Connecting rational homotopy type singularities |
| |
Authors: | Robert Hardt Tristan Rivière |
| |
Institution: | (1) Department of Mathematics, Rice University, P.O. Box 1892, Houston, TX 77251, USA;(2) Department of Mathematics, Swiss Federal Institute of Technology, CH-8092 Zürich, Switzerland |
| |
Abstract: | Let N be a compact simply connected smooth Riemannian manifold and, for p ∈ {2,3,...}, W
1,p
(R
p+1, N) be the Sobolev space of measurable maps from R
p+1 into N whose gradients are in L
p
. The restriction of u to almost every p-dimensional sphere S in R
p+1 is in W
1,p
(S, N) and defines an homotopy class in π
p
(N) (White 1988). Evaluating a fixed element z of Hom(π
p
(N), R) on this homotopy class thus gives a real number Φ
z,u
(S). The main result of the paper is that any W
1,p
-weakly convergent limit u of a sequence of smooth maps in C
∞(R
p+1, N), Φ
z,u
has a rectifiable Poincaré dual
. Here Γ is a a countable union of C
1 curves in R
p+1 with Hausdorff -measurable orientation and density function θ: Γ→R. The intersection number between and S evaluates Φ
z,u
(S), for almost every p-sphere S. Moreover, we exhibit a non-negative integer n
z
, depending only on homotopy operation z, such that even though the mass may be infinite. We also provide cases of N, p and z for which this rational power p/(p + n
z
) is optimal. The construction of this Poincaré dual is based on 1-dimensional “bubbling” described by the notion of “scans”
which was introduced in Hardt and Rivière (2003). We also describe how to generalize these results to R
m
for any m ⩾ p + 1, in which case the bubbling is described by an (m–p)-rectifiable set with orientation and density function determined by restrictions of the mappings to almost every oriented
Euclidean p-sphere. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|