Closed square-integrable adjoint one-forms on a cusped hyperbolic three-manifold are exact

Put $G={\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$, $K=\mathrm{P}\mathrm{S}\mathrm{U}(2)$ and $V={\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$. Choose $X_1=\frac{1}{2}\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)$, $X_2=\frac{1}{2}\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$ and $X_3=\frac{1}{2}\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right)$ in $\mathfrak{k}$, and put $Y_j=i{X}_j$. These $Y_j$ are an orthonormal tangent basis for curvature $-1$ hyperbolic space; for example $\exp (t{Y}_1)$ moves the basepoint distance $\mathrm{\mid }t\mathrm{\mid }$, and the $K$ action rotates this basis. The matrix commutators give $[X_i,X_j]={ϵ}_{ijk}{X}_k$ and $[Y_i,Y_j]=i{ϵ}_{ijk}{Y}_k$. On $V$ let $R_j=\mathrm{ad}{Y}_j$. Each $Y_j$ is Hermitian, so $R_j$ is self-adjoint for $⟨A,B⟩=\mathrm{tr}(A{B}^{\ast })$, by cyclicity of trace. The $K$ action is unitary for this product.

Represent $E$-valued forms by horizontal $K$-equivariant $V$-valued forms on $Q=\mathrm{\Gamma }\mathrm{\backslash }G$, expressing their coefficients in the left Maurer–Cartan frame. Indeed a frame $g$ identifies the fiber over $gK$ with $V$ by $\mathrm{Ad}g$, and replacing $g$ by $gk$ changes these coefficients by $\mathrm{Ad}{k}^{-1}$ and rotates the tangent indices. In these coefficients write ${\epsilon }_j$ for exterior multiplication by the dual of $Y_j$ and ${\iota }_j$ for contraction by $Y_j$. The ordinary covariant differential of the $K$ connection is $D={\sum }_j{\epsilon }_j{Y}_j$, because the horizontal part of $[Y_i,Y_j]$ is zero. Differentiating the fiber change $\mathrm{Ad}g$ gives $d_E=D+T$, where $T={\sum }_j{\epsilon }_j{R}_j$. These are actual coefficient formulas on all degrees, not definitions of a different connection.

The invariant volume on $Q$ is the product of hyperbolic volume and normalized compact-fiber volume. Integration in a local bundle chart and then a subordinate smooth partition of unity [F2] shows that integration of a $K$-invariant coefficient norm on $Q$ equals its integration on $M$. The left invariant fields $Y_j$ have zero divergence for this volume: $G$ is unimodular, since $\mathrm{tr}(\mathrm{ad}Z)=0$ for the displayed Lie algebra basis, hence for all $Z$, and $\det \mathrm{Ad}(\exp Z)=1$. Right translation therefore preserves its Haar volume. Integration by parts on compact supports now gives $D^{\ast }=-{\sum }_j{\iota }_j{Y}_j$ and $T^{\ast }={\sum }_j{\iota }_j{R}_j$. The relations ${\epsilon }_i{\iota }_j+{\iota }_j{\epsilon }_i={\delta }_{ij}$ show that $D{T}^{\ast }+T^{\ast }D+T{D}^{\ast }+D^{\ast }T=0$: the two anticommutator sums are respectively ${\sum }_j{R}_j{Y}_j$ and its negative, since coefficient differentiation commutes with the constant matrices $R_j$. Thus $d_E{\delta }_E+{\delta }_E{d}_E=D{D}^{\ast }+D^{\ast }D+H$, where $H=T{T}^{\ast }+T^{\ast }T$. All statements are $K$-equivariant and descend from $Q$ to $M$.

On one-form coefficients $a=(a_1,a_2,a_3)\in V^3$, the exterior-contraction relations give $(Ha{)}_i={\sum }_j{R}_j^2{a}_i+{\sum }_j[R_i,R_j]a_j$. This finite-dimensional self-adjoint operator is positive definite. If $Ha=0$, positivity of $T{T}^{\ast }+T^{\ast }T$ gives $Ta=0$, namely $R_i{a}_j=R_j{a}_i$ for all $i,j$. Since $[R_i,R_j]=i{ϵ}_{ijk}{R}_k$, the commutator sum in the displayed formula is a multiple of $R_k{a}_j-R_j{a}_k$ and vanishes. Consequently $0=⟨Ha,a⟩={\sum }_{i,j}\mathrm{\parallel }{R}_j{a}_i{\mathrm{\parallel }}^2$. Each $a_i$ commutes with all $Y_j$, which span ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ over $\mathbb{C}$. A matrix commuting with all traceless matrices is scalar, and a traceless scalar is zero. Thus $a=0$. Compactness of the unit sphere in $V^3$ supplies $c>0$ with $H\ge cI$.

For a compactly supported smooth one-form, integration by parts in step 3.1 and step 4.1 give $\mathrm{\parallel }{d}_E\alpha {\mathrm{\parallel }}_2^2+\mathrm{\parallel }{\delta }_E\alpha {\mathrm{\parallel }}_2^2=\mathrm{\parallel }D\alpha {\mathrm{\parallel }}_2^2+\mathrm{\parallel }{D}^{\ast }\alpha {\mathrm{\parallel }}_2^2+(H\alpha ,\alpha )\ge c\mathrm{\parallel }\alpha {\mathrm{\parallel }}_2^2$. We next extend this inequality to the joint weak graph domain. On each cusp choose a smooth cutoff equal to one for $t\le R$, zero for $t\ge 2R$, with derivative bounded by $C\mathrm{/}R$, and set it equal to one on the compact core. These cutoffs ${\chi }_R$ have compact support and $\mathrm{\mid }d{\chi }_R\mathrm{\mid }\le C\mathrm{/}R$. For a weak graph-domain form $v$, the product formulas $d_E({\chi }_{R}v)={\chi }_R{d}_{E}v+d{\chi }_R\wedge v$ and ${\delta }_E({\chi }_{R}v)={\chi }_R{\delta }_{E}v-{\iota }_{\mathrm{\nabla }{\chi }_R}v$ show convergence to $v$ in the joint graph norm. The same conclusion holds separately for each differential and codifferential and for forms of every degree. In the compact case take the cutoff identically one.

