Harmonic sections of a Riemannian foliation. (Sections harmoniques d’un feuilletage Riemannien.) (French. English summary) Zbl 1291.58014
Summary: Let \((M,\mathcal F)\) be a \(q\)-codimensional foliation with compact leaves on a compact Riemannian manifold \((M,h)\) of dimension \(m=p+q\).
It is well known that the space of all leaves \(\mathcal B=M/\mathcal F\) admits a structure of \(q\)-dimensional Satake manifold.
If the leaves have nonpositive sectional curvature and if \(u:\mathcal B\to M\) is a \(C^1\)-section then the vertical heat equation \(\begin{cases}\tau^v & (u_t)=\frac{\partial u_t}{\partial t}\\ & u_0=u\end{cases}\) admits a global solution defined on \(\mathbb R_+\times\mathcal B\). Furthermore, this solution convergences uniformly as \(t\) goes to infinity to a harmonic section which is homotopic to \(u\).
It is well known that the space of all leaves \(\mathcal B=M/\mathcal F\) admits a structure of \(q\)-dimensional Satake manifold.
If the leaves have nonpositive sectional curvature and if \(u:\mathcal B\to M\) is a \(C^1\)-section then the vertical heat equation \(\begin{cases}\tau^v & (u_t)=\frac{\partial u_t}{\partial t}\\ & u_0=u\end{cases}\) admits a global solution defined on \(\mathbb R_+\times\mathcal B\). Furthermore, this solution convergences uniformly as \(t\) goes to infinity to a harmonic section which is homotopic to \(u\).
MSC:
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53C12 | Foliations (differential geometric aspects) |
| 49J53 | Set-valued and variational analysis |
| 58E20 | Harmonic maps, etc. |
