arXiv subjects

Alec Payne

Publications and source records attributed to Alec Payne.

9 recordsLinked to original sources

arXiv2026

The Spacetime Positive Mass Theorem with Multiple Time Dimensions

We generalize the spacetime positive mass theorem to include multiple time dimensions. In particular, we show that the mass remains nonnegative in the sense that the energy $E$ is bounded from below by the trace norm of the linear momenta $J^1,...,J^m$. Equality in this energy inequality implies a foliation by flat submanifolds of a generalized initial data set. Moreover, under an additional umbilicity assumption, we find that the initial data set isometrically embeds into a generalized pp-wave.

gr-qc↗

arXiv2025

A parabolic flow for the large volume heterotic $G_2$ system

We introduce a geometric flow of conformally coclosed $G_2$-structures, whose fixed points are large volume solutions of the heterotic $G_2$ system, with vanishing scalar torsion class $\tau_0 = 0$. After conformal rescaling, it becomes a flow of coclosed $G_2$-structures, related to Grigorian's modified $G_2$ coflow, which is coupled to a flow for a dilaton function. Our main results establish fundamental short-time existence and Shi-type smoothing properties of this flow, as well as a classification of its fixed points. By a classical rigidity result in the string theory literature, the fixed points on a compact manifold correspond to torsion-free $G_2$-structures, that is, to metrics with holonomy contained in $G_2$. Thus, we establish in the affirmative a folklore question in the special holonomy community, about the existence of a well-posed flow for coclosed $G_2$-structures with fixed points given by torsion-free $G_2$-structures. The flow also satisfies a monotonicity formula for the $G_2$-dilaton functional (volume scale in string theory), which allows us to strengthen the rigidity result with an alternative proof. The monotonicity of the $G_2$-dilaton functional, combined with the Shi-type estimates, leads to a general result on the convergence of nonsingular solutions. A dimension reduction analysis reveals an interesting link with natural flows for $SU(3)$-structures, previously introduced in the literature.

math.DG↗

arXiv2025

An Intersection Principle for Mean Curvature Flow

The avoidance principle says that mean curvature flows of hypersurfaces remain disjoint if they are disjoint at the initial time. We prove several generalizations of the avoidance principle that allow for intersections of hypersurfaces. First, we prove that the Hausdorff dimension of the intersection of two mean curvature flows is non-increasing over time, and we find precise information on how the dimension changes. We then show that the self-intersection of an immersed mean curvature flow has non-increasing dimension over time. Next, we extend the intersection dimension monotonicity to Brakke flows and level set flows which satisfy a localizability condition, and we provide examples showing that the monotonicity fails for general weak solutions. We find a localization result for level set flows with finitely many singularities, and as a consequence, we obtain a fattening criterion for these flows which depends on the behavior of intersections with smooth flows.

math.DG↗

arXiv2023

Closed $G_2$-Structures with Negative Ricci Curvature

We study existence problems for closed $G_2$-structures with negative Ricci curvature, and we prove the $G_2$-Goldberg conjecture for noncompact manifolds. We first show that no closed manifold admits a closed $G_2$-structure with negative Ricci curvature. In the noncompact setting, we show that no complete manifold admits a closed $G_2$-structure with Ricci curvature pinched sufficiently close to a negative constant. As a consequence, an Einstein closed $G_2$-structure on a complete manifold must be torsion-free. In addition, when the Einstein metric is incomplete, we find restrictions on lengths of geodesics.

math.DG↗

arXiv2023

Warped Tori with Almost Non-Negative Scalar Curvature

For sequences of warped product metrics on a $3$-torus satisfying the scalar curvature bound $R_j \geq -\frac{1}{j}$, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrinsic Flat sense to a flat $3$-torus.

math.DG↗

arXiv2022

Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow

We prove a uniqueness result for asymptotically conical (AC) gradient shrinking solitons for the Laplacian flow of closed G_2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G_2-cone, then their G_2-structures are equivalent, and in particular, the two solitons are isometric. The proof extends Kotschwar and Wang's argument for uniqueness of AC gradient shrinking Ricci solitons. We additionally show that the symmetries of the G_2-structure of an AC shrinker end are inherited from its asymptotic cone; under a mild assumption on the fundamental group, the symmetries of the asymptotic cone extend to global symmetries.

math.DG↗

arXiv2021

Nonconvex Surfaces which Flow to Round Points

In this article, we extend Huisken's theorem that convex surfaces flow to round points by mean curvature flow. We construct certain classes of mean convex and non-mean convex hypersurfaces that shrink to round points and use these constructions to create pathological examples of flows. We find a sequence of flows that exist on a uniform time interval, have uniformly bounded diameter, and shrink to round points, yet the sequence of initial surfaces has no subsequence converging in the Gromov-Hausdorff sense. Moreover, we find a sequence of flows which all shrink to round points, yet the initial surfaces converge to a space-filling surface. Also constructed are surfaces of arbitrarily large area which are close in Hausdorff distance to the round sphere yet shrink to round points.

math.DG↗

arXiv2020

Mass Drop and Multiplicity in Mean Curvature Flow

Brakke flow is defined with a variational inequality, which means it may have discontinuous mass over time, i.e. have mass drop. It has long been conjectured that the Brakke flow associated to a nonfattening level set flow has no mass drop and achieves equality in the Brakke inequality. Under natural assumptions, we show that a flow has no mass drop if and only if it satisfies the multiplicity one conjecture $\mathcal{H}^n$-a.e. One application is that there is no mass drop for level set flows with mean convex neighborhoods of singularities, and a generic flow has no mass drop until there is a higher multiplicity planar tangent flow. Also, if a nonfattening flow has no higher multiplicity planes as limit flows, then each limit flow has no mass drop. We upgrade these results to equality in the Brakke inequality for certain important cases. We show that nonfattening flows with three-convex blow-up type are Brakke flows with equality. This includes flows with generic singularities in dimension three and flows with mean convex neighborhoods of singularities in dimension four.

math.DG↗

arXiv2019

Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces

We construct embedded ancient solutions to mean curvature flow related to certain classes of unstable minimal hypersurfaces in $\mathbb{R}^{n+1}$ for $n \geq 2$. These provide examples of mean convex yet nonconvex ancient solutions that are not solitons, meaning that they do not evolve by rigid motions or homotheties. Moreover, we construct embedded eternal solutions to mean curvature flow in $\mathbb{R}^{n+1}$ for $n \geq 2$. These eternal solutions are not solitons, are $O(n)\times O(1)$-invariant, and are mean convex yet nonconvex. They flow out of the catenoid and are the rotation of a profile curve which becomes infinitely far from the axis of rotation. As $t \to \infty$, the profile curves converge to a grim reaper for $n \geq 3$ and become flat for $n=2$. Concerning these eternal solutions, we also show they are asymptotically unique up to scale among the embedded $O(n)\times O(1)$-invariant, eternal solutions with uniformly bounded curvature and a sign on mean curvature.

math.DG↗