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Gabriel Paternain

Publications and source records attributed to Gabriel Paternain.

7 recordsLinked to original sources
arXiv2026

Conformal boundary rigidity from null geodesic travel times

The gravitational field of a distant, isolated system is manifested by the conformally invariant Weyl tensor. Thus the conformal structure far from the system encodes the system's gravitational mass. It also encodes the causal structure, thereby linking it to the mass. For asymptotically anti-de Sitter (AdS) spacetimes, this link led to a novel positive mass theorem of Page, Surya, and the second author \cite{PSW} which did not rely on any traditional energy condition. Here we ask whether that theorem has a rigidity case. Specifically, we consider all null geodesics in an asymptotically AdS spacetime that depart from the Penrose conformal infinity, travel through spacetime, and return to conformal infinity. If all such geodesics from a given point refocus at an antipodal point at infinity, is the spacetime conformal to anti-de Sitter space? It is easy to answer the question if the asymptotically AdS spacetime either (i) obeys the null energy condition in 3 or 4 spacetime dimensions, or (ii) is static (in any dimension), and we give simple proofs in those cases. We then answer the question in the case of globally stationary, asymptotically AdS spacetimes, by applying the theory of magnetic geodesics on the Riemannian manifold-with-boundary obtained by quotienting by the stationary Killing vector field. The question has an analogue for asymptotically flat spacetimes, which we also discuss.

math.DG↗
arXiv2021

On some information-theoretic aspects of non-linear statistical inverse problems

Results by van der Vaart (1991) from semi-parametric statistics about the existence of a non-zero Fisher information are reviewed in an infinite-dimensional non-linear Gaussian regression setting. Information-theoretically optimal inference on aspects of the unknown parameter is possible if and only if the adjoint of the linearisation of the regression map satisfies a certain range condition. It is shown that this range condition may fail in a commonly studied elliptic inverse problem with a divergence form equation, and that a large class of smooth linear functionals of the conductivity parameter cannot be estimated efficiently in this case. In particular, Gaussian `Bernstein von Mises'-type approximations for Bayesian posterior distributions do not hold in this setting.

math.ST↗
arXiv2000

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an F-structure (possibly of rank 0) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial circle action. As a corollary we obtain that the simplicial volume of a colsed manifold admitting an F-structure is zero. We also show that if M admits an F-structure then it collapses with curvature bounded from below. This is turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative. We show that F-structures of rank zero appear rather frequently:every compact complex elliptic surface admits one as well as any simply connected 5-manifold. We use these results to study the minimal entropy problem. We show the following two theorems: suppose M is obtained by taking connected sums of copies of CP^2 (with any orintation), S^2 \times S^2 and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a metric with zero topological entropy if and only if M is diffeomorphic to S^4, CP^2, S^2 \times S^2, CP^2#CP^2 or CP^2#(-CP^2). Finally, suppose that M is a closed simply connected 5-manifold. Than M has zero minimal entropy. Moreover, M admits a metric with zero topological entropy if and only if M is diffeomorphic to S^5, S^3 \times S^2, the non-trivial S^3-bundle over S^2 or the Wu manifold SU(3)/SO(3).

math.DG↗
arXiv2000

Einstein manifolds of non-negative sectional curvature and entropy

We find obstructions to the existence of Einstein metrics of non-negative sectional curvature on a smooth closed simply connected manifold of any dimension. The results are achieved by combining the classical Morse theory of the loop space with a new upper bound for the topological entropy of the geodesic flow in terms of the curvature tensor.

math.DG↗
arXiv1996

Counting geodesics on a Riemannian manifold and topological entropy of geodesic flows

Let $M$ be a compact $C^{\infty}$ Riemannian manifold. Given $p$ and $q$ in $M$ and $T>0$, define $n_{T}(p,q)$ as the number of geodesic segments joining $p$ and $q$ with length $\leq T$. Mañé showed that the exponential growth rate of the integral of $n_{T}(p,q)$ over $M \times M$ is the topological entropy of the geodesic flow of $M$. In the present paper we exhibit an open set of metrics on the two-sphere for which the exponential growth rate of $n_{T}(p,q$ is less than the topological entropy of the geodesic flow for a positive measure set of $(p,q)\in M\times M$. This answers in the negative questions raised by Mañé.

math.DS↗