Abstract
Given a Hermitian line bundle $$L\rightarrow M$$
L
→
M
over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as $$\epsilon \rightarrow 0$$
ϵ
→
0
, of couples $$(u_\epsilon ,\nabla _\epsilon )$$
(
u
ϵ
,
∇
ϵ
)
critical for the rescalings $$\begin{aligned} E_\epsilon (u,\nabla )=\int _M\Big (|\nabla u|^2+\epsilon ^2|F_\nabla |^2+\frac{1}{4\epsilon ^2}(1-|u|^2)^2\Big ) \end{aligned}$$
E
ϵ
(
u
,
∇
)
=
∫
M
(
|
∇
u
|
2
+
ϵ
2
|
F
∇
|
2
+
1
4
ϵ
2
(
1
-
|
u
|
2
)
2
)
of the self-dual Yang–Mills–Higgs energy, where u is a section of L and $$\nabla $$
∇
is a Hermitian connection on L with curvature $$F_{\nabla }$$
F
∇
. Under the natural assumption $$\limsup _{\epsilon \rightarrow 0}E_\epsilon (u_\epsilon ,\nabla _\epsilon )<\infty $$
lim sup
ϵ
→
0
E
ϵ
(
u
ϵ
,
∇
ϵ
)
<
∞
, we show that the energy measures converge subsequentially to (the weight measure $$\mu $$
μ
of) a stationary integral $$(n-2)$$
(
n
-
2
)
-varifold. Also, we show that the $$(n-2)$$
(
n
-
2
)
-currents dual to the curvature forms converge subsequentially to $$2\pi \Gamma $$
2
π
Γ
, for an integral $$(n-2)$$
(
n
-
2
)
-cycle $$\Gamma $$
Γ
with $$|\Gamma |\le \mu $$
|
Γ
|
≤
μ
. Finally, we provide a variational construction of nontrivial critical points $$(u_\epsilon ,\nabla _\epsilon )$$
(
u
ϵ
,
∇
ϵ
)
on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren’s existence result for (nontrivial) stationary integral $$(n-2)$$
(
n
-
2
)
-varifolds in an arbitrary closed Riemannian manifold.
Boundary separated and clustered layer positive solutions for an elliptic Neumann problem with large exponent
