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arxiv: 2601.02642 · v4 · submitted 2026-01-06 · 🧮 math.AP

Quasiconvexity in the Riemannian setting

Aurora Corbisiero , Chiara Leone , Carlo Mantegazza This is my paper

Pith reviewed 2026-05-16 17:47 UTC · model grok-4.3

classification 🧮 math.AP
keywords quasiconvexitylower semicontinuityRiemannian manifoldintegral functionalweak-star topologycalculus of variationsW^{1,∞} maps
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The pith

Riemannian quasiconvexity characterizes weak-star lower semicontinuity of integral functionals

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

This paper introduces a notion of quasiconvexity for continuous functions defined on the vector bundle of linear maps from the tangent space of a Riemannian manifold to Euclidean space. It generalizes the standard quasiconvexity from the calculus of variations in Euclidean domains. The central result is that this quasiconvexity is necessary and sufficient for the associated integral functional to be sequentially lower semicontinuous with respect to the weak-star topology in the space of bounded Lipschitz maps. The characterization holds for the functional restricted to any bounded open subset of the manifold. A sympathetic reader would care because it provides the right convexity-type condition to apply existence theorems for minimizers in variational problems on curved spaces.

Core claim

The authors define a quasiconvexity condition for functions f on the vector bundle of linear maps between tangent spaces of (M, g) and R^m. This condition characterizes the sequential lower semicontinuity of F(u, Ω) = ∫_Ω f(du) dμ with respect to the weak* topology of W^{1,∞}(Ω, R^m) for every bounded open Ω ⊆ M.

What carries the argument

The quasiconvexity condition on the integrand f, which requires that the average of f over perturbations by gradients is at least f at the base point, extended via the tangent bundle of the Riemannian manifold.

Load-bearing premise

The manifold is smooth Riemannian and the integrand f is continuous on the bundle of linear maps.

What would settle it

An explicit integrand that is not quasiconvex on a manifold such as the sphere but for which the integral functional remains sequentially lower semicontinuous in the weak-star topology would disprove the characterization.

read the original abstract

We introduce a notion of quasiconvexity for continuous functions $f$ defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold $(M,g)$ and $\mathbb{R}^m$, naturally generalizing the classical Euclidean definition. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functional \[ F(u, \Omega) = \int_{\Omega} f(du) \, d\mu \] with respect to the weak$^*$ topology of $W^{1,\infty}(\Omega, \mathbb{R}^m)$, for every bounded open subset $\Omega\subseteq M$.

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 / 3 minor

Summary. The paper introduces a notion of quasiconvexity for continuous functions f on the Hom-bundle over a smooth Riemannian manifold (M,g), generalizing the classical Euclidean definition. It proves that this condition is necessary and sufficient for the sequential weak* lower semicontinuity of the integral functional F(u, Ω) = ∫_Ω f(du) dμ with respect to the W^{1,∞} weak* topology, for every bounded open Ω ⊆ M.

Significance. If the result holds, it furnishes a complete intrinsic characterization of lower semicontinuity for variational integrals on Riemannian manifolds, extending the fundamental theorems of Morrey and Acerbi-Fusco. The reduction to the Euclidean case via charts, together with the intrinsic definition via compactly supported test sections, supplies a clean and usable tool for geometric variational problems; the manuscript earns credit for making the argument self-contained once the Euclidean theory is invoked.

minor comments (3)
  1. [§2] §2, Definition of Riemannian quasiconvexity: the statement that the condition reduces exactly to the Euclidean one when (M,g) is flat should be recorded explicitly, with a one-line verification that the volume form dμ becomes Lebesgue measure and the test sections become standard test functions.
  2. [§4] Proof of necessity (likely §4): the passage from weak* convergence of du_n to the integral inequality uses only continuity of f and the definition; a short remark confirming that no curvature term arises would clarify the localization argument.
  3. [Throughout] Notation: the symbol du is used both for the differential and for its measurable representative; a single sentence distinguishing the two would prevent any ambiguity for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in extending the classical results of Morrey and Acerbi-Fusco to the Riemannian setting, and recommendation for minor revision. We are pleased that the intrinsic definition and self-contained argument were viewed favorably.

Circularity Check

0 steps flagged

No significant circularity; characterization uses independent arguments

full rationale

The paper introduces an intrinsic definition of quasiconvexity on the Hom-bundle that directly generalizes the Euclidean case, then proves both necessity and sufficiency of this condition for sequential weak* lower semicontinuity of F by localizing via charts to the Euclidean setting. The two directions rely on separate arguments involving test sections with compact support, the volume measure dμ, and weak* convergence of du; no step reduces a prediction to a fitted input, renames a known result, or depends on a load-bearing self-citation. The smoothness of (M,g) and continuity of f serve only to make du measurable and F well-defined, leaving the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The central claim rests on the newly introduced definition of quasiconvexity together with standard properties of smooth Riemannian manifolds and weak-star topology in Sobolev spaces; no numerical parameters are fitted.

axioms (3)
  • domain assumption M is a smooth Riemannian manifold with metric g
    Required to define tangent spaces TM and the differential du as a bundle map
  • domain assumption f is continuous on the vector bundle of linear maps
    Ensures the integrand f(du) is measurable and the functional is well-defined
  • standard math Weak-star topology on W^{1,∞}(Ω, R^m)
    Standard notion from functional analysis used for the convergence mode in the theorem
invented entities (1)
  • Riemannian quasiconvexity no independent evidence
    purpose: Generalize classical Euclidean quasiconvexity to functions on the Hom(TM, R^m) bundle
    New definition introduced to characterize the semicontinuity property

pith-pipeline@v0.9.0 · 5394 in / 1528 out tokens · 48249 ms · 2026-05-16T17:47:09.859841+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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    RENATOCACCIOPPOLI

    L. Tonelli,La semicontinuit` a nel calcolo delle variazioni, Rend. Circ. Mat. Palermo44(1920), 167–249. (Aurora Corbisiero) DEPARTMENT OFMATHEMATICS ANDAPPLICATIONS“RENATOCACCIOPPOLI”, UNIVERSIT `A DI NAPOLIFEDERICOII Email address:au.corbisiero@studenti.unina.it (Chiara Leone) DEPARTMENT OFMATHEMATICS ANDAPPLICATIONS“RENATOCACCIOPPOLI”, UNIVERSIT `A DINA...

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