Here are the local smoothing details needed to finish that graph-density argument. In a finite collection of trivializing charts around the support of ${\chi }_{R}v$, every differential or codifferential is a first-order matrix operator $P={\sum }_j{A}_j(x){\mathrm{\partial }}_j+B(x)$ with smooth coefficients. Multiplication by a subordinate smooth partition [F2] reduces to forms supported inside individual charts. For convolution with a smooth unit-mass kernel ${\rho }_{ϵ}$, the difference $P(v\ast {\rho }_{ϵ})-(Pv)\ast {\rho }_{ϵ}$ is a sum of integral operators with kernels $(A_j(x)-A_j(y)){\mathrm{\partial }}_j{\rho }_{ϵ}(x-y)+({\mathrm{\partial }}_j{A}_j)(y){\rho }_{ϵ}(x-y)$ and a zero-order coefficient commutator. The $L^1$ kernel bounds are uniform on the compact chart, by the mean-value bound for $A_j$ and scaling of ${\rho }_{ϵ}$; thus these commutators are uniformly bounded on $L^2$. They tend to zero on smooth functions by differentiation of convolutions, hence on all $L^2$ functions by the smooth-density fact in [F2] and the uniform bound. Therefore local mollification converges in each asserted graph norm, also for the direct sum operator $(d_E,{\delta }_E)$. The partition contributes only fixed zero-order commutators, so a finite sum completes the approximation by compactly supported smooth forms. Together with step 5.1 this proves the coercive inequality on the maximal joint graph domain. It also proves that maximal differentials are the adjoints of maximal formal codifferentials, since their compact-support integration-by-parts identities extend by this graph density.

Let ${\mathcal{H}}_p=L^2({\mathrm{\Lambda }}^p{T}^{\ast }M\otimes E)$ and let $d_p$ be the maximal weak differential. It is closed: convergence of a form and its differential in $L^2$ preserves the weak test-function identity. Flatness gives $d_1{d}_0=0$ distributionally. Put $Z=\ker d_1$, a closed subspace of ${\mathcal{H}}_1$, and $\delta =d_0^{\ast }$. Every $w\in Z^{\perp }$ belongs to $\mathrm{Dom}\delta$ with $\delta w=0$, because it is orthogonal to $\mathrm{ran}{d}_0\subseteq Z$. Hence the orthogonal projection $P_Z$ maps $\mathrm{Dom}\delta$ into itself and $\delta P_{Z}v=\delta v$. On $Z\cap \mathrm{Dom}\delta$, steps 5.1 and 6.1 give $\mathrm{\parallel }v{\mathrm{\parallel }}_2\le c^{-1\mathrm{/}2}\mathrm{\parallel }\delta v{\mathrm{\parallel }}_2$. In particular $A=\delta {\mathrm{\mid }}_{Z\cap \mathrm{Dom}\delta }$ is injective with closed range: a Cauchy sequence of its images is, by this estimate, the image under the closed operator $\delta$ of a convergent sequence in $Z$.

Fix a weakly closed $\alpha \in {\mathcal{H}}_1$, so $\alpha \in Z$. On the closed range of $A$ define $\mathrm{\ell }(Av)=⟨v,\alpha ⟩$. It is well defined and bounded, with norm at most $c^{-1\mathrm{/}2}\mathrm{\parallel }\alpha {\mathrm{\parallel }}_2$, by step 7.1. Extend it to ${\mathcal{H}}_0$ by zero on the orthogonal complement of that range and use [F1] to obtain $b\in {\mathcal{H}}_0$ with $⟨Av,b⟩=⟨v,\alpha ⟩$ and $\mathrm{\parallel }b{\mathrm{\parallel }}_2\le c^{-1\mathrm{/}2}\mathrm{\parallel }\alpha {\mathrm{\parallel }}_2$. For arbitrary $v\in \mathrm{Dom}\delta$, replace $v$ by $P_{Z}v$; its $\delta$ image is unchanged and its pairing with $\alpha$ is unchanged. Thus $⟨\delta v,b⟩=⟨v,\alpha ⟩$ on the full domain of $\delta$. By the adjoint definition and the graph-density conclusion of step 6.1, $b\in \mathrm{Dom}{d}_0$ and $d_{0}b=\alpha$. This proves the primitive assertion in the weak sense without an external closed-range or harmonic-representative theorem.

If $\alpha$ is smooth, the primitive $b$ is smooth. In a flat parallel frame on a coordinate ball, $d_{E}b=\alpha$ is the ordinary componentwise distributional equation $db=\alpha$. A smooth closed one-form on a smaller star-shaped coordinate ball has a smooth primitive $b_0$, given by integrating along straight radial segments; differentiating the integral and using closedness verifies $d{b}_0=\alpha$. Then $d(b-b_0)=0$ distributionally. Mollification on still smaller balls makes this difference a sequence of constants, so its distributional limit is constant as well. Thus $b=b_0+\text{constant}$ is smooth locally and hence globally. Its $L^2$ norm and the bound from step 8.1 remain unchanged.