IMAR 75
Invariants and obstructions in low regularity Sobolev spaces to manifolds
dedicated to the memory of Haim Brézis
Petru Mironescu
Université Claude Bernard Lyon 1, France
Université Claude Bernard Lyon 1, France
Abstract:
We present a number of objects that 'hear' the homotopy properties or the singularities
of Sobolev maps to manifolds. A simple starting point is the following: when $M$ is a
metric measure space and $N$ is a smooth compact manifold, we may naturally associate
with a vanishing mean oscillation ($\text{VMO}$) map $f:M\to N$ a homotopy class, which
is a 'robust' object. Another example is $\int_M f^\ast\omega$, if $f\in \text{VMO}(M ; N)$, $M$ is $k$-dimensional,
Lipschitz, and compact, and $\omega$ is a closed $k$-form on $N$.
This is again a robust (homotopy invariant) object.
In higher dimensions, say $M={\mathbb R}^d$, with $d>k$, one may define the current $ d (f^\ast\omega)$, when $f$ is in one of the 'critical' Sobolev spaces $W^{s,p}({\mathbb R}^d ; N)$ with $sp=k$. This is straightforward when $s\ge 1$, but more delicate when $0$<$s$<$1$. We explain the existence of this current, and its robust character. When $N$ has a simple topology (e.g., when it is $(k-1)$-connected), the above currents detect all the topological the obstructions to the approximation with smooth maps with values into $N$.
These results generalize previous works of Bethuel, Bourgain, Bousquet, Brezis, Coron, Demengel, Hélein, Giaquinta, Mucci, and the author. Joint work with Antoine Detaille and Kai Xiao (Université Claude Bernard Lyon 1).
In higher dimensions, say $M={\mathbb R}^d$, with $d>k$, one may define the current $ d (f^\ast\omega)$, when $f$ is in one of the 'critical' Sobolev spaces $W^{s,p}({\mathbb R}^d ; N)$ with $sp=k$. This is straightforward when $s\ge 1$, but more delicate when $0$<$s$<$1$. We explain the existence of this current, and its robust character. When $N$ has a simple topology (e.g., when it is $(k-1)$-connected), the above currents detect all the topological the obstructions to the approximation with smooth maps with values into $N$.
These results generalize previous works of Bethuel, Bourgain, Bousquet, Brezis, Coron, Demengel, Hélein, Giaquinta, Mucci, and the author. Joint work with Antoine Detaille and Kai Xiao (Université Claude Bernard Lyon 1).