09-08 Notes

Let \(X\) be smooth manifold of dimension \(n\), let \(E\) be a vector bundle over \(X\), and let \(A\) be a connection on \(E\). WE define the Sobolev Space \(L^p_{k,A}(X,E)\) to be the completion of \(C^\infty(X,E)\) with respect to the norm

\[||s||^p_{p,k,A}=\sum_{i=0}^k\int_{X}|\nabla_A\overset{i \mathrm{ times}}{\circ\cdots\circ}\nabla_A s|^p\mathrm{ d\mathrm{vol}}\]

If no vector bundle is specified, then the trivial bundle, i.e. the set of smooth functions on \(X\), is implied. Other notations for this space are \(W^p_{k,A}\) and in the case \(p=2\), \(H_{k,A}\). We would like to know when \(L^p_k(X)\subset L^q_l(X)\) holds. Define the conformal weight of \(L^p_k\), to be \(k-n/p\). Then for \(f\in L^p_k\), \(||f(tx)||_{p,k}\simeq t^{k-n/p}||f(x)||\), so for such an inclusion to hold, we must have \(k-n/p\leq l-n/q\). If \(X\) is compact, then this, and the condition that \(k\geq l\) are sufficient for \(L^p_k(X)\subset L^q_l(X)\). Furthermore, Rellich’s theorem tells us that if both inequalities are strict, this inclusion is a compact embedding,

Let \(C^{l,\alpha}(X)\) be the space of functions with \(l\) derivatives, such that the \(l^{th}\) derivative is Holder continuous with exponent \(\alpha\). If \(0<\alpha<1\), then \(L^p_k(X)\subset C^{l,\alpha}\iff k\geq l\), and \(k-n/p\geq l+\alpha\). The case \(\alpha=0\) is much more subtle. For example, if \(\Sigma\) is a Riemann surface, \(L^2_1(\Sigma)\not\subset C^0(\Sigma)\), and if \(X\) is a 4-manifold, \(L^4_1(X)\not\subset C^0(X)\).

Sobolev Multiplication

Multiplication gives a map \(L^p_k\times L^q_l\rightarrow L^r_m\) if \[\mathrm{a)\,\,\, }(k-n/p)+(l-n/q)>m-n/r\,\,\mathrm{ and }\,\,k,l\geq m,\mathrm{ or}\] \[\mathrm{b)\,\,\, }k-n/p>0\,\,\mathrm{ and }\,\,k-n/p,l-n/q\geq m-n/r\]

The proofs of these two results follow from the embedding theorem, and Holder’s inequality. For a), write \(\nabla^m(fg)=\sum_{i=0}^{m}c_i\nabla^if\nabla^{m-i}g\) where the \(c_i\) only depend on \(m\). By Holder’s inequality, we have \(||\nabla^if\nabla^{m-i}g||_r\leq||\nabla^if||_a||\nabla^{m-i}g||_b\) when \(1/a+1/b\leq1/r\). This is equivalent to \(m-n/r\leq(i-n/a)+(m-i-n/b)\). Since \(k\geq m\geq i\), \(l\geq m\geq m-i\), we can pick \(a\) so that \(k-n/p=i-n/a\) if \(k-n/p\leq i\) or \(a\geq 2r\) if \(k-n/p>i\) (And similarly for b), and apply Sobolev Embedding term by term.

For b), note that the hypotheses imply the hypotheses of a).

Differentiation on Banach Spaces

Let \(X,Y\) be Banach spaces, \(U=subset X\) an open subset, and \(f:U\rightarrow Y\) a map. We say that \(f\) is differentiable at \(x\in U\) if there is \(C>0\) and a bounded linear operator \(A:X\rightarrow Y\) such that \(||f(x+h)-f(x)-A\dot h||_Y\leq C||h||^2_X\) (this exponent 2 here is not quite the correct definition, but it will do for our purposes). In this case, we write \(d_x f\) for \(A\), and call this the derivative of \(f\) at \(x\).

We say that \(f\) is continuously differentiable if \(x\mapsto A_x:U\rightarrow \mathrm{Hom}(X,Y)\) is a continuous map. We say that \(f\) is twice differentiable at \(x\) is it is continuously differentiable at \(x\), and the map \(x'\mapsto d_{x'}f\) is differentiable at \(x\). Iterating this definition allows us to define higher derivatives. In particular, \(f\) is \(C^\infty\) if it has derivatives of all orders.

It turns out that if \((f,g)\mapsto \mu(f,g)=f\cdot g:L^p_k\times L^q_l\rightarrow L^r_m\) is continuous, then it is \(C^\infty\). In fact, if we let \(\delta\) denote a small variation, and letting \(d_{\mu(f,g)}(\delta f,\delta g)=f\delta g+g\delta f\), we have

\[||\mu(f+\delta f,g+\delta g)-\mu(f,g)-d_{\mu(f,g,)}(\delta f,\delta g)||_{r.m}\]

\[=||\delta f\cdot\delta g||_{r,m}\leq C(||\delta f||_{p,k}||\delta g||_{l,q})^2\]

where the last inequality follows from the multiplication theorem. The derivative \(d_{\mu(f,g)}(\delta f,\delta g)\) only depends on the product \(f\cdot g\), so the higher derivatives all vanish.

Application to the Space of Connections

Let \(\pi:P\rightarrow X\) be a principal \(G\)-bundle, and let \(\mathfrak{g}\) be the Lie algebra of \(G\). Let \(A\) denote a connection on \(P\), \(F_A\in\Omega^2(X,\mathrm{ad }P)\) its curvature. If, \(\tau\) is a local trivialization, we let \(a^\tau\) denote the one-form associated to the connection, \(A\mapsto F_A\) corresponds to \(a^\tau\mapsto da^\tau+[a^\tau\wedge a^\tau]\). Recall the following definition:

\[\mathcal{A}^p_k=\{\mathrm{connections }A|\mathrm{ if }\tau:\pi^{-1}(U)\rightarrow U\times P\mathrm{ is a }C^\infty\mathrm{ trivialization such that if }\] \[\phi\in C^\infty_0(U)\mathrm{ we have } a_\tau\in\Omega^1(U,\mathfrak{g}), \phi a_\tau\in L^p_k(U,T^pU\otimes\mathfrak{g})\}\]

Let \(\mathcal{F}:\mathcal{A}^p_k\rightarrow L^q_l(X,\Lambda^2\otimes\mathrm{ad }P)\) be the map sending a connection to its curvature. We would like to know when \(\mathcal{F}\) is continuous. Clearly, \(a^\tau\mapsto da^\tau\) is continuous into \(L^p_{k-1}\). From the multiplication theorem, \(a^\tau\mapsto [a^\tau\wedge a^\tau]\) is continuous when \((k-n/p)+(k-n/p)>(k-1)-n/p\), i.e. \(k+1\geq n/p\). In fact, in this case it is \(C^\infty\). For example, for \(n=4\), \((p,k)=(2,1)\), \((q,l)=(4,0)\), \(a^\tau\mapsto [a^\tau\wedge a^\tau]\) is continuous.

We specialize to the the situation where \(G=O(n),U(n),Sp(n)\), amd where \(P\) is the frame bundle of some vector bundle \(E\). A smooth gauge transformation \(u\) is a smooth bundle endomorphism of \(E\) such that \(u\) is an orthogonal (resp. unitary or symplectic) isomorphism on each fiber. In other words, \(u\in C^\infty(X,\mathrm{End}(E))\), and \(u^*u=1\), where \(u^*\) denotes the transpose if \(G=O(n)\), the conjugate transpose if \(G=U(n)\), etc. We define

\[\mathcal{G}^p_k=\{u\in L^p_k(X,\mathrm{End}(E))|u^*u=1\mathrm{ almost everywhere}\}\]

We would like \(\mathcal{G}^p_k\) to be a group, and ideally, a Banach Lie group.

Theorem 1. If \((k-n/p)>0\), then \(\mathcal{G}^p_k\) is a Banach Lie group, and \(L^p_k(X,\mathrm{End}(E))\) is a Banach space and algebra

Proof. Define \(\Phi:L^p_k(X,\mathrm{End}(E))\rightarrow L^p_k(X,\mathrm{End}(E))\) by \(\Phi(s)=s^*s\). Then \(\Phi\) is a \(C^\infty\) map, and \(d_s\Phi(t)=s^*t+t^*s\). Using \(s^*s=1\), once easily sees that \(d_s\Phi(sw/2)=w\), so that \(d_s\Phi\) is surjective for all \(s\in\mathcal{G}^p_k\). It is also clear that \(\mathrm{ker}(d_s\Phi)=\{sr|r=-r^*\}\). These two facts allows us to apply the implicit function theorem to conclude that \(\Phi^{-1}(1)\subset\mathcal{G}^p_k\) is a Banach Lie group. ◻

We would next like to determine when \(\mathcal{G}^p_k\) acts on \(\mathcal{A}^q_l\). The multiplication theorem implies that \(L^p_{k-1}\) is an \(L^p_k\) module if \((k-n/p)\)>0. The argument at the end of section 3 shows that this multiplication is smooth, so that \(\mathcal{G}^p_k\times\mathcal{A}^p_{k-1}\rightarrow \mathcal{A}^p_{k-1}\) is a smooth.

Let \(\mathcal{B}^p_{k-1}=\mathcal{A}^p_{k-1}/\mathcal{G}^p_k\). We want to know when this is a Hausdorff space. Recall the following criterion: If \(G\) is a topological group, that acts continuously on a topological space \(X\), then \(X/G\) is Hausdorff if and only if the graph \(\Gamma=\{(x,xg)|x\in X,g\in G\}\) is closed in \(X\times X\).

Theorem 2. If \((k-n/p)>0\), then \(\mathcal{B}^p_{k-1}\) is Hausdorff

Proof. Suppose we have a sequence \((A_i,u_iA_i)\) that converges to \((A,A')\) in \(L^p_{k-1}\). Then \(a^\tau_i\) converges to \(a^\tau\) in \(L^p_k\), and \(u^{-1}_idu_i+u^{-1}_ia^\tau_iu_i\rightarrow a'^\tau\). Rearranging this gives \(du_i=u_ia'^\tau-a^\tau_iu_i\). Suppose \(k=1\), \(p=n+\epsilon\). Then since \(a^\tau_i\) converges, and \(u_i\) are all in \(\mathcal{G}^p_k\), this implies that \(du_i\) is uniformly bounded. If we pass to a subsequence where \(u_i\) converges, say to \(u\), in \(C^0\), then \(du_i\) converges in \(L^{n+\epsilon}\), so that \(u_i\) converges to \(u\) in \(L^{n+\epsilon}_1\). For \(k>1\), one applies a bootstrap argument. ◻