A smooth Reshetnyak-harmonic sphere with unbounded distortion

Abstract

We construct a compact smooth Riemannian sphere and a smooth homeomorphism from the round sphere which locally minimizes Reshetnyak energy against every same-trace Sobolev competitor, but has essentially unbounded infinitesimal distortion. This gives a negative answer for the Reshetnyak member of the general-energy harmonic-sphere question posed in the inspected 2025 texts of Meier–Vikman–Wenger. Scalar calibration on a flat belt and an exact two-form calibration on conformal caps prove the full local comparison. The map does not minimize energy in its homotopy class and is not Korevaar–Schoen harmonic. Separately, an angular-stress argument gives the nonsharp bound 4 + √17 for locally Korevaar–Schoen-minimizing spheres into complete metric spaces; the sharper classical SL(2)-minimal bound also applies. Classical ingredients are credited; first priority is not claimed.

The idea in plain English

An energy-minimizing map can be thought of as a way of placing one surface onto another that cannot be improved by changing a small patch while keeping its boundary fixed. Whether that condition prevents severe stretching depends on which energy is being minimized.

The paper constructs a smooth map from the round sphere onto a smooth Riemannian sphere. It is a homeomorphism: it is one-to-one, onto, and has a continuous inverse. It locally minimizes Reshetnyak energy, which measures the square of the greatest infinitesimal stretch, yet its stretching can be arbitrarily unequal in different directions.

Near the equator, one direction is compressed by a factor proportional to the square of the distance from the equator while the other is left unchanged. The ratio of the two stretches exceeds every finite bound on a region of positive area. This is a distortion phenomenon, even though the map and both surfaces are smooth.

The choice of energy matters. The example does not minimize the usual Dirichlet, or Korevaar–Schoen, energy and does not minimize energy globally within its homotopy class. A separate argument in the paper establishes bounded infinitesimal distortion for locally Korevaar–Schoen-minimizing spheres into complete metric spaces.

The precise result

There is a compact smooth Riemannian surface X diffeomorphic to S² and a smooth homeomorphism u from the round S² to X that locally minimizes Reshetnyak energy E₊² but has essentially unbounded infinitesimal distortion.

Local minimization has the full fixed-trace meaning: each source point has a neighborhood U such that, for every Lipschitz domain Ω compactly contained in U, u has no larger energy than any W¹,²(Ω, X) competitor with the same boundary trace. Competitors need not be continuous, remain in the chosen image patch, or share a prescribed homotopy class.

The target is geodesic, satisfies a local quadratic isoperimetric inequality, and every sufficiently small-diameter continuous sphere in it is null-homotopic. Thus the counterexample satisfies the target assumptions of the general-energy harmonic-sphere question in the inspected Meier–Vikman–Wenger arXiv v1 and Oberwolfach texts.

On a punctured equatorial band the ratio of largest to smallest infinitesimal stretch is 1/t² in cylindrical coordinates. Every proposed finite distortion bound fails on a set of positive area. The inverse homeomorphism need not be smooth.

Separately, for a complete metric target in the standard metric-Sobolev framework, a locally Korevaar–Schoen-minimizing W¹,² map from S² has infinitesimal distortion at most 4 + √17. The stress argument also gives SL(2)-minimality, so the sharper established Lytchak–Wenger bound 2√2 + √6 applies. The displayed elementary constant is not claimed as an improvement or an optimum.

Why the result holds

The target sphere has a flat cylindrical belt and smooth conformal caps. The map compresses one belt coordinate by h(t), with h′(t) = t² near the equator, while preserving the angular coordinate. It becomes conformal away from the compressed region.

On the belt, a globally 1-Lipschitz extension of the angular coordinate turns every competitor into a scalar Sobolev function with the same boundary data. Its Dirichlet energy gives a lower bound attained by the constructed map. This comparison still works when a competitor leaves the local image patch.

On the caps, an exact two-form with comass at most one agrees with the area form near the image. Pullback and trace-gluing identities make its integral depend only on the fixed boundary trace, giving the required lower bound for arbitrary Sobolev competitors.

For the separate KS result, inner variations yield an angular stress tensor. On the sphere it defines a holomorphic quadratic differential and therefore vanishes. The resulting balance of the infinitesimal seminorm bounds distortion; the established SL(2)-minimal theory gives the sharper constant.

Scope and limitations

The counterexample uses Reshetnyak energy, the integral of the squared operator norm. It is not Korevaar–Schoen harmonic, where the smooth-target energy instead sums the squares of both singular values. Comparable energies can have different minimizers.

The map is a local fixed-trace minimizer, not a global minimizer in its homotopy class. A conformal homeomorphism in the same class has strictly lower total energy, so the example does not contradict the established theory of global homotopy minimizers.

The negative answer is tied to the Reshetnyak member of the general-energy question in the inspected 2025 arXiv version and workshop contribution. The revised journal body was not available for comparison, and neither unchanged final wording nor continued open status is asserted.

The positive Korevaar–Schoen statement gives bounded distortion in general metric targets, not conformality of every tangent seminorm. The smooth Riemannian-target conformality phenomenon and the sharper SL(2)-minimal bound are classical ingredients.

First priority and optimal constants are not claimed. Finite algebra checks and numerical diagnostics support the construction but do not certify the universal Sobolev comparison or the global stress argument.

Status and AI use

AI tools were used extensively in development, verification, adversarial review, and writing. The author is responsible for the claims. AI-assisted reviews are not independent human or journal peer review, and the accompanying checks are not formal proof certification.

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Collaborate on this result

I welcome substantive collaboration on an independent geometric-analysis review, especially the unrestricted Sobolev competitor comparison and the metric angular-stress argument, or on developing a journal submission. Expertise in metric Sobolev maps, energy minimization, and quasiconformality would be especially useful.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). A smooth Reshetnyak-harmonic sphere with unbounded distortion (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23356680

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Background and prior work

• Meier, Vikman and Wenger, Energy minimizing harmonic 2-spheres in metric spaces: Question 1.9 asks about infinitesimal quasiconformality of harmonic spheres. Definition 6.1 supplies the local fixed-trace condition; the allowed energies include Reshetnyak and Korevaar–Schoen energy.
• Meier, joint work with Vikman and Wenger, Oberwolfach Report 30/2025: Question 1 on printed page 1625 asks the corresponding question after fixing an admissible energy. The preceding global homotopy-minimizer result has a stronger hypothesis.
• Lytchak and Wenger, Energy and area minimizers in metric spaces: Provides the established energy/area and SL(2)-minimal seminorm framework. Section 3.3 records the sharper Korevaar–Schoen quasiconformality constant 2√2 + √6.
• Korevaar and Schoen (1993), Sobolev spaces and harmonic maps for metric space targets: Foundational metric-Sobolev and harmonic-map theory used by the separate Korevaar–Schoen statement. The paper does not identify this energy with Reshetnyak energy.
• Sacks and Uhlenbeck (1981), The existence of minimal immersions of 2-spheres: Classical harmonic-sphere theory provides the smooth-target background for the holomorphic quadratic-differential argument. That classical conformality mechanism is credited, not presented as new.