| The Yang-Mills energy of a connection A on a principal fiber bundle is defined as ${cal YM}(A) = {1over2}intvert F(A)vertsp2dV,$ where F(A) is the curvature of A. The $Lsp2$-gradient flow for the Yang-Mills energy functional is given by the Yang-Mills heat equation$$dA/dt + Dsbsp{A}{*}F(A) = 0,eqno(i)$$where A is a time-dependent connection. We study the existence, uniqueness, regularity, stability and asymptotic behavior of the initial value problem for this equation on any closed 2 or 3-dimensional Riemannian manifold M.;Our approach to existence, uniqueness, regularity and stability is similar in spirit to D. DeTurck's proof of short time existence for Hamilton's Ricci flow (J. Diff. Geom., 18 (1983), 157). If A satisfies (i) then A and $Omega = F(A)$ satisfy the system$$cases{dA/dt + D$sbsp{A}{*}Omega$ &= 0crcr$dOmega/dt$ + $Deltasb{A}Omega$ &= 0.cr}eqno(ii)$$We prove that this system can be solved with arbitrary initial data $Asb0in Hsp1$ and $Omegasb0in Lsp2,$ even if $Omegasb0ne F(Asb0).$ The leading terms of (ii) are$$left(matrix{d/dt&Dsp*crcr0&d/dt + Deltacr}right).$$This defines an invertible operator between suitable Sobolev spaces and the initial value problem (ii) can be solved using the implicit function theorem. If $(A,Omega)$ is a solution to (ii) with initial data $(Asb0,Omegasb0)$ such that $Omegasb0$ = $F(Asb0)$, then $Omega(t)$ = F(A(t)) for all t and A is a solution to (i). It follows that the initial value problem for (i) can be solved with initial data any connection $Asb0in Hsp1$. Our main regularity result is that $Ain Csbsp{rm loc}{0}(Hsp1)$ and $F(A)in Csbsp{rm loc}{0}(Lsp2)cap Lsbsp{rm loc}{2}(Hsp1)$. The solution A depends smoothly on the initial data $Asb0$ in these topologies. In particular, the Yang-Mills heat equation defines a continuous flow on the space of connections of Sobolev class $Hsp1$.;Using L. Simon's real analytical technique (Ann. Math., 118 (1983), 525) we prove that the solutions converge as $ttoinfty$.;These results fill in the missing analytical details in M. F. Atiyah's and R. Bott's study of the Morse theory for the Yang-Mills functional over a Riemann surface (Phil. Trans. Roy. Soc. London A, 308 (1982), 524). |
