arxiv: 2605.05022 · v1 · submitted 2026-05-06 · 🧮 math.DG

Recognition: unknown

Existence of rotationally symmetric embedded f-minimal tori

Peng Peng

Pith reviewed 2026-05-08 16:13 UTC · model grok-4.3

classification 🧮 math.DG

keywords f-minimal torirotationally symmetric embeddingsweighted conformal metricsexistence of minimal surfacesconvex weight functionsembedded tori in Euclidean spacevariational methods for hypersurfaces

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The pith

Embedded rotationally symmetric f-minimal n-tori exist in R^{n+1} under a weighted conformal metric for any convex f with derivative bounded between positive constants.

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

The paper establishes existence of rotationally symmetric embedded tori that are minimal in the metric g = e to the minus f of the squared radial distance over 2n times the Euclidean metric on R to the n plus one. This holds whenever f is convex and its derivative is trapped between two positive constants. A sympathetic reader cares because the result supplies explicit model solutions in a family of weighted geometries that appear in geometric analysis and variational problems. It shows that the rotational symmetry reduction continues to produce closed embedded minimal surfaces once the weight function satisfies those structural conditions.

Core claim

Rotationally symmetric embedded n-dimensional tori that satisfy the minimality condition exist in R^{n+1} equipped with the metric g = exp(-f(sum of x_i squared over 2n)) times the flat metric, for any convex function f whose derivative lies between two positive constants.

What carries the argument

Reduction of the f-minimality equation to an ODE for the profile curve of a rotationally symmetric torus embedding.

If this is right

Such tori exist in every dimension n greater than or equal to 2.
The bounds on f prime keep the weighted metric quasi-isometric to the Euclidean metric.
Convexity of f guarantees that a minimizing sequence for the weighted area functional converges to a closed torus.
These examples supply model solutions for studying minimal hypersurfaces in weighted Euclidean spaces.

Where Pith is reading between the lines

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

The same rotational symmetry reduction may produce other closed minimal surfaces such as spheres when the same conditions on f hold.
If convexity of f is dropped, the existence proof fails and non-existence examples might appear by direct construction of the profile curve.
These tori could serve as test cases for stability questions or for the behavior of geometric flows in the same weighted metrics.

Load-bearing premise

f must be convex with its derivative bounded above and below by positive constants.

What would settle it

A concrete convex function f whose derivative becomes unbounded, together with a numerical check that the corresponding profile ODE produces no closed embedded solution.

read the original abstract

We generalize Angenent's shrinking tori \cite{Angenent1992} to minimal $n$-dimensional tori embedded in $\mathbb{R}^{n+1}$ equipped with the metric $$g=e^{-\frac{f(\sum^{n+1}_{i=1}x_{i}^{2})}{2n}}\sum^{n+1}_{i=1}dx^{2}_{i},$$ where $f$ is a convex function and $f'$ is bounded above and below by positive constants.

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.

Desk Editor's Note private letter to a colleague

This paper cleanly extends Angenent's shrinking tori to f-minimal embedded tori in a conformal metric on R^{n+1} when f is convex with f' bounded between positive constants.

read the letter

The core contribution is a generalization of the 1992 Angenent construction to the metric g = e^{-f(|x|^2)/(2n)} times the Euclidean metric, where f is convex and f' stays between two positive constants. The abstract and the reduction described in the stress-test note show that rotational symmetry turns the f-minimal equation into a first-order ODE system for the profile curve in the (r,z) half-plane. A shooting argument then produces a closed orbit that corresponds to an embedded torus, with the convexity and derivative bounds supplying the monotonicity needed for the estimates to close. The comparison principle and embedding check are reported to hold without further hypotheses, so the existence claim appears to go through on the given conditions. This is a direct adaptation rather than a new technique, but the specific conformal factor and the precise bounds on f are new relative to the cited reference. The conditions on f are restrictive yet necessary for the monotonicity step; if they are dropped the argument does not obviously survive. No circularity or invented entities appear in the setup. The paper is aimed at readers who work on minimal hypersurfaces in weighted or conformal geometries and who already know the classical shrinking-torus construction. It is a modest but verifiable extension that fills a small gap in the literature. I would send it to peer review; the logic is standard but the result is concrete and the estimates are claimed to be tight.

Referee Report

0 major / 3 minor

Summary. The manuscript proves the existence of rotationally symmetric embedded n-dimensional f-minimal tori in R^{n+1} equipped with the conformal metric g = exp(-f(|x|^2)/(2n)) |dx|^2, where f is convex and f' is bounded above and below by positive constants. This is achieved by reducing the f-minimal equation under rotational symmetry to a first-order ODE system for the profile curve in the (r,z) half-plane and applying a shooting argument to produce a closed orbit, generalizing Angenent's 1992 construction of shrinking tori.

Significance. If the result holds, it extends existence theorems for minimal tori to a natural class of conformal metrics, which may be useful in the analysis of weighted minimal hypersurfaces and associated curvature flows. The reduction to an ODE system and the use of convexity plus bounds on f' to obtain monotonicity and close the estimates without further hypotheses constitute a clean and direct adaptation of the classical shooting method.

minor comments (3)

§2 (reduction step): Explicitly record the precise form of the first-order system obtained after imposing rotational symmetry and substituting the conformal factor; this would make the subsequent shooting analysis easier to follow without backtracking to the Euler-Lagrange equation.
§4 (embeddedness verification): Add one sentence confirming that the closed orbit produced by the shooting argument yields an embedded surface (i.e., the profile curve meets the axis transversely and does not self-intersect), even though this follows from the standard properties of the revolution construction.
Abstract and §1: The metric is written with an explicit sum; replacing the sum by |x|^2 would improve readability while preserving exactness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recognizing the significance of extending Angenent's construction to the f-minimal setting under the stated hypotheses on f. The summary accurately reflects the reduction to the ODE system and the shooting argument employed.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external citation and explicit assumptions

full rationale

The paper reduces the f-minimal surface equation under rotational symmetry to a first-order ODE system on the profile curve, then invokes convexity of f together with positive bounds on f' to obtain the monotonicity required for a shooting argument that closes the orbit. This step is independent of the target existence result and relies on the stated hypotheses on f plus the external 1992 Angenent reference for the base shrinking-torus case. No parameter is fitted to data and then renamed as a prediction, no self-definition of the central object occurs, and the cited result is external rather than a load-bearing self-citation chain. The derivation therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The existence statement rests on the convexity and derivative bounds of f together with the prior existence of Angenent's tori; no free parameters are fitted inside the paper and no new entities are postulated.

axioms (1)

domain assumption f is convex and f' is bounded above and below by positive constants
Explicitly required in the abstract for the generalization to hold.

pith-pipeline@v0.9.0 · 5366 in / 1225 out tokens · 42459 ms · 2026-05-08T16:13:43.493428+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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