arXiv subjects

Kelvin Lam

Publications and source records attributed to Kelvin Lam.

4 recordsLinked to original sources

arXiv2026

Conformal boundary rigidity for simple Finsler metrics

In this paper we prove that simple Finsler manifolds are conformally stable. Given a simple Finsler manifold and a class of conformal factors, we characterize the singularity of their induced boundary distance functions which allows us to define an appropiate $H^2$ norm. We then obtain a stability estimate with respect to this Sobolev norm and the $L^2$ norm on the class of the conformal factors. To prove the main theorems, we adapt the classical integration by parts technique employed by Mukhometov \cite{Muhometov} to the Finsler setting.

math.DG↗

arXiv2025

Transcendence and algebraic independence of a family of $p$-adic valuation generating functions

We show that $T_p(z)=\prod_{j=1}^{\infty}(1-z^{p^{j}})^{-1/p^{j}}$ is transcendental over $\overline{\mathbb{Q}}(z)$, and establish the transcendence of its values at nonzero algebraic points inside the unit disk. Furthermore, we obtain an algebraic independence result for multiplicatively independent algebraic arguments. In summary, this paper extends Mahler's method beyond the classical automatic setting by studying the function $T_p(z)$, whose coefficients are governed by the unbounded arithmetic function $ν_p(n)$.

math.NT↗

arXiv2023

Determination of DN map from the scattering relation for simple surfaces at low regularity

In this paper we prove that on a simple surface where the metric is $C^{17}$, the scattering relation determines the Dirichlet to Neumann map (DN map) - a known result for the case when the metric is smooth. For metrics with finite differentiability we had to modified each technical result used in the original proof; such as properties of the exit time function and the characterization of $C_α$ space. Moreover, surjectivity of $I^*$ in the original proof required the use of microlocal analysis on the normal operator $I^*I$ - which is not a standard pseudodifferential operator for metrics with finite regularity. Finally, using the injectivity of $I$ on Lipschitz one forms for simple $C^{1,1}$ manifolds we prove an equivalent characterization of harmonic conjugacy using operators defined by the scattering relation to prove the titular result. We also prove that the boundary distance function determines the metric at the boundary (which in turns determines the scattering relation) for a closed disk even when the metric is only $C^{1,1}$ and the exponential map is only Lipschitz and does not preserve tangent vectors or differentials pointwise.

math.DG↗

arXiv2023

Microlocal analysis of the X-ray transform in non-smooth geometry

We prove that the geodesic X-ray transform is injective on $L^2$ when the Riemannian metric is simple but the metric tensor is only finitely differentiable. The number of derivatives needed depends explicitly on dimension, and in dimension $2$ we assume $g\in C^{10}$. Our proof is based on microlocal analysis of the normal operator: we establish ellipticity and a smoothing property in a suitable sense and then use a recent injectivity result on Lipschitz functions. When the metric tensor is $C^k$, the Schwartz kernel is not smooth but $C^{k-2}$ off the diagonal, which makes standard smooth microlocal analysis inapplicable.

math.AP↗