AbstractWe exhibit a family of convex functionals with infinitely many equal-energy $$C^1$$
C
1
stationary points that (i) occur in pairs $$v_{\pm }$$
v
±
satisfying $$\det \nabla v_{\pm }=1$$
det
∇
v
±
=
1
on the unit ball B in $${\mathbb {R}}^2$$
R
2
and (ii) obey the boundary condition $$v_{\pm }=\text {id}$$
v
±
=
id
on $$ \partial B$$
∂
B
. When the parameter $$\epsilon $$
ϵ
upon which the family of functionals depends exceeds $$\sqrt{2}$$
2
, the stationary points appear to ‘buckle’ near the centre of B and their energies increase monotonically with the amount of buckling to which B is subjected. We also find Lagrange multipliers associated with the maps $$v_{\pm }(x)$$
v
±
(
x
)
and prove that they are proportional to $$(\epsilon -1/\epsilon )\ln |x|$$
(
ϵ
-
1
/
ϵ
)
ln
|
x
|
as $$x \rightarrow 0$$
x
→
0
in B. The lowest-energy pairs $$v_{\pm }$$
v
±
are energy minimizers within the class of twist maps (see Taheri in Topol Methods Nonlinear Anal 33(1):179–204, 2009 or Sivaloganathan and Spector in Arch Ration Mech Anal 196:363–394, 2010), which, for each $$0\le r\le 1$$
0
≤
r
≤
1
, take the circle $$\{x\in B: \ |x|=r\}$$
{
x
∈
B
:
|
x
|
=
r
}
to itself; a fortiori, all $$v_{\pm }$$
v
±
are stationary in the class of $$W^{1,2}(B;{\mathbb {R}}^2)$$
W
1
,
2
(
B
;
R
2
)
maps w obeying $$w=\text {id}$$
w
=
id
on $$\partial B$$
∂
B
and $$\det \nabla w=1$$
det
∇
w
=
1
in B.
On the Topology of Warped Product Pointwise Semi-Slant Submanifolds with Positive Curvature
