Feeds:
Posts
Suppose $M$ is a submanifold of $S^4(1/h)$ and we diagonalize the second fundamental form $h_{ij}$ to obtain three real-valued functions $\lambda_1(x), \lambda_2(x), \lambda_3(x)$.  Now use the eigenvectors on a coframe adapted to $M$, i.e. $\omega_1(x), \omega_2(x), \omega_3(x), \omega_4(x) = N(x)$ the last being the dual of the unit normal.  If the $\lambda_i$ never vanish, then by the Poincare-Hopf theorem the Euler characteristic is zero for $M$; this is automatic since $M$ is three dimensional.  Any orientable closed three-manifold is parallelizable, so this is not a giant restriction.  Now consider cohomology $H^1(M,\mathbf{R})$.  If this is zero, we have a homology sphere which is diffeomorphic to $S^3$ by Poincare Conjecture.  In this case, we can look at the Hodge decomposition of 2-forms $\Omega^1(M) = d\Omega^1(M) \oplus \delta \Omega^3(M)$.  There is a unique 3-form $G$ such that $d*d A = d\delta G$.  Now consider the heat flow $(\partial_t - d\delta) u = 0$ starting at $u(0,x) = G$.  This will lead to a harmonic solution as $t\rightarrow \infty$.