The lecture will give a survey on existence and properties of foliations or -- more generally -- laminations (i.e. foliations of closed subsets) of a Riemannian manifold by volume-minimizing subvarieties. This topic has evolved out of research in different fields, in particular in differential geometry (M. Gromov [G]), in dynamical systems (J. Mather [Ma]), in partial differential equations (J. Moser [Mo 1]) and in solid state physics (S. Aubry [AuDa]). A unified picture emerged due to Gromov's ideas and there are many promising and fundamental open questions some of which will be addressed in the lecture.
To be concrete let us consider an important case: the $n$-torus $T^n$. For a flat metric we have the canonical totally geodesic foliations of $T^n$ which are the projections of the foliations of ${\bf R}^n$ by parallel $k$-dimensional affine subspaces. If we deform the flat metric to an arbitrary metric on $T^n$ we can ask if similar foliations or laminations by volume-minimizing (in the universal cover) $k$-dimesnional subvarieties exist and if they are unique. Results of this type will be presented for the case $k=1$ where we look at length-minimizing (in the universal cover) geodesics, in the hypersurface case $k=n-1$ and -- in the context of pseudholomorphic curves -- in the case $k=2$, $n=2m$. In all these cases a version of KAM theory applies for small perturbations of the flat metric, [Mo 2] and [Mo 3]. But also for arbitrary metrics the known results and examples provide a relatively clear picture in the cases $k=1$ and $k=n-1$, cf. [Ba 1--3].
The case of geodesics $(k=1)$ is closely related to Aubry-Mather's theory of minimal orbits of monotone twist maps and its higher-dimensional generalizations, cf. the lecture by S. V. Bolotin at this congress which treats questions related to this.
An important notion in this context is the "stale norm" defined in geometric measure theory [F] on the real homology vector spaces of a compact Riemannian manifold. Its properties -- e.g. strict convexity or smoothness of its unit ball -- are closely related to the existence and the properties of our laminations, cf. [Ba 2] and [S].
Geometrically interesting rigidity questions arise when we assume that we have a Riemannian metric $g$ on $T^n$ such that for all (or for sufficiently many) canonical foliations of $T^n$ there exist conjugate minimal foliations of $(T^n,g)$. For $k=1$ this leads to the E. Hopf conjecture that $n$-tori without conjugate points are flat. This was recently proved by D. Burago and S. Ivanov [BuIv].
In hyperbolic $n$-space $H^n$ a similar question is the "asymptotic Plateau problem" which asks for a volume minimizing subvariety with prescribed boundary in the ideal boundary of $H^n$, cf. [An]. More generally one can solve this problem for every metric which is Lipschitz equiavlent to a complete metric whose sectional curvature is negative, bounded and bounded away from zero [BaLa].
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