths in curvature $-1$, with value $+\infty$ if there are none. On a closed surface with a hyperbolic metric this is a positive attained minimum, and a minimizing geodesic is simple. For fixed closed genus $g\ge 2$, a common positive lower bound gives relative compactness in the unmarked moduli space of Riemann surfaces by Mumford's compactness theorem.

The metric area of a hyperbolic surface. In the Poincaré half-plane model the density is $dxdy/y^2$. A finite-area torsion-free quotient of genus $g$ with $r$ cusps has area $2\pi (2g-2+r)$ by the Gauss-Bonnet theorem.