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arxiv: 2606.03659 · v1 · pith:UAZO2TN3new · submitted 2026-06-02 · 🧮 math.DG

Normal curvature bounds for immersions into Riemannian domains

Matteo Raffaelli This is my paper

Pith reviewed 2026-06-28 08:22 UTC · model grok-4.3

classification 🧮 math.DG
keywords normal curvatureimmersionsRiemannian domainsn-trace convexityCartan-Hadamard manifoldssubmanifoldscurvature boundsgeodesic balls
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The pith

Closed submanifolds immersed in Riemannian domains have average normal curvature bounded below by the domain's optimal n-trace convexity invariant.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a lower bound on the average normal curvature of closed immersed submanifolds inside a Riemannian domain. The bound is stated using an invariant that quantifies the optimal n-trace convexity of the domain after unit-gradient normalization. A reader would care because the result recovers and extends an earlier bound of Petrunin from Euclidean balls to geodesic balls in Cartan-Hadamard manifolds and to other domains satisfying suitable convexity conditions. In the Cartan-Hadamard case the paper also derives equality rigidity under an assumption on average scalar curvature.

Core claim

Our main result is a lower bound for the average normal curvature of a closed submanifold immersed in a Riemannian domain. The bound is expressed in terms of an invariant measuring the optimal n-trace convexity of the domain under a unit-gradient normalization. As applications we recover and extend Petrunin's lower bound for closed submanifolds immersed in Euclidean balls to geodesic balls in Cartan-Hadamard manifolds and more generally to Riemannian domains satisfying suitable convexity conditions. In the Cartan-Hadamard setting, under a natural assumption on the average scalar curvature, equality forces the submanifold to lie minimally in the boundary sphere and the radial sectional curvat

What carries the argument

The optimal n-trace convexity invariant of the Riemannian domain under unit-gradient normalization, which supplies the explicit lower bound on average normal curvature.

If this is right

  • The bound recovers Petrunin's result for submanifolds in Euclidean balls.
  • The bound extends to geodesic balls in Cartan-Hadamard manifolds.
  • Sharper estimates hold for immersions into hyperbolic balls and Euclidean tubes.
  • In Cartan-Hadamard domains with controlled average scalar curvature, equality implies the submanifold is minimal in the boundary sphere and radial sectional curvature vanishes along it.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same convexity invariant may yield curvature bounds for submanifolds in other classes of manifolds once the invariant can be computed.
  • Equality cases suggest possible rigidity theorems linking minimal submanifolds in spheres to vanishing curvature conditions.
  • The normalization by unit gradients may allow comparison of the invariant across different Riemannian metrics on the same domain.

Load-bearing premise

The Riemannian domain admits a well-defined positive optimal n-trace convexity invariant under unit-gradient normalization.

What would settle it

Exhibit a closed immersed submanifold whose average normal curvature lies strictly below the value given by the domain's optimal n-trace convexity invariant.

read the original abstract

We study Gromov's problem on the minimal normal curvature of immersions. Our main result is a lower bound for the average normal curvature of a closed submanifold immersed in a Riemannian domain. The bound is expressed in terms of an invariant measuring the optimal $n$-trace convexity of the domain under a unit-gradient normalization. As applications, we recover and extend Petrunin's lower bound for closed submanifolds immersed in Euclidean balls to geodesic balls in Cartan-Hadamard manifolds and, more generally, to Riemannian domains satisfying suitable convexity conditions. In the Cartan-Hadamard setting, under a natural assumption on the average scalar curvature, we show that equality forces the submanifold to lie minimally in the boundary sphere and that the radial sectional curvature vanishes along it. We also obtain sharper estimates for immersions into hyperbolic balls and Euclidean tubes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper addresses Gromov's problem on minimal normal curvature for immersions. Its central result is a lower bound on the average normal curvature of a closed submanifold immersed in a Riemannian domain, expressed via an invariant that quantifies the optimal n-trace convexity of the domain under unit-gradient normalization. The bound is applied to recover and extend Petrunin's result from Euclidean balls to geodesic balls in Cartan-Hadamard manifolds and to more general convex Riemannian domains; in the Cartan-Hadamard case, equality is characterized by the submanifold lying minimally in the boundary sphere together with vanishing radial sectional curvature along it. Sharper estimates are also derived for hyperbolic balls and Euclidean tubes.

Significance. If the derivation of the convexity invariant and the resulting curvature bound hold, the work supplies a flexible, invariant-based framework that unifies and extends existing lower bounds on normal curvature. The explicit construction of the invariant (via infimum over admissible unit-gradient functions) and the verification of its positivity under the stated convexity hypotheses are strengths, as are the equality-case analyses and the applications to Cartan-Hadamard geodesic balls and tubes.

minor comments (2)
  1. The abstract and introduction refer to the 'optimal n-trace convexity invariant' without an early pointer to its precise definition (presumably in §2 or §3); adding a forward reference would improve readability for readers focused on the main theorem.
  2. In the equality-case discussion for Cartan-Hadamard domains, the radial curvature vanishing condition is stated; it would help to indicate explicitly whether this condition is necessary only under the average scalar curvature hypothesis or more generally.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our manuscript, as well as the recommendation for minor revision. The referee's description accurately reflects the paper's contributions to Gromov's problem via the n-trace convexity invariant and its applications.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit invariant definition

full rationale

The paper defines the optimal n-trace convexity invariant explicitly via infimum over admissible unit-gradient functions on the domain, then derives the average normal curvature lower bound as a theorem from this definition under the stated convexity hypotheses. Applications to Cartan-Hadamard balls and tubes follow by verifying the invariant is positive, with equality cases handled by direct curvature conditions. No step reduces a prediction to a fitted input, renames a known result, or relies on load-bearing self-citation; the chain is independent and externally falsifiable via the domain's geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities can be extracted or verified from the provided text.

pith-pipeline@v0.9.1-grok · 5661 in / 1065 out tokens · 23562 ms · 2026-06-28T08:22:59.309358+00:00 · methodology

discussion (0)

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Reference graph

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