arXiv subjects

Adam Martens

Publications and source records attributed to Adam Martens.

7 recordsLinked to original sources

arXiv2026

A note on Ricci flow from small curvature concentration and a Morrey-type condition

In \cite{ChauMartens} the authors proved the long-time existence of Ricci flow starting from complete bounded curvature Riemannian manifolds with scale-invariant integral curvature bounded by a dimensional constant times the inverse of the Sobolev constant. We generalize this result by replacing the bounded curvature assumption with the assumption that $g$ is only equivalent to a complete bounded curvature metric $h$ while satisfying a Morrey-type condition on the gradient of $g$ relative to $h$: a local integral condition on the covariant derivative $\nabla_h g$. The Morrey-type condition was first considered in \cite{LeeLiu} in the context of Ricci flow on non-compact manifolds, and in particular allows the possibility for $g$ to have unbounded curvature on $M$. As in \cite{ChauMartens}, our long-time solution enjoys curvature decay estimates implying in particular that $M$ is diffeomorphic to $\mathbb{R}^n$.

math.DG↗

arXiv2024

Removing scalar curvature assumption for Ricci flow smoothing

In recent work of Chan-Huang-Lee, it is shown that if a manifold enjoys uniform bounds on (a) the negative part of the scalar curvature, (b) the local entropy, and (c) volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially (depending only on these a priori bounds). In this work, we show that the bound on scalar curvature assumption (a) is redundant. We also give some applications of this quantitative short-time existence, including a Ricci flow smoothing result for measure space limits, a Gromov-Hausdorff compactness result, and a topological and geometric rigidity result in the case that the a priori local bounds are strengthened to be global.

math.DG↗

arXiv2024

Sharpening a gap theorem: nonnegative Ricci and small curvature concentration

We sharpen a gap theorem of Chan & Lee for nonnegative Ricci curvature manifolds that have positive asymptotic volume ratio and small enough scale-invariant integral curvature (so-called "curvature concentration"), by showing that the curvature concentration need only depend linearly on the asymptotic volume ratio. We prove the result by exhibiting a long-time Ricci flow solution with faster than $1/t$ curvature decay, which allows us to shift the limiting contradiction argument to time infinity and thus obtain an explicit bound on the size of the gap.

math.DG↗

arXiv2024

Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature

We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the curvature is sub-critical $L^p$-integrable, this flow converges locally smoothly to a limiting metric $g(\infty)$ on $M$ with $(M,g(\infty))$ isometric to the standard flat $\mathbb{R}^n$, which implies topological rigidity of $M$. This generalizes work of Chen \cite{ChenEric}, who proved analogous results for asymptotically flat manifolds. We also prove a long-time Ricci flow existence (and likewise topological rigidity) result for unbounded curvature initial data, assuming the initial data is a locally smooth limit of bounded curvature manifolds as described above.

math.DG↗

arXiv2023

Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows

Lai (2021) used singular Ricci flows, introduced by Kleiner and Lott (2017), to construct a nonnegative Ricci curvature Ricci flow $g(t)$ emerging from an arbitrary 3D complete noncompact Riemannian manifold $(M^3, g_0)$ which has nonnegative Ricci curvature. We show $g(t)$ is complete for positive times provided $g_0$ satisfies a volume ratio lower bound that approaches zero at spatial infinity. Our proof combines a pseudolocality result of Lai (2021) for singular flows, together with a pseudolocality result of Hochard (2016) and Simon and Topping (2022) for nonsingular flows. We also show that the construction of complete nonnegative complex sectional curvature flows by Cabezas-Rivas and Wilking (2015) can be adapted here to show $g(t)$ is complete for positive times provided $g_0$ is a compactly supported perturbation of a nonnegative sectional curvature metric on $\mathbb{R}^3$.

math.DG↗

arXiv2021

On the Bartnik mass of non-negatively curved CMC spheres

Let $g$ be a smooth Riemannian metric on $\mathbb{S}^2$ and $H>0$ a constant. We establish an upper bound for the corresponding Bartnik mass $\mathfrak m_B(\mathbb{S}^2, g, H)$ assuming that the Gauss curvature $K_g$ is non-negative. Our upper bound approaches the Hawking mass $\mathfrak m_H(\mathbb{S}^2, g, H)$ when either $g$ becomes round or else $H\to 0$, the bound is zero for $H$ sufficiently large, and in any case the bound is not more than $r/2=\mathfrak m_H(\mathbb{S}^2, g, 0)$. We obtain upper bounds on $\mathfrak m_B(\mathbb{S}^2, g, H)$ as well in the case when $g$ is arbitrary and $H$ is sufficiently large depending on $g$.

math.DG↗

arXiv2020

Exterior Schwarzschild initial data for degenerate apparent horizons

In this note we show that if $g$ is a smooth Riemannian metric on $\mathbb{S}^2$ such that the first eigenvalue of the operator $L_g:=-\Delta_g +K_g$ satisfies $\lambda_1(L_g)=0$ then $(\mathbb{S}^2, g)$ arises as an apparent horizon in an asymptotically flat initial data set with ADM mass arbitrarily close to the associated Hawking mass $\sqrt{\text{area}(\mathbb{S}^2, g)/16\pi}$. In particular, this determines the Bartnik quasilocal mass (introduced by Bartnik \cite{Bartnik} in 1989) associated with $(\mathbb{S}^2, g)$ in this setting. We prove these by modifying the construction of Mantoulidis-Schoen \cite{MS} who proved the same results in the case $\lambda_1(L_g)>0$. It follows that $\lambda_1(g)\geq 0$ is necessary and sufficient for $(\mathbb{S}^2, g)$ to arise from an apparent horizon in an asyptotically flat space-time under the dominant energy condition and in the time symmetric setting, and that the Bartnik mass of the horizon is $\sqrt{\text{area}(\mathbb{S}^2, g)/16\pi}$.

math.DG↗