NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

98 GUILLARMOU and TZOU
Using the fact that u is real-valued, that ϕ is harmonic, and that N
j=0 |dϕj| 2 is
uniformly bounded away from zero, we see that
2
h 〈∂xu, u∂xϕɛ〉+ 2
h 〈∂yu, u∂yϕɛ〉 = 1
h 〈u, uϕɛ〉 ≥ C
ɛ u2
for some C>0, and therefore, by possibly modifying C>0, wehave
e −ϕɛ/h ϕɛ/h 2 C
e u ≥ du
ɛ
2 + 1
h2 u|dϕɛ| 2 + C
ɛ u2
+ boundary terms. (8)
Now if the diameter of the support of u is chosen small (with size depending only on
|Hessϕ0|(p)) with a unique critical point p of ϕ0 inside, one can use integration by
parts and the fact that the critical point is nondegenerate to obtain
¯∂u 2 + 1
h2 u|∂ϕ0| 2 ≥ 1
∂¯z(u
h
2
)∂zϕ0dxdy
≥ 1
u
h
2 ∂2 z ϕ0
dxdy
≥ C′′
h u2
(9)
for some C ′′ > 0. Clearly, the same estimate holds trivially if does not contain a
critical point of ϕ0. Using a partition of unity (θj)j in and absorbing terms of the
form ||u¯∂θj|| 2 into the right-hand side, one obtains (9) for any function u supported
in and vanishing at the boundary. Thus, combining with (8), there are positive
constants C, C0,C1 such that, for h small enough, we have
C
du
ɛ
2 + 1
≥ C1
ɛ
h 2 u|dϕɛ| 2 + C
ɛ u2
du 2 + 1
h 2 u|dϕ0| 2 + 1
h u2
Combining now with (8)gives
≥ C
du
ɛ
2 + 1
.
h2 u|dϕ0| 2 − C0
ɛ
u2
2
e −ϕɛ/h
ϕɛ/h 2
C1
e u ≥ du
ɛ
2 + 1
h2 u|dϕ|2 + 1
h u2
+ boundary terms.
Let us now discuss the boundary terms in (7). If ϕj aretakensothat∂νϕj = 0 on Ɣ0,
then ∂νϕɛ = 0 on Ɣ0 and ∂νϕɛ = ∂νϕ + O(h/ɛ) on Ɣ, and thus
1
|∂νu|
h
2 |∂νϕɛ| ≤ C2
|∂νu|
h
2
∂M0
for some constant C2 > 0. We finally claim that B∂τ A − A∂τ B = 1 on ∂M0 ∩ .
Indeed, since the chart near a connected component can be taken to be an interior
neighborhood of the circle |z| =1 in C, one has A + iB = e −it , where t ∈ S 1
parameterize the boundary component, so that B∂τ A − A∂τ B = 1 since ∂τ = ∂t
Ɣ