Paper · Jul 22, 2026

The Penrose conjecture for initial data sets satisfying a -convexity condition

Conghan Dong

math.DGWider flowsarXiv:2607.20741v2
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Abstract

Let be a smooth, connected, asymptotically flat initial data set with connected outermost past apparent horizon . We prove the Penrose conjecture, namely that , under the assumptions of the dominant energy condition and the -convexity condition that the sum of the two smallest eigenvalues of is nonnegative. The main tool is the -inverse mean curvature flow, together with a monotonicity formula developed in.

Comments: v2: connected horizon assumption added; references added;

Mean curvature flowOther curvature flowsInverse mean curvature flowBoundaries & non-compactEigenvalues & spectrumPhysics

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.