Scalar Calderón maps and conditional uniqueness — Examples and Counterexamples

The boundary of $M$ is the level set $x_1^2+4\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^2=1$. Its defining function has gradient $(2x_1,8x_2,8x_3)$, which cannot vanish on that level set, so $M$ is a compact smooth three-dimensional domain. Every point of $M$ has $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }\le 1\mathrm{/}2<1$, hence $M⋐\mathbb{R}\times D^{\circ }$; the ambient metric is exactly $d{x}_1^2\oplus g_D$. By [F1], this is a transversally anisotropic product domain.

The disk has strictly convex boundary: its outward unit normal is $x^{\mathrm{\prime }}$ on $\mathrm{\mid }{x}^{\mathrm{\prime }}\mathrm{\mid }=1$ and $g({\mathrm{\nabla }}_w{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}},w)=\mathrm{\mid }w{\mathrm{\mid }}^2>0$ for every nonzero tangent $w$. Its curvature is zero, so a normal Jacobi field has scalar coefficient $a+bt$; if it vanishes at two distinct times then $a=b=0$, proving no conjugate points. Every unit-speed straight line exits the bounded disk in finite forward and backward time, and the segment between two points in the disk is unique. Thus $D$ is simple by [F2].

Apply [F3] to this simple surface. Its unweighted scalar geodesic ray transform is injective on the stated compactly supported smooth real functions. This verifies the injective-transversal condition for $M$.

For every integer $m\ge 0$, differentiation gives $\frac{d^m}{d{t}^m}{e}^{-1\mathrm{/}t}=P_m(1\mathrm{/}t)e^{-1\mathrm{/}t}$ on $t>0$ for a polynomial $P_m$, by induction on $m$. For every polynomial $P$, $P(s)e^{-s}\to 0$ as $s\to \mathrm{\infty }$; for each power $s^N$, this follows from $e^s\ge s^{N+1}\mathrm{/}(N+1)!$. Thus every right derivative tends to zero at $t=0$, so extension by zero to $t\le 0$ is smooth and flat there. Composition with $t=1-4\mathrm{\mid }x{\mathrm{\mid }}^2$ proves $\psi \in C_c^{\mathrm{\infty }}(B(0,1))$, with compact support in $\overline{B(0,1\mathrm{/}2)}$. It is nonzero at $x=0$, and $\mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$.

The bound $\mathrm{\mid }ϵ\mathrm{\mid }\mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}<1$ gives ${\gamma }_2(x)\ge 1-\mathrm{\mid }ϵ\mathrm{\mid }\mathrm{\parallel }\psi {\mathrm{\parallel }}_{\mathrm{\infty }}>0$, so it satisfies the positivity condition in [F1]. Since $\psi$ vanishes on a neighborhood of $\mathrm{\partial }\mathrm{\Omega }$, ${\gamma }_1$ and ${\gamma }_2$ agree on that neighborhood and all their boundary derivatives agree. At the origin, ${\gamma }_2(0)-{\gamma }_1(0)=ϵe^{-1}\mathrm{\ne }0$. Thus identical boundary jets do not force interior equality.

The set $\mathrm{\Gamma }$ is nonempty because it contains the north pole and is open in the sphere because it is the inverse image of $(1\mathrm{/}2,\mathrm{\infty })$ under the continuous coordinate $x\mapsto x_3$. It is proper because the south pole is not in it. A nonzero smooth bump supported in a small lower cap defines boundary data outside $C_c^{\mathrm{\infty }}(\mathrm{\Gamma })$, so the allowed inputs form a proper subset of the full boundary-data space.

By [F1] and [F2], the full-data theorem requires the map on every $H^{1\mathrm{/}2}(\mathrm{\partial }\mathrm{\Omega })$ input and the corresponding full boundary output. The stated protocol restricts both input and output to $\mathrm{\Gamma }$, so it does not satisfy that hypothesis. This classifies the configuration as outside the theorem's scope and asserts no uniqueness or nonuniqueness conclusion.