NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 95
(as in the proof of Lemma 2.3.5) that no point on Ɣ0 ⊂ Ɣ ′ 0 can be an accumulation
point for critical points. Now ∂M0\Ɣ0 is contained in the interior of O, and therefore
no points on ∂M0\Ɣ0 can be an accumulation point of critical points. Since is Morse
in the interior of O, there are no degenerate critical points on ∂M0\Ɣ0. This ends the
proof.
3. Carleman estimate for harmonic weights with critical points
In this section, we prove a Carleman estimate using harmonic weight with nondegenerate
critical points in a way similar to [19]. Let = ϕ + iψ be a holomorphic Morse
function with no critical points on Ɣ0, and which is purely real on Ɣ0. In particular,
this implies that ∂νϕ|Ɣ0 = 0.
PROPOSITION 3.1
Let (M0,g) be a smooth Riemann surface with boundary, and let ϕ : M0 → R be a
C k (M0) harmonic Morse function for k large. Then for all V ∈ L ∞ (M0) there exist
C>0 and h0 > 0 such that, for all h ∈ (0,h0) and u ∈ C ∞ (M0) with u|∂M0 = 0,we
have
1
h u2
L 2 (M0)
1
+ u|dϕ|2
h2 L2 (M0) +du2 L2 2
(M0) +∂νuL2 (Ɣ0)
≤ C e −ϕ/h (g + V )e ϕ/h u 2
L2 1 2
(M0) + ∂νuL h 2
(Ɣ) ,
where ∂ν is the exterior unit normal vector field to ∂M0.
We start by modifying the weight as follows. If ϕ0 := ϕ : M0 → R is a real-valued
harmonic Morse function with critical points {p1,...,pN} in the interior of M0,then
we let ϕj : M0 → R be harmonic functions such that pj is not a critical point of
ϕj for j = 1,...,N, their existence is ensured by Lemma 2.2.4. Forallɛ>0, we
define the convexified weight ϕɛ := ϕ − (h/2ɛ)( N j=0 |ϕj| 2 ). To prove the estimate,
we will localize in charts j covering the surfaces. These charts will be taken so that
if j ∩ ∂M0 = ∅,thenj∩∂M0 S1 is a connected component of ∂M0. Moreover,
by Riemann’s mapping theorem (e.g., [25, Lemma 3.2]), this chart can be taken to be
a neighborhood of |z| =1 in {z ∈ C; |z| ≤1} and such that the metric g is conformal
to the Euclidean metric |dz| 2 .
LEMMA 3.2
Let be a chart of M0 as above, and let ϕɛ : → R be as above. Then there are
constants C, C ′ > 0 such that, for all u ∈ C ∞ (M0) supported in and h>0 small
(3